TOWARDS A FRAMEWORK FOR EFFICIENT
RESOURCE ALLOCATION IN WIRELESS NETWORKS:
QUALITY-OF-SERVICE AND DISTRIBUTED DESIGN
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Bin Li, MS
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2014
Dissertation Committee:
Atilla Eryilmaz, Advisor
Ness B. Shroff
Eylem Ekici
c Copyright by
Bin Li
2014
ABSTRACT
With the fast growing deployment of smart mobile devices and increasingly
demanding multimedia applications, the future wireless networks must provide highquality services to mobile users under resource-limited conditions. This necessitates
the design of efficient and distributed algorithms with various key characteristics: high
throughput, low energy consumption, fast convergence, low delay, and regular service.
Earlier works extensively study the first-order metrics, such as throughput, fairness,
and energy consumption, and very few of them address the second-order metrics,
such as convergence speed, delay and service regularity — critical for the growing
time-sensitive and dynamic applications penetrating the wireless networks. Also,
it is important to consider the distributed implementations of theoretically proven
efficient algorithms. To that end, this dissertation mainly focuses on the following
two aspects: (i) the performance and optimization of the second-order metrics; (ii)
the distributed algorithm design.
In the first part of this dissertation, we first develop a cross-layer algorithm that
achieves the optimal convergence speed at which the running average of the received
service rates and the network utility over a finite time horizon converges to their
respective limits under the discrete transmission rate selections — a typical feature
of wireless networks. This result is important in two aspects, in revealing a previouslyunknown limit on how fast the service rates can approach an optimal point, and in
providing a new algorithm that achieves the fastest possible speed. Then, we focus
ii
on the efficient algorithm design for overcoming unavoidable temporary overloads.
We develop a novel “queue reversal” approach that relates the metrics in unstable
systems to the metrics in stable systems, for which a rich set of tools and results
exists. Furthermore, to support widely popular real-time applications, we develop a
scheduling algorithm that simultaneously achieves maximum throughput, minimum
delay in heavy-traffic regimes, and service regularity guarantees.
In the second part of this dissertation, we first explore the throughput limitations
of randomized schedulers that are widely used in distributed implementations. This
systematic study is important in helping network designers to use or avoid certain
types of probabilistic scheduling strategies depending on the network topology. Then,
we turn to the distributed design of an optimal scheduling algorithm for serving both
elastic and inelastic traffic over wireless fading channels. The corresponding result
provides one of the first promising means of effectively handling changing conditions
in distributed resource allocation algorithm design. Noting the high energy cost and
operational difficulty for all users to continuously estimate the channel quality before
each transmission, we further design efficient and distributed joint channel probing
and scheduling strategies under energy constraints.
Overall, this thesis develops methodologies to study the metrics beyond the traditional first-order requirements (e.g., throughput, fairness, average energy consumption, etc.), and to design distributed resource allocation algorithms, both of which
enable the incorporation of sharp quality-of-service requirements into the efficient
and distributed algorithm design. It also opens an interesting and new avenue to
the performance analysis and optimization of the second or higher order metrics of
complex networks, including smart power grids and cloud computing.
iii
To my parents, Jinjie Li and Jinzhao Chen,
my sister, Juan Li,
and my wife, Xiaozhen Wang.
iv
ACKNOWLEDGMENTS
I am sincerely grateful to my advisor, Prof. Atilla Eryilmaz, for his invaluable
guidance, generous support, and constant encouragement throughout my Ph.D. study.
He taught me how to identify and attack problems — essential for the research. His
broad knowledge, diligence and passion for research set up an excellent example for
me and will no doubt have a great impact on my future career.
I also would like to thank my doctoral committee members, Prof. Ness B. Shroff
and Prof. Eylem Ekici. My thanks also go to Prof. C. Emre Koksal and Prof. Wei
Zhang for serving in my candidacy committee. Their insightful comments and helpful
suggestions improve the content of this thesis. Also, I would like to thank all other
professors at The Ohio State University who taught, helped or encouraged me during
my Ph.D. study. I am also grateful to Prof. R. Srikant at The University of Illinois
Urbana-Champaign for his help and advice.
I would like to express my great gratitude to my family, especially my parents
and my wife, Xiaozhen Wang, for their unconditional love, support and encouragement. Their care and understanding allowed me to pursue my dreams, and will surely
continue helping me in all my future endeavors.
I would like to acknowledge all my current and former colleagues in the IPS lab
– Arun, Bo, Dongyue, Fangzhou, Gene, Irem, Jia, John, Justin, Karim, Ming, Onur,
Ozan, Ozgur, Ruogu, Shengbo, Shuang, Swapna, Wenzhuo, Yang, Yousi, Zhoujia,
v
Zizhan, and many others for many fruitful conversations on both technical and nontechnical subjects. Also, I would like to thank the lab secretary, Jeri, and the graduate
academic counselor, Tricia, for their patient and helpful attitude in answering my
numerous questions.
vi
VITA
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.S. in Electronic and Information Engineering, Xiamen University
2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M.S. in Communication and Information
System, Xiamen University
2009 — Present . . . . . . . . . . . . . . . . . . . . . . . Presidential Fellow, Graduate Research Associate, Graduate Teaching Associate —
Electrical and Computer Engineering, The
Ohio State University
PUBLICATIONS
Journal Papers:
Bin Li, Ruogu Li, Atilla Eryilmaz, On the Optimal Convergence Speed of Wireless
Scheduling for Fair Resource Allocation, accepted by IEEE/ACM Transactions on
Networking.
Bin Li, Atilla Eryilmaz, Non-Derivative Algorithm Design for Efficient Routing over
Unreliable Stochastic Networks, Elsevier’s Performance Evaluation, 71: 44-60, 2014.
Bin Li, Atilla Eryilmaz, Optimal Distributed Scheduling under Time-Varying Conditions: A Fast-CSMA Algorithm with Applications, IEEE Transactions on Wireless
Communications, 12(7): 3278-3288, 2013.
Bin Li, Atilla Eryilmaz, Exploring the Throughput Boundaries of Randomized Schedulers in Wireless Networks, IEEE/ACM Transactions on Networking, 20(4):11121124,2012.
vii
Conference Papers:
Bin Li, Atilla Eryilmaz, R. Srikant, Leandros Tassiulas, On Optimal Routing in
Overloaded Parallel Queues, In Proc. IEEE conference on Decision and Control
(CDC), Florence, Italy, December, 2013.
Bin Li, Ozgur Dalkilic, Atilla Eryilmaz, Exploring the Tradeoff between Waiting Time
and Service Cost in Non-Asymptotic Operating Regimes, In Asilomar Conference on
Signals, Systems and Computers, Pacific Grove, California, November, 2013.
Bin Li, Ruogu Li, Atilla Eryilmaz, Heavy-Traffic-Optimal Scheduling Design with
Regular Service Guarantees in Wireless Networks. In ACM International Symposium
on Mobile Ad Hoc Networking and Computing (MOBIHOC), Bangalore, India, July,
2013.
Bin Li, Atilla Eryilmaz, Ruogu Li, Wireless Scheduling for Utility Maximization with
Optimal Convergence Speed, In Proc. IEEE International Conference on Computer
Communications (INFOCOM), Turin, Italy, April, 2013.
Ruogu Li, Atilla Eryilmaz, Bin Li, Throughput-Optimal Scheduling with Regulated
Inter-Service Times, In Proc. IEEE International Conference on Computer Communications (INFOCOM), Turin, Italy, April, 2013.
Bin Li, Atilla Eryilmaz, Optimal Constant Splitting for Efficient Routing over Unreliable Networks, In Proc. IEEE conference on Decision and Control (CDC), Maui,
Hawaii, December, 2012.
S. Lakshminarayana, Bin Li, M. Assaad, A. Eryilmaz, M. Debbah, A Fast-CSMA
Based Distributed Scheduling Algorithm under SINR Model, In Proc. IEEE International Symposium on Information Theory (ISIT), Cambridge, MA, July, 2012.
Bin Li, Atilla Eryilmaz, Distributed Channel Probing for Efficient Transmission
Scheduling over Wireless Fading Channels, In Proc. IEEE International Conference on Computer Communications (INFOCOM) mini-Conference, Orlando, Florida,
USA, March, 2012.
Bin Li, Atilla Eryilmaz, A Fast-CSMA Algorithm for Deadline Constraint Scheduling
over Wireless Fading Channels, In workshop on Resource Allocation and Cooperation
in Wireless Networks (RAWNET), Princeton, NJ, May, 2011.
Bin Li, Atilla Eryilmaz, On the Limitations of Randomization for Queue-LengthBased Scheduling in Wireless Networks, In Proc. IEEE International Conference on
Computer Communications (INFOCOM), Shanghai, China, April, 2011.
viii
Bin Li, Atilla Eryilmaz, On the Boundaries of Randomization for Throughput-Optimal
Scheduling in Switches, In Proc. Allerton Conference on Communication, Control,
and Computing (Allerton), Monticello, IL, Sept. 2010.
FIELDS OF STUDY
Major Field: Electrical and Computer Engineering
Specialization: Networking
ix
TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
CHAPTER
1
I
2
PAGE
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Resource Allocation for Time-Sensitive Applications . . . . . . . .
1.2 Distributed Resource Allocation Design . . . . . . . . . . . . . . .
1.3 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
6
8
Resource Allocation for Time-Sensitive Applications
11
Efficient Resource Allocation with Optimal Convergence Speed . . . . .
12
2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . .
2.2 A Motivating Example . . . . . . . . . . . . . . . . . . . . .
2.3 Convergence Speed in Rate Deviation . . . . . . . . . . . . .
2.3.1 A Lower Bound on the Expectation of Rate Deviation
2.3.2 A Rate Deviation Optimal Policy . . . . . . . . . . .
2.4 Convergence Speed in Utility Benefit . . . . . . . . . . . . . .
2.4.1 An Upper Bound on the Utility Benefit . . . . . . . .
2.4.2 Utility Benefit Optimality of the RDO Algorithm . .
2.4.3 Utility Benefit Optimality of the Dual Algorithm . .
2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Non-fading Scenario . . . . . . . . . . . . . . . . . . .
2.5.2 Fading Scenario . . . . . . . . . . . . . . . . . . . . .
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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II
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Algorithm Design with Optimal Convergence Speed in Overloaded Systems 39
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Lower Bound Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Queue Reversal Theorem . . . . . . . . . . . . . . . . . . .
3.2.2 A Lower Bound on the Cumulative Unused Service . . . .
3.3 Overload Analysis of RR and JSQ policies . . . . . . . . . . . . . .
3.3.1 Lower and Upper Bounds under the RR Policy . . . . . . .
3.3.2 Optimality of the JSQ Policy in Symmetric Conditions . .
3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 The Impact of Overload Level on Mean Unused Services
3.4.2 The Impact of Server Number L on Mean Unused Service .
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Resource Allocation Algorithm for Achieving Maximum Throughput,
Heavy-Traffic Optimality, and Service Regularity Guarantee . . . . . .
56
4.1 Problem Formulation . . . . . . . . . . .
4.2 Regular Service Scheduler . . . . . . . . .
4.3 Service Regularity Performance Analysis .
4.3.1 Lower Bound Analysis . . . . . .
4.3.2 Upper Bound Analysis . . . . . .
4.4 Tradeoff Between Mean Delay and Service
4.5 Heavy-Traffic Optimality Analysis . . . .
4.6 Simulation Results . . . . . . . . . . . . .
4.6.1 Throughput Performance . . . . .
4.6.2 Service Regularity Performance .
4.6.3 Heavy-Traffic Performance . . . .
4.7 Summary . . . . . . . . . . . . . . . . . .
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Regularity
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Distributed Resource Allocation Algorithm Design
80
Limitations of Randomization for Distributed Resource Allocation . . .
5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Overview of Main Results . . . . . . . . . . . . . . . . . . . . . .
5.3 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 f -Throughput-Optimality of the RSOF Scheduler . . . .
5.3.2 Throughput-Optimality of RMOF and RFOS Schedulers
5.4 Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Throughput Performance . . . . . . . . . . . . . . . . . .
5.5.2 Delay Performance . . . . . . . . . . . . . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
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6
Optimal Distributed Scheduling Design under Time-Varying Conditions 104
6.1 The Principle of Fast-CSMA design . . . . . . . . .
6.2 Scenario 1: Scheduling Elastic Traffic with CSI . . .
6.2.1 FCSMA Algorithm Implementation . . . . .
6.2.2 Simulation Results . . . . . . . . . . . . . .
6.3 Scenario 2: Scheduling Elastic Traffic without CSI .
6.3.1 FCSMA Algorithm Implementation . . . . .
6.3.2 Simulation Results . . . . . . . . . . . . . .
6.4 Scenario 3: Scheduling Inelastic Traffic with CSI . .
6.4.1 Basic Setup . . . . . . . . . . . . . . . . . .
6.4.2 FCSMA Algorithm Implementation . . . . .
6.4.3 Simulation Results . . . . . . . . . . . . . .
6.5 Scenario 4: Scheduling Inelastic Traffic without CSI
6.5.1 FCSMA Algorithm Implementation . . . . .
6.5.2 Simulation Results . . . . . . . . . . . . . .
6.6 Practical Implementation Suggestions . . . . . . . .
6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . .
7
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Efficient Distributed Channel Probing and Scheduling Design . . . . . . 136
7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 A Motivating Scenario . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Optimal Centralized Probing and Transmission . . . . . . . . . . .
7.3.1 Characterization of the Capacity Region . . . . . . . . . .
7.3.2 An Optimal Joint Probing and Transmission Algorithm . .
7.4 Sequential Greedy Probing Policy and Analysis . . . . . . . . . . .
7.4.1 A Sequential Greedy Probing Algorithm . . . . . . . . . .
7.4.2 Optimality of the SGP Algorithm for Symmetric Channels
7.5 The Modified SGP Policy and Analysis . . . . . . . . . . . . . . .
7.6 Distributed Implementation with Fast-CSMA . . . . . . . . . . . .
7.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.1 The Impact of Iterative Steps . . . . . . . . . . . . . . . .
7.7.2 The Impact of Using Delayed Queue Length Information .
7.7.3 The Performance of Greedy Probing Algorithms . . . . . .
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
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Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . 164
Appendix A: Proofs for Chapter 2
A.1
A.2
A.3
A.4
Proof
Proof
Proof
Proof
of
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Lemma 2.3.2 . .
Proposition 2.3.3
Lemma 2.3.7 . .
Lemma 2.3.8 . .
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A.5
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Proof
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Proposition 2.4.3
Proposition 2.4.6
Proposition 2.4.9
Lemma A.8.1 . .
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Appendix B: Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 185
B.1
B.2
B.3
B.4
B.5
B.6
B.7
B.8
B.9
Proof of Inequality (4.2.4) . . . . . . . . .
Proof of Proposition 4.2.3 . . . . . . . . .
Proof of Inequality (B.2.1) . . . . . . . .
Proof of Lemma B.4 . . . . . . . . . . . .
Proof of Inequality (B.2.20) . . . . . . . .
Proof of Lemma 4.3.1 . . . . . . . . . . .
Proof of Proposition 4.3.3 . . . . . . . . .
Proof of Equation (B.7.5) . . . . . . . . .
Detailed Heavy-Traffic Analysis . . . . . .
B.9.1 State-Space Collapse . . . . . . .
B.9.2 Proof of Heavy-Traffic Optimality
B.10Proof of Lemma B.9.2 . . . . . . . . . . .
B.11Proof of Lemma B.9.3 . . . . . . . . . . .
B.12Proof of Proposition B.9.1 . . . . . . . . .
B.13Proof of Lemma B.12.2 . . . . . . . . . .
B.14Proof of Lemma B.12.1 . . . . . . . . . .
B.15Proof of Inequality (B.9.11) . . . . . . . .
B.16Proof of Inequality (B.9.13) . . . . . . . .
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Appendix C: Proofs for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . 228
C.1 Properties of Functional Classes . . . . . . . . . . . . . . . . . . . 228
C.2 Proof of Lemma 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 229
C.3 Proof of Inequality (5.3.4) . . . . . . . . . . . . . . . . . . . . . . . 230
Appendix D: Proofs for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 234
D.1 Proof of Lemma 6.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . 234
D.2 Proof of Proposition 6.4.8 . . . . . . . . . . . . . . . . . . . . . . . 235
Appendix E: Proofs for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . 241
E.1
E.2
E.3
E.4
E.5
E.6
Proof of Proposition 7.2.1 . . . . . .
Proof of Lemma 7.3.1 . . . . . . . .
Proof of Lemma 7.3.2 . . . . . . . .
Proof of Proposition 7.3.4 . . . . . .
Some Properties of Function f (E, e)
Proof of Basic Iterative Equation . .
xiii
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E.7 Proof
E.8 Proof
E.9 Proof
E.10Proof
E.11Proof
E.12Proof
of
of
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of
f (D∗ , d) ≤ f (B, d) in Lemma 7.4.2
Lemma 7.5.3 . . . . . . . . . . . .
Lemma 7.5.4 . . . . . . . . . . . .
Lemma 7.5.5 . . . . . . . . . . . .
Proposition 7.5.6 . . . . . . . . . .
Proposition 7.6.3 . . . . . . . . . .
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253
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264
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
xiv
LIST OF FIGURES
FIGURE
PAGE
2.1
Relationship between r() and r∗
. . . . . . . . . . . . . . . . . . . .
16
2.2
The convergence speed of an i.i.d Bernoulli sequence. . . . . . . . . .
19
2.3
The relationship between R() and R. . . . . . . . . . . . . . . . . .
22
2.4
The operation of the RDO Algorithm at link l. . . . . . . . . . . . .
25
2.5
The utility benefit of an algorithm in class P. . . . . . . . . . . . . .
29
2.6
Dual Algorithm performance with varying . . . . . . . . . . . . . .
35
2.7
Variants of the RDO Algorithm . . . . . . . . . . . . . . . . . . . . .
37
3.1
Routing to parallel queues. . . . . . . . . . . . . . . . . . . . . . . .
42
3.2
(a) Forward queue; (b) Reverse queue . . . . . . . . . . . . . . . . .
46
3.3
Lower bounding system. . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.4
The impact of on the expected cumulative unused service. . . . . .
54
3.5
The impact of L on the expected cumulative unused service. . . . . .
54
4.1
Delay and service regularity performance of the RSG Algorithm . . .
68
4.2
Geometric structure of capacity region . . . . . . . . . . . . . . . . .
70
4.3
Throughput performance of the RSG Algorithm . . . . . . . . . . . .
74
4.4
Tradeoff between mean queue length and service regularity . . . . . .
76
4.5
Heavy-traffic performance in the symmetric case . . . . . . . . . . . .
77
4.6
Asymmetric arrivals in the asymmetric case . . . . . . . . . . . . . .
78
5.1
The relationship between classes A, B and C. . . . . . . . . . . . . .
86
xv
5.2
Throughput performance of the RSOF Scheduler. . . . . . . . . . . .
88
5.3
Throughput performance of the RFOS Scheduler. . . . . . . . . . . .
89
5.4
Throughput performance validation of the randomized schedulers . . 100
5.5
Delay performance comparison of the randomized schedulers with different functional forms . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1
(a) Markov chain for a CSMA algorithm (b) Markov chain for a FCSMA algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2
Performance of FCSMA for scheduling elastic traffic with CSI . . . . 114
6.3
Performance comparison between FCSMA and QCSMA . . . . . . . 115
6.4
Performance of FCSMA for scheduling elastic traffic without CSI . . 117
6.5
Performance of FCSMA for scheduling inelastic traffic with CSI . . . 125
6.6
Performance of FCSMA for scheduling inelastic traffic without CSI . 132
6.7
Performance comparison between FCSMA and its discrete-time version 135
7.1
Maximum throughput under different number of users . . . . . . . . 141
7.2
Throughput performance of RP policy . . . . . . . . . . . . . . . . . 143
7.3
The directed graph G = (X , E) when L = 3 . . . . . . . . . . . . . . 149
7.4
Impact of iterative steps . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.5
Impact of using delayed queue length information . . . . . . . . . . . 161
7.6
Impact of asymmetric channel statistics . . . . . . . . . . . . . . . . 162
7.7
Impact of asymmetric channel rates . . . . . . . . . . . . . . . . . . . 162
A.1
B.1
An example when rl = 12 , Sl ∈ {0, 1} and c = 18 . . . . . . . . . . . . 169
P
P
(δ) (δ)
The relationship between Ll=1 Ul0 (rl )rl and Ll=1 Ul0 (rl∗ )rl∗ . . . . . 177
E.1
The relations among all sets . . . . . . . . . . . . . . . . . . . . . . . 254
A.2
()
Markov Chain {X[t]}t≥0 . . . . . . . . . . . . . . . . . . . . . . . . . 206
xvi
CHAPTER 1
INTRODUCTION
Wireless networks typically have restrictive resources (e.g., time, frequency, space and
power) and thus all network users need to share these precious resources. This necessitates the design of efficient resource allocation algorithms that determine when and
how to allocate limited resources for network users. Traditional resource allocation
algorithms (e.g., [91, 92, 89, 37, 59, 54, 88, 16, 14, 55, 69, 70, 4, 34, 67], see [87, 21, 66]
for an overview) aim to the optimization of first-order metrics such as throughput,
fairness, average energy consumption, etc. These works play an important role in
developing systematic approaches for resource allocation algorithm designs with longterm efficiency guarantees, most notably Lyapunov-drift-minimization-based design
(e.g., [91, 70, 67, 66]) and optimization-based design (e.g., [37, 59, 54, 88, 16, 56, 87]).
Yet, optimizing these first-order metrics is not a sufficient objective for increasingly many network applications, which are time-sensitive and possess highly dynamic
user behavior. In addition, wireless networks may experience temporary overloads
during peak demand hours or at hotspots serving large number of users, which requires the resource allocation algorithms to quickly respond to these unavoidable
transient phases of overloads. All these features require the design of resource allocation algorithms with fast convergence characteristic. Previous works either focus
on the convergence aspect of algorithm designs (e.g., [37, 59, 16, 14, 55, 69]), or the
1
convergence speed of particular algorithms (e.g., [18, 30, 93, 57]) in the stable networked systems. There does not exist an extensive study of algorithm design with
optimal convergence speed. This motivates us to systematically analyze and design
algorithms in terms of their convergence speed in both underloaded and overloaded
networked systems.
Moreover, to support real-time applications, resource allocation algorithms not
only achieve high throughput or satisfy fairness criteria, but also meet the requirements of Quality-of-Service (QoS) including packet delivery ratio (or throughput),
delay, and service regularity. Recently, a large body of works focus on the design of algorithms that improve various aspects of the QoS, especially on the delay
performance of the algorithms. For example, some works consider designing algorithms with low end-to-end delay performance, such as [7, 97, 95]. Constant delay
bounds (e.g. [68]) and delivery ratio requirements for deadline-constrained traffic (e.g.
[27, 28, 29, 31, 44]) are some of the other QoS metrics considered in the literature.
However, none of these works simultaneously addresses the throughput, delay and
service regularity — critical metrics for real-time applications. This motivates us to
consider the scheduling design that achieves maximum throughput, minimum delay
and best service regularity.
In addition, low-complexity implementation of theoretically efficient resource allocation algorithms is strongly desirable in the presence of many users appearing
in real-world networked systems, and has been a topic of extensive research activity
(e.g., [90, 55, 13, 63, 94, 25, 53, 35, 13, 43]). One such thread leads to the development
of a class of evolutionary randomized algorithms (also named pick and compare algorithms) with throughput-optimality characteristics (see [90, 13, 84]). Another thread
leads to the development of distributed but suboptimal randomized/greedy strategies
(see [55, 35, 6]). Relatively recently, another exciting thread of results have emerged
2
that can guarantee throughput-optimality by cleverly utilizing queue-length information in the context of carrier sense multiple access (CSMA) (e.g., [60, 33, 80, 72]).
Therefore, randomized algorithms have the potential to possess excellent network
performance, and more importantly, they can be implemented distributively.
While randomization allows flexibilities in the distributed implementation of algorithms, it causes inefficient operation and may be hurtful if it is not performed carefully. Despite the presence of a variety of earlier works (e.g., [33, 72]) on the design
and analysis of particular randomized schedulers, there does not exist an extensive
study on the limitations of randomization in scheduling designs. This motivates us
to develop a common framework for the modeling and analysis of randomized schedulers. Furthermore, it is important to consider the distributed scheduling algorithm
design under time-varying conditions and limited energy constraints, under which the
most network applications operate and existing distributed algorithms (e.g., [33, 72])
do not work efficiently.
In this thesis, we address all these challenging issues by developing a rigorous
theoretical foundation for efficient and distributed algorithm design supporting diverse applications in wireless networks. In particular, our dissertation research focuses on the following two main aspects: (i) efficient resource allocation algorithm
design for time-sensitive applications; (ii) distributed resource allocation algorithm
design. The results presented in this thesis have been published (or submitted) in
[45, 49, 46, 52, 48, 47, 39, 41, 43, 40, 44, 42, 38]. Next, we briefly summarize our
main contributions.
1.1
Resource Allocation for Time-Sensitive Applications
In this research effort, we first focus on the convergence speed of cross-layer algorithms
in wireless networks, which are dominated by the dynamics of incoming and outgoing
3
users as well as the time-sensitive applications. However, the design of controllers
with fast convergence speed in most wireless networks is complicated by two natural
constraints: (i) interference constraints leading to discrete link scheduling choices; and
(ii) a finite set of choices for the transmission rate selection over the scheduled links.
The latter constraint is caused by both digital communication (e.g., modulation,
coding, etc.) and hardware design principles. For example, in IEEE 802.11b standard,
there are only four transmission rates: 1Mpbs, 2Mpbs, 5.5Mpbs and 11Mbps.
Previous works focus either on the design and analysis of policies with optimal
long-term behavior (e.g., [37, 59, 16, 14, 55, 69]), or on the design of distributed
Interior Point (e.g., [18]) and Newton’s methods (e.g., [30, 93, 57]) for fast convergence
in wired networks. They do not incorporate an important feature of wireless networks,
namely the discreteness in the transmission rate selections. We tackle this challenge
by explicitly incorporating such discrete constraints to understand their impact on
the convergence speed at which the running average of the received service rates and
the network utility converges to their respective limits. By providing universal bounds
on the convergence speed of any scheme, we establish the limits of convergence speed
under discrete scheduling and transmission rate constraints. Using this bound, we
develop an algorithm that achieves the optimal convergence speed in both metrics.
Somewhat surprisingly, we also show that even a first-order method such as the wellknown dual algorithm can achieve the aforementioned optimal convergence speed in
terms of its utility value. These results are important in two aspects: (i) it reveals
a previously-unknown limit on how fast the service rates can approach an optimal
point; (ii) it provides a new algorithm that achieves the fastest possible speed. As
such, it provides the means of quickly serving the demands of dynamically arriving
and leaving users, as in many wireless mobile networks.
4
Then, we consider the convergence speed of efficient algorithms in overloaded systems. Several interesting works (e.g., [22, 8, 86, 50]) have analyzed the performance of
well-known policies in overloaded conditions. In particular, these works have studied
the performance and optimization of the metric of queue overflow rates, and the related metric of the departure rates of served packets. One drawback of the currently
used performance metrics of overflow rate and departure rate is that, being based
on long-term time averages, they may not be able to differentiate between policies
in terms of their convergence speeds to the same limit. Noting the complexity in
the performance analysis of overloaded systems, we consider the convergence speed
analysis in a simple classic problem of routing random arrivals to parallel queues
with random services in overloaded regimes. We propose and analyze the metric of
Cumulative Unused Service (CUS) over time to analyze the performance of routing
policies in overloaded systems. This metric not only measures the amount of underutilization in the multi-server system over time, but also captures the speed at which
the running-average of the departure rates converges to their limiting value.
The proposed CUS metric is difficult to analyze in both stable and unstable systems due to its non-stationary nature. To tackle this challenge, we establish a novel
“queue reversal” result that equates the expected cumulative unused service in the
unstable system to the expected queue-length of a related (in fact, reversed) stable
system. With this connection, we can obtain the mean cumulative unused service
metric by studying the mean queue-length of a stable Markov chain, for which a rich
set of tools and results exists. Using this result for a single-server queue, we obtain
a lower bound on the expected unused service in the parallel queueing system for
any feasible routing policy. We then compare this lower bound to the performance of
two simple routing policies: Randomized and Join-the-Shortest-Queue (JSQ) routing.
Through numerical studies, we reveal two interesting properties of the JSQ policy: (i)
5
the JSQ policy achieves the optimal convergence speed in lightly overloaded regimes,
i.e., when the arrival rate approaches the total service rates; (ii) the convergence
speed under the JSQ policy is independent of number of servers compared to the
derived low bound on the convergence speed under any feasible policy.
To support real-time applications, we then consider the design of scheduling strategies that maximize system throughput, minimize mean delay, and provide regular service for all users. We develop a parametric class of maximum-weight type scheduling
algorithms, called Regular Service Guarantee (RSG) Algorithm, where each user’s
weight consists of its own queue-length and a counter that tracks the time since the
last service. The RSG Algorithm not only is throughput-optimal, but also achieves a
tradeoff between the service regularity performance and the mean delay under the RSG
Algorithm, i.e., the service regularity performance of the RSG Algorithm improves at
the cost of increasing mean delay.
This further motivates us to investigate whether satisfactory service regularity
and low mean-delay can be simultaneously achieved by the RSG Algorithm by carefully selecting its design parameter. To that end, we show that the RSG Algorithm
can minimize the total mean queue-length to establish mean delay optimality under heavily-loaded conditions as long as its design parameter scales slowly with the
network load. To the best of our knowledge, this is the first work that provides regular service while also achieving maximum throughput and heavy-traffic optimality
in mean queue-lengths.
1.2
Distributed Resource Allocation Design
Distributed algorithm design is strongly desirable for implementing theoretically
proven efficient algorithms in practical networks. Randomization is a powerful and
6
pervasive strategy for developing distributed scheduling algorithms in interferencelimited wireless networks. Yet, despite the presence of a variety of earlier works on
the design and analysis of particular randomized schedulers (e.g., [33, 72]), there
does not exist an extensive study of the limitations of randomization on the efficient
scheduling in wireless networks. To that end, we develop a common framework for
the modeling and analysis of randomized schedulers. In particular, we reveal that
the performance of randomized schedulers may especially be sensitive to the network topology and the functional form used in assigning priority to network users.
Then, we establish necessary and sufficient conditions on the network topology and
the functional forms for maximum throughput of randomized schedulers. This extensive understanding of the limitations of randomization is important in revealing the
vulnerabilities and strengths of a wide range of scheduling strategies, and it equips
network designers with the machinery for determining the efficient scheduling rules
for the network they will operate.
After understanding the limitations of randomization for distributed algorithms,
we focus on the distributed scheduling algorithm design under time-varying conditions,
under which the most network applications operate. Recently, low-complexity and
distributed Carrier Sense Multiple Access (CSMA)-based scheduling algorithms (e.g.,
[33, 73, 23, 81]) have attracted extensive research interests due to their throughputoptimal characteristics in general network topologies. However, these algorithms are
not well-suited for serving deadline-constrained traffic, such as those generated by
voice or video streaming applications, over time-varying channels due to the large
convergence time of the underlying system dynamics. This motivates us to attack
the problem of distributed scheduling for both elastic and inelastic traffic over timevarying channels. Specifically, we propose a Fast-CSMA (FCSMA) algorithm that
7
converges much faster than the existing CSMA algorithms and thus yields significant advantages for time-sensitive applications. This result provides one of the first
promising means of effectively handling changing conditions in distributed resource
allocation algorithm design.
Yet, all these distributed algorithms presume the knowledge of channel state information (CSI) at the beginning of each transmission. We note that it is highly
energy-consuming and operationally difficult for all network users to continuously
acquire CSI before each data transmission decision. This further motivates us to
investigate the question of whether and how throughput gains can still be achieved
with significant reductions in channel probing requirements and without centralized
coordination amongst the competing users. Earlier works in the design of joint probing and transmission strategies (e.g., [24, 51, 9, 74]) are not suitable for distributed
operation in large-scale networks, since they assume centralized controllers that utilize the whole system state information. We tackle this challenge by first providing
an optimal centralized joint probing and transmission algorithm under the probing
constraints. Noting the difficulties in the implementation of the centralized solution,
we then develop a novel Sequential Greedy Probing (SGP) algorithm by using the
maximum-minimums identity, which is naturally well-suited for physical implementation and distributed operation. The resulting SGP algorithm can achieve explicit
performance guarantees that are tight in certain regimes of interest. We further discuss the distributed implementation of the greedy solution by using the Fast-CSMA
technique.
1.3
Network Model
Here, we introduce the network model used throughout this thesis. We consider a
wireless network with a set L = {1, 2, ..., L} of links, where a link represents a pair of
8
a transmitter and a receiver that are within the transmission range of each other. We
assume that the system operates in slotted time with normalized slots t ∈ {1, 2, ...}.
Due to the interference-limited nature of wireless transmissions, the success or failure
of a transmission over a link depends on whether an interfering link is also active in
the same slot, which is called the link-based conflict model. We call a set of links that
can be active simultaneously as a feasible schedule and denote it as S[t] = (Sl [t])Ll=1 ,
where Sl [t] = 1 if the link l is scheduled in slot t and Sl [t] = 0, otherwise. We use S
to denote the set of all feasible schedules.
We capture the channel fading over link l via a non-negative random variable
Cl [t], with Cl [t] ≤ Cmax , ∀l, t, for some Cmax < ∞, which measures the maximum
amount of service available in slot t, if the link l is scheduled. We assume that
C[t] = (Cl [t])Ll=1 , ∀t ≥ 0, are independently and identically distributed (i.i.d.) over
time. Let S (c) , {Sc : S ∈ S} denote the set of feasible rate vectors when the channel
is in state c, where ab = (al bl )Ll=1 denotes the component-wise product of two vectors
a and b. Then, the capacity region is defined as
R,
X
Pr{C[t] = c} · CH{S (c) },
(1.3.1)
c
where CH{A} denotes a convex hull of the set A, and the summation is a Minkowski
addition of sets.
We assume a per-link traffic model, where Al [t] denotes the number of packets
arriving at link l in slot t that are independently distributed over links, and i.i.d.
over time with finite mean λl > 0, and Al [t] ≤ Amax , ∀l, t, for some Amax < ∞.
Accordingly, a queue is maintained for each link l with Ql [t] denoting its queue length
at the beginning of time slot t. Then, the evolution of queue l is described as follows:
Ql [t + 1] = (Ql [t] + Al [t] − Cl [t]Sl [t])+ , ∀l,
9
(1.3.2)
where (x)+ = max{x, 0}. We say that the queue l is strongly stable (see [21]) if it
satisfies
lim sup
T →∞
T
1X
E[Ql [t]] < ∞.
T t=1
(1.3.3)
We call system stable if all queues are strongly stable. We consider the policies under
which the system evolves as a Markov Chain. We call an algorithm throughputoptimal if it makes all queues strongly stable for any arrival rate vector λ = (λl )Ll=1
that lies strictly within the capacity region.
The rest of the dissertation is organized as follows: In Chapter 2, we study the
design of cross-layer algorithms with optimal convergence speed in wireless networks.
In Chapter 3, we turn our attention to the convergence speed analysis of routing
algorithms in overloaded parallel queueing systems. In Chapter 4, we consider the
scheduling design for achieving maximum throughput, heavy-traffic optimality and
service regularity guarantee. In Chapter 5, we explore the throughput limitations of
randomized schedulers in wireless networks. In Chapter 6, we turn to the optimal
scheduling design for time-varying applications. We further design distributed joint
probing and scheduling algorithms under the limited probing rates in Chapter 7. We
then conclude in Chapter 8.
10
Part I
Resource Allocation for
Time-Sensitive Applications
11
CHAPTER 2
EFFICIENT RESOURCE ALLOCATION WITH
OPTIMAL CONVERGENCE SPEED
In this chapter, we consider the convergence speed of cross-layer algorithms in wireless
networks, which are dominated by the dynamics of incoming and outgoing users as
well as the time-sensitive applications. As we discussed in Chapter 1, the design of
controllers with fast convergence speed in most wireless networks is complicated by
two natural constraints: (i) interference constraints leading to discrete link scheduling
choices; and (ii) a finite set of choices for the transmission rate selection over the
scheduled links.
Previous works mainly focus on the design and analysis of policies with optimal
limiting behavior. A large body of works (e.g. [37, 59, 16, 14, 55, 69]) has utilized dual
and primal-dual methods to develop cross-layer policies with long-term optimality
guarantees. Such solutions are amenable to distributed implementation due to their
natural decomposition into loosely coupled components. However, being first-order
methods, they suffer from the slow convergence speed shared by all such methods
(e.g. [71, 5, 1]).
This speed deficiency of dual methods has recently spurred an exciting thread of
research activity in the design of distributed Interior Point (e.g. [18]) and Newton’s
(e.g. [30, 93, 57]) methods for network utility maximization. However, these works
12
do not incorporate two aforementioned features of wireless networks, namely the discreteness in the scheduling and transmission rate selections. We explicitly incorporate
these intrinsic characteristics of wireless networks in our analysis and algorithm design. To the best of our knowledge, this is the first work that systematically analyzes
and designs algorithms in terms of their converge speed in wireless networks with
such discrete constraints. Next, we list our main contributions, along with references
on where they appear in this chapter.
• We show that the convergence speed1 at which the running average of the
received service rates (cf. Section 2.3.1) and their utility (cf. Section 2.3.1) over
1
. This fundamental limitation on the
T time slots cannot be faster than Ω
T
convergence speed is caused by the discrete nature of the allowable transmission
rates under the operation of any stabilizing and asymptotically optimal flow control
and scheduling policy.
• We develop a generic algorithm that can work with a range of flow rate controllers, and achieves the optimal convergence speed in both rate (cf. Section 2.3.2)
and utility (cf. Section 2.4.2) metrics.
• Somewhat surprisingly, we also show that even a first-order method such as
the well-known dual algorithm can achieve the aforementioned optimal convergence
speed in terms of its utility value (cf. Section 2.4.3).
• These results collectively reveal that, under wireless networks subject to discrete scheduling and rate constraints, the convergence speed of cross-layer algorithms
is dominated by the convergence speed of the scheduling component, and not the
flow rate controller. As such, the speed improvements in the flow rate convergence,
1
The following standard notations are used to describe the rates of growth of two real-valued
sequences {an } and {bn }:an = O(bn ) if ∃c > 0 such that |an | ≤ c|bn |; an = Ω(bn ) if bn = O(an ).
13
unfortunately, cannot extend to the received service rates or utilities in wireless networks. On the bright side, however, with careful design we can achieve the optimal
convergence speed under such constraints.
2.1
Problem Formulation
We consider a multi-hop fading wireless network with L links. Due to modulation,
coding, as well as other practical constraints, each link has to transmit at one of a
finite set of rates2 . We use R[t] = (Rl [t])Ll=1 to denote the service rate vector offered
to the links in slot t, which must be selected from a feasible set of transmission
rates. We note that the capacity region R is a polyhedron due to the discreteness of
the transmission rate choices, and hence can be written as R = {y ≥ 0 : Hy ≤ b},
where y ∈ RL and H is some non-negative matrix. Note that H has L columns and
the number of rows in H is equal to the dimension of the vector b associated with
the number of interference constraints.
To capture the heterogeneous and potentially inter-dependent preferences of users,
we define a utility function U : RL+ → R+ that measures the total network utility
when link l receives an average service rate of rl , where r = (rl )Ll=1 . We assume
U (r) to be a strictly concave function that is non-decreasing in each coordinate. The
objective of Network Utility Maximization (NUM), then, is to design a congestion
control and scheduling algorithm such that the average service rate vector r solves
the following optimization problem:
2
For example, IEEE 802.11a standard uses OFDM transmission technique and can support rates
in Mega bits per second selected from the finite set {6, 9, 12, 18, 24, 36, 48, 54}; In CDMA2000
1xEV-DO specification, the forward link transmission rate in kilo bits per second is chosen from
the finite set {38.4, 76.8, 153.6, 307.2, 614.4, 921.6, 1228.8, 1843.2, 2457.6}.
14
Definition 2.1.1. (Network Utility Maximization (NUM))
max
r=(rl )L
l=1
U (r)
(2.1.1)
Subject to r ∈ R,
(2.1.2)
where R is defined in (1.3.1).
The strict concavity of U (·) together with the convexity of R guarantees the
uniqueness of the solution of NUM, which is denoted as r∗ = (rl∗ )Ll=1 . Also, due to
the non-decreasing nature of U (·), r∗ must lie on the boundary of R.
It is important to note that r∗ is the optimal average offered service rate to the
links. The purpose of the flow rate controller, however, is to determine the optimal
injection rate of traffic into the network while maintaining network stability. Recall
that Ql [t] denotes the queue-length at link l ∈ L at the beginning of slot t. Let Gl [t]
denote the amount of injected data into Queue-l in slot t under a given flow rate
controller, and recall that Rl [t] denotes the service rate offered to link l in slot t
under a given scheduler. Then, the evolution of Ql can be expressed as
Ql [t + 1] = (Ql [t] + Gl [t] − Rl [t])+ ,
t ≥ 1.
In this chapter, we are interested in the convergence speed of a broad class of joint
flow rate control and scheduling policies P that are both stabilizing and asymptotically rate optimal. To define this class of policies abstractly, we introduce
the parameter > 0 as a generic term to characterize the performance of the joint
policy under specific design choices. Accordingly, the average injection rate3 of a
given policy under parameter is r() . Similarly, we will use the superscript ·() over
(Ql [t])l , (Gl [t])l , (Rl [t])l , etc. to express the queue-lengths, injections, offered service
3
For each parameter > 0, the system is stable and has a steady-state distribution. Thus, the
long-term average injection rate is well-defined.
15
rates, etc. under the policy with parameter . The stability condition requires that
r() is strictly within the capacity region R for all > 0, and the asymptotic rate optimality condition requires that lim r() = r∗ , i.e., the asymptotically optimal policy
↓0
achieves the optimal service rate vector in the limit. Figure 2.1 shows the relationship
between r() and r∗ in a two-dimensional case.
Figure 2.1: Relationship between r() and r∗
Thus, the parameter captures the closeness of the injection rate to the optimal
service rate r∗ under the class of joint policies parametrized by . We note that
this abstraction includes a wide range of joint control and scheduling policies in the
literature. For example, in the well-known subgradient-based designs (e.g., [37, 59, 16,
14, 55]) the generic term maps to the particular design parameter that corresponds
to the step-size on the subgradient iteration.
The stability condition of the joint flow rate control and scheduling policies in P
implies that the running average of departures over time must also converge to r() .
16
Since the running average of departures4 up to time T is the real measure of received
service until that time, we are interested in its convergence speed to r() . To be more
()
()
()
precise, for the policy with parameter we use Dl [t] , min(Rl [t], Ql [t]) to denote
the departures in slot t for link l ∈ L, and define its running average until T ≥ 1 as
()
dl [T ] ,
()
T
1 X ()
D [t],
T t=1 l
∀l ∈ L,
(2.1.3)
()
and use d [T ] , (dl [T ])l . Next, we introduce the metrics of interest in our study of
convergence speed, both in the running average departure rate and its corresponding
utility value.
Definition 2.1.2. (Metrics of Interest) For any policy in P with parameter , we
()
()
define the rate deviation φ(d [T ], r() ) between d [T ] and r() at time T as
()
()
φ(d [T ], r() ) , d [T ] − r() ,
(2.1.4)
()
and the utility benefit received until time T as U (d [T ]), where kyk is the l2 norm
of the vector y.
In the rest of chapter, we will: (i) provide an example showing the fundamental
speed limitation exerted by the discrete choice of transmission rates (cf. Section 2.2);
()
(ii) establish fundamental limits on the speed at which E[φ(d [T ], r() )] converges to
zero as T increases (cf. Section 2.3.1); (iii) develop joint flow control and scheduling policy with provably optimal convergence speed in terms of rate deviation (cf.
Section 2.3.2); (iv) derive fundamental limits on the speed at which the utility benefit converges to the optimal utility value of NUM when sources of randomness are
eliminated (cf. Section 2.4.1); (v) show that our proposed algorithm, as well as the
4
Due to the discreteness of transmission rate choices, it is unlikely that the departure rate in slot
T converges to r() , as T increases.
17
well-known dual algorithm, achieves the optimal convergence speed in terms of utility benefit (cf. Section 2.4.2, 2.4.3); and finally (vi) provide the detailed comparison
between the proposed algorithms and the traditional dual algorithm in terms of the
convergence speed and the average delay through simulations (cf. Section 2.5).
2.2
A Motivating Example
In this section, we study a simple example to see how the convergence speed of a
sequence is limited by the discreteness of its elements. In particular, we consider
the convergence speed of any zero-one sequence converging to 0.5. For any zero-one
sequence {D[t] : D[t] ∈ {0, 1}}t≥0 , we have
T
1 X
1
D[t] − 0.5 =
T
2T
t=1
Noting that both 2
T
X
T
X
D[t] − T .
2
(2.2.1)
t=1
D[t] and T are integers, we have
t=1
T
1 X
φ(d[T ], 0.5) = D[t] − 0.5
T
 t=1
T
X

1


D[t];

 ≥ 2T , if T 6= 2
t=1
=
T
X




=
0,
if
T
=
2
D[t].

(2.2.2)
t=1
1
2Tk
and thus the convergence speed of any zero-one sequence cannot be faster than
1
Ω
. To validate this result, we consider an independently and identically disT
tributed (i.i.d.) Bernoulli random sequence with mean 0.5. Figure 2.2 shows one re-
Hence, the subsequence {φ(d[Tk ], 0.5) : Tk is odd} is always lower-bounded by
alization of this random sequence. From this figure, we can see that φ(d[T ], 0.5) hits
0 for some T , and is always non-zero when T is odd. The subsequence {φ(d[Tk ], 0.5) :
1
Tk is odd} is always lower-bounded by
.
2Tk
18
0.5
φ(d[T ], 0.5)
0.45
φ(d[Tk ], 0.5)
1
2T
Rate deviation
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5
10
15
20
25
30
35
40
45
50
Time step T
Figure 2.2: The convergence speed of an i.i.d Bernoulli sequence.
This simple example suggests that the discreteness in the choice of elements in the
sequence exerts a fundamental limitation on the speed with which its running average
over time can approach its limit. In what follows, we will show that this observation
indeed holds even in the wider context of a multi-hop fading wireless network with a
finite selection of transmission rates.
2.3
Convergence Speed in Rate Deviation
In this section, we study the optimal convergence speed in terms of rate deviation
over wireless fading channels. To that end, we first give the fundamental lower bound
on the expected rate deviation for any algorithm. Then, we provide an algorithm that
can achieve this lower bound and establish the optimality of the proposed algorithm.
19
2.3.1
A Lower Bound on the Expectation of Rate Deviation
In this subsection, we show that for any policy in P, the convergence speed of expected
1
. To that end, we need the following integer assumption on
rate deviation is Ω
T
the transmission rate, which measures the number of packets that can be transmitted
in one time slot.
Assumption 2.3.1. The service rate Rl for each link l ∈ L is selected from a finite
and nonnegative-integer-valued set {Bl,1 , Bl,2 , ..., Bl,Kl }, where 0 ≤ Bl,1 < Bl,2 < ... <
Bl,Kl and Kl is some positive integer.
Next, we give the following key lemma, which will also be useful in the later
section.
Lemma 2.3.2. Let I , {a1 , a2 , ..., aK }, where 0 ≤ a1 < a2 < ... < aK and K is
some positive integer. If r ∈ (ai , ai+1 ) for some i = 1, ..., K − 1, then for any sequence
r − ai ai+1 − r
{I[t] : I[t] ∈ I}t≥1 , there exists a constant cr ∈ 0, min
,
such
2
2
T
c
1 X
r
I[t] − r ≤ , then
that if T
T
t=1
T +1
1 X
cr
I[t]
−
r
.
≥
T + 1 t=1
T +1
(2.3.1)
Remark: Note that K can be as large as ∞.
Proof. See Appendix A.1 for the proof.
Proposition 2.3.3. Under Assumption 2.3.1, for any policy in P with parameter ,
if r() is not a vector with all integer-valued coordinates, then convergence speed of the
1
expected rate deviation to zero is Ω
, i.e., there exists a strictly positive constant
T
c and a positive integer-valued increasing sequence {Tk }∞
k=1 such that
()
φ(d [Tk ], r() ) ≥
c
,
Tk
20
∀k ≥ 1,
(2.3.2)
holds for any sample path of departure rate vector sequence {D() [t]}t≥1 , which also
implies that
()
E[φ(d [Tk ], r() )] ≥
c
,
Tk
∀k ≥ 1.
(2.3.3)
Remark: If the optimal rate vector r∗ has at least one non-integer-valued coordinate, then the condition for Proposition 2.3.3 holds when is sufficiently small.
Moreover, since the region R is compact, there are finitely many rate vectors with all
coordinates being integer in R. Thus, Proposition 2.3.3 holds in almost all cases.
Proof. See Appendix A.2 for the proof.
Proposition 2.3.3 indicates that the discrete structure of the transmission rates
intrinsically limits the convergence speed for any algorithm in class P. Thus, the
search for higher-order numerical optimization methods cannot overcome this fundamental limitation in wireless networks. Despite pessimism of this observation, we are
still interested in designing an algorithm that can achieve this fundamental bound
and establish the optimality of this algorithm in terms of its convergence speed. To
that end, we define the rate deviation optimality for an algorithm in class P.
Definition 2.3.4. (Rate Deviation Optimality) An algorithm in class P with
parameter is called rate deviation optimal, if its departure rate vector sequence
{D() [t]}t≥1 satisfies
()
()
E[φ(d [T ], r() )] ≤
()
where F1
F1
,
T
∀T ≥ 1,
()
is a positive constant and d [T ] is defined in (2.1.3).
Next, we propose an algorithm with rate deviation optimality.
21
(2.3.4)
.
2.3.2
A Rate Deviation Optimal Policy
In this subsection, we propose a rate deviation optimal algorithm with parameter
> 0 that converges to the injection rate r() solving the following optimization
problem.
Definition 2.3.5. (-NUM)
max
U (r)
r=(rl )L
l=1
Subject to r ∈ R() ,
(2.3.5)
(2.3.6)
where R() , {y ≥ 0 : Hy ≤ b − }.
Since U is strictly concave and R() is convex, r() is unique. In addition, as → 0,
r() converges to the optimal rate vector r∗ . Without loss of generality, we assume
kr() − r∗ k ≤ ρ() , where lim ρ() = 0. The relationship between R() and R in a
↓0
two-dimensional case is shown in Figure 2.3.
İ
Figure 2.3: The relationship between R() and R.
22
()
Each link maintains a data queue and a virtual queue. Let Yl [t] denote the
virtual queue length at link l in each slot t. In each slot t, the amount of arrival to
()
the virtual queue l is rl [t]. The service for each virtual queue is determined by the
scheduling policy defined below.
Algorithm 2.3.6. (Rate Deviation Optimal (RDO) Algorithm with parameter ): In each time slot t,
()
Flow control: {r() [t] = (rl [t])l }t≥1 is a sequence generated by a numerical
optimization algorithm solving -NUM. Note that r() [t] ∈ R() , ∀t ≥ 1.
Arrival generation: For each link l,
()
(1) if t = 1, then Gl [t] = Bl,Kl ;
(2) else if
t−1
X
()
Gl [i] <
i=1
t−1
X
()
()
()
rl [i], then Gl [t] = Bl,Kl ; Gl [t] = 0, otherwise.
i=1
()
Then, inject Gl [t] packets into each data queue l and increase virtual queue length
()
()
Yl [t] by rl [t].
Scheduling: Perform Maximum Weight Scheduling (MWS) algorithm among
virtual queues, that is,
()
R [t] ∈
argmax
L
X
(C[t])
η=(ηl )L
l=1 ∈S
l=1
()
Yl [t]ηl .
(2.3.7)
Use R() [t] to serve data queues.
()
()
Queue evolution: Let Ql [1] = Yl [1] = 0, ∀l, and for t ≥ 2, update the data
queue length and virtual queue length as follows:
()
Ql [t
()
+ 1] =
Yl [t + 1] =
()
Ql [t]
+
()
()
Gl [t]
()
−
()
Rl [t]
()
Yl [t] + rl [t] − Rl [t]
23
+
+
, ∀l,
(2.3.8)
, ∀l.
(2.3.9)
Remarks: (1) Recent advances in the design of distributed Newton’s method (e.g, [93],
[57]) show the promise in generating sequence {r() [t]}t≥1 in quick and distributed way.
In addition, we can also use Gradient Projection method to solve -NUM.
(2) The purpose of maintaining the virtual queue is to help show the stability of
data queues. In fact, directly performing MWS among data queues does not hurt
the convergence speed, as we will see in the simulations. However, the traditional
one-step Lyapunov drift argument to show the stability of the proposed algorithm
does not work in such a case, since the deterministic arrivals to each link l are alternating between 0 and Bl,Kl , which leads to the potential positiveness of the one-step
Lyapunov drift given the current queue length state.
(3) The key step for establishing the optimal convergence speed of the RDO
()
Algorithm is to show that the generated arrival sequence {G() [t] = (Gl [t])l }t≥0
satisfies
T
X
E (G() [t] − r() [t]) < ∞.
(2.3.10)
t=1
This can be seen in the equation (A.5.1). The purpose of the deterministic arrival
generation is to make (2.3.10) true and to stabilize the system, because the simple
independent random arrival generation does not meet the requirement of (2.3.10).
Figure 2.4 shows the operation of the RDO Algorithm at link l. Next, we show
that the RDO Algorithm can achieve rate deviation optimality if the generated sequence {r() [t]}t≥1 converges fast enough. To that end, we need the following lemma
exhibiting that the generated arrivals closely track the generated sequence {r() [t]}t≥1 .
Lemma 2.3.7. For each link l, we have
T
X
()
()
(Gl [t] − rl [t]) ≤ Bl,Kl ,
t=1
24
∀T ≥ 1.
(2.3.11)
rl(İ) t
Data queue Ql(İ) t
Gl(İ) t
Virtual queue Yl(İ) t
Rl(İ) t
Rl(İ) t
Figure 2.4: The operation of the RDO Algorithm at link l.
Proof. See Appendix A.3 for the proof.
Based on Lemma 2.3.7, we can show that for each link, the data queue length
is upper-bounded by the sum of some constant and the virtual queue length for all
sample paths, which is useful in establishing the rate deviation optimality of the RDO
Algorithm.
Lemma 2.3.8. For each link l, the data queue length is upper-bounded by the sum
of the virtual queue length and 2Bl,Kl for all sample paths, i.e.,
()
()
Ql [T ] ≤ Yl [T ] + 2Bl,Kl ,
∀T ≥ 1,
(2.3.12)
holds for all sample paths.
Proof. See Appendix A.4 for the proof.
We are now ready to establish the rate deviation optimality of the RDO Algorithm.
Proposition 2.3.9. For the RDO Algorithm with parameter > 0, as long as the
T
1 X ()
W1
flow controller satisfies
kr [t] − r() k ≤
, for all T ≥ 1, the generated link
T t=1
T
departure sequence {D() [t]}t≥1 satisfies
()
W
E[φ(d [T ], r )] ≤ 1 ,
T
25
()
()
∀T ≥ 1,
(2.3.13)
()
where W1 and W1
are some positive constants.
Remark: {r() [t]}t≥1 generated by the distributed Newton’s method (e.g., [30, 93,
57]) satisfies the condition for Proposition 2.3.9.
Proof. See Appendix A.5 for the proof.
Proposition 2.3.10. For the RDO Algorithm with parameter > 0, if at least one
coordinate of r() is a non-integer and the same condition in Proposition 2.3.9 holds,
then the RDO Algorithm is rate deviation optimal (cf. Definition 2.3.4).
Proof. The result directly follows from Propositions 2.3.3, 2.3.9 and the definition of
rate deviation optimality.
So far, we have observed that the discrete choice of transmission rates significantly
1
limits the convergence speed to Ω( ) and provided an algorithm that can achieve the
T
optimal convergence speed in terms of rate deviation. In [64], the authors showed that
for dual algorithm, the convergence speed of the running average of primal variables
1
over T slots can be as fast as Ω
in terms of utility benefit. To the best of
T
our knowledge, there does not exist a convergence speed analysis of dual methods in
terms of rate deviation metric due to the non-smoothness of the dual function (see
[3]). This motivates us to investigate the optimality of dual algorithm in terms of its
convergence speed of the utility benefit metric under additional assumptions of nonrandomness. These assumptions are necessary to establish the fundamental upper
bound on the utility benefit under random environment, since the aggregation over
links and the randomness (such as random arrivals, randomized scheduling or channel
fading) distort the discrete structure. Thus, we focus on the deterministic system in
next section, where there is no randomness in the system. It is still quite difficult and
non-trivial to establish the convergence speed optimality in terms of utility benefit in
such a system.
26
2.4
Convergence Speed in Utility Benefit
In this section, we mainly consider the deterministic system, where there is no randomness (such as channel fading, random arrivals and randomized scheduling) in the
system. We first establish the fundamental upper bound on the utility benefit for
any algorithm in such a deterministic system. Then, we show that both deterministic
version of the RDO Algorithm and the well-known dual algorithm can achieve this
upper bound and establish their optimality under utility benefit metric.
2.4.1
An Upper Bound on the Utility Benefit
()
In this subsection, we establish an upper bound on the utility benefit U (d [T ]) for
any algorithm in class P with parameter . We do not require the integer Assumption
2.3.1 for the deterministic system. Without loss of generality, we assume that each
link has a finite set of transmission rates Fl , {bl,1 , bl,2 , ..., bl,Kl }, where 0 ≤ bl,1 <
bl,2 < ... < bl,Kl . To establish the fundamental upper bound on the utility benefit, we
need the following assumption on the scheduling:
Assumption 2.4.1. Each link l with queue length less than bl,Kl is not to be scheduled.
Remark: This scheduling assumption helps establish the fundamental bound on
the utility benefit. Removing this assumption does not speedup the convergence,
which is validated through simulations.
We also need the following assumptions on the utility function:
Assumption 2.4.2. (1) The utility function U (r) is additive, that is, U (r) =
L
X
Ul (rl ),
l=1
where Ul (y) is a concave and non-decreasing function of y;
(2) hmin ≤ Ul0 (y) ≤ hmax , ∀y, where 0 < hmin < hmax < ∞;
(3) −βmax ≤ Ul00 (y) ≤ −βmin , ∀y, where 0 < βmin < βmax .
27
(y + γ)1−m
,
1−m
where m and γ are positive constants. Now, we are ready to establish the fundamental
Examples of such utility functions include Ul (y) = log(y +γ) and Ul (y) =
upper bound on the utility benefit for any algorithm in class P.
Proposition 2.4.3. (1) Under Assumption 2.4.1 and 2.4.2, for any δ ∈ (0, max kr −
r∈R
∗
r k) and any policy in P with parameter , there exists a constant c
(δ)
> 0 and a
positive integer-valued increasing sequence {Tk }∞
k=1 such that
√
1
c(δ)
()
,
U (d [Tk ]) ≤ U (r∗ ) − βmin Lδ 2 −
2
Tk
√
where H , L(2βmax max bl,Kl + hmax ).
∀Tk ≤
c(δ)
,
Hδ
(2.4.1)
l∈L
(2) If we further have
L
X
Ul0 (rl∗ )rl∗ ∈
/ H, then for a sufficiently small δ > 0, we
l=1
have
c(δ)
1
= min
4 y∈H
!
L
X
Ul0 (rl∗ )rl∗ − y − Gδ > 0,
(2.4.2)
l=1
where G > 0 is some constant. This implies that c(0) , lim c(δ) > 0 and lim
δ↓0
δ↓0
c(δ)
= ∞,
Hδ
and (2.4.1) becomes
c(0)
,
U (d [Tk ]) ≤ U (r ) −
Tk
()
where H ,
( L
X
∗
∀k ≥ 1,
(2.4.3)
)
Ul0 (rl∗ )Il : I = (Il )Ll=1 ∈ R and Il ∈ Fl , ∀l .
l=1
Remark: The finiteness of the set Fl implies that the set H also has a finite
L
X
number of elements. Thus, it is unlikely that
Ul0 (rl∗ )rl∗ is in the set H in practice.
l=1
Proof. See Appendix A.6 for the proof.
The first part of Proposition 2.4.3 establishes a fundamental bound on how close
the utility benefit can be to the optimal utility level within a finite range of time. The
second part, then, shows that, under an additional mild assumption on r∗ , the range
28
over which the bound holds can be made to extend to infinity by letting the error go
to zero. To illustrate the nature of this result, Figure 2.5 shows the utility benefit
of an algorithm in class P over time. It shows that the utility benefit repeatedly
c(δ)
, which, from the second part of the
falls below the fundamental bound until time
Hδ
proposition, goes to infinity as δ vanishes.
Upper Bound
U(d(H )[T])
One possible utility benefit
U (r *)
1
U (r*) E min LG 2
2
0
T1 T2
T3
T9
T10 c ( G )
HG
T
Figure 2.5: The utility benefit of an algorithm in class P.
Note that it is impossible for any policy in class P with parameter > 0 that
(2.4.1) holds for all T ≥ 1, since the “good” policy (e.g., where is sufficiently
()
small) can achieve the optimal value at arbitrary accuracy and thus U (d [T ]) will
√
1
exceed U (r∗ ) − βmin Lδ 2 eventually. In addition, inequality (2.4.3) implies that
2
the utility benefit of any algorithm cannot be beyond the optimal value. Thus, these
fundamental upper bounds on the utility benefit motivate the definition of utility
benefit optimality of an algorithm given next.
29
Definition 2.4.4. (Utility benefit optimality) For any δ > 0, an algorithm in
class P with parameter > 0 is called utility benefit optimal, if its generated departure
rate vector sequence {D() [t]}t≥1 satisfies
()
√
1
F
()
(δ)
U (d [T ]) ≥ U (r∗ ) − βmin Lδ 2 − 3 , ∀T ≤ F4 ,
2
T
()
(δ)
(δ)
where F3 > 0 and F4 > 0 with lim F4 = ∞.
δ→0
Next, we first investigate the utility benefit optimality of the deterministic version
of the RDO Algorithm.
2.4.2
Utility Benefit Optimality of the RDO Algorithm
In this subsection, we show that the deterministic version of the RDO Algorithm is
utility benefit optimal under Assumptions 2.4.1 and 2.4.2.
The deterministic version of the RDO Algorithm works as follows:
Algorithm 2.4.5. (Deterministic RDO (DRDO) Algorithm with parameter > 0): At each time slot t,
()
Flow control: {r() [t] = (rl [t])l }t≥1 is a sequence generated by a numerical
optimization algorithm solving -NUM.
()
Arrival: Inject rl [t] amount of data into each queue l;
Scheduling: Perform the MWS algorithm among links whose queue length are
no less than bl,Kl , that is,
R() [t] ∈ argmax
η=(ηl )L
l=1 ∈R
L
X
()
Ql [t]1{Q() [t]≥b
l
l=1
l,Kl }
ηl ;
(2.4.4)
Queue evolution: Update the queue length as follows:
()
()
()
()
Ql [t + 1] = (Ql [t] + rl [t] − Rl [t])+ ,
30
∀l.
(2.4.5)
Next, we give a lower bound of the DRDO Algorithm under utility benefit metric.
Proposition 2.4.6. Under Assumption 2.4.2 on the utility function U , for the DRDO
T
1 X ()
W2
Algorithm with parameter > 0, if
kr [t] − r() k ≤
, for all T ≥ 1, then its
T t=1
T
departure sequence {D() [t]}t≥1 satisfies
√
U (d [T ]) ≥ U (r ) − hmax Lρ() −
()
()
where W2 and W2
√
∗
()
Lhmax W2
, ∀T ≥ 1,
T
are some positive constants.
Proof. The proof first establishes the connection between the utility benefit of the
DRDO Algorithm and its rate deviation by using the concavity of the utility function.
Then, we establish the upper bound on the rate deviation by using similar technique
in Proposition 2.3.9. The details can be found in Appendix A.7.
Proposition 2.4.7. Under Assumptions 2.4.1 and 2.4.2, the DRDO Algorithm is
utility benefit optimal (c.f. Definition 2.4.4), i.e., for any δ > 0, by choosing > 0
βmin δ 2
such that ρ() ≤
, the DRDO Algorithm can achieve the upper bound in (2.4.1).
2hmax
Proof. The proof immediately follows from Propositions 2.4.3, 2.4.6 and the definition
of utility benefit optimality.
Next, we study the utility benefit optimality of the well-known dual algorithm.
2.4.3
Utility Benefit Optimality of the Dual Algorithm
In this subsection, we establish the utility benefit optimality of the well-known dual
algorithm (e.g., [37, 59, 14, 55]). The dual algorithm can be obtained by Lagrangian
relaxation and naturally decomposes the network function into the two main components: the congestion control and the scheduling. Next, we give the definition of the
dual algorithm for completeness.
31
Algorithm 2.4.8. (Dual Algorithm with parameter > 0):
()
Flow control: Given Q() [t] = (Ql [t])Ll=1 , solve the following optimization problem:
X ()
1
r [t] ∈ argmax U (w) −
Ql [t]wl ,
0≤w≤M l=1
L
()
(2.4.6)
where M is the maximum injection rate;
Scheduling: Perform MWS algorithm among links whose queue length are no less
than bl,Kl , that is,
()
R [t] ∈ argmax
L
X
()
ηl ;
(2.4.7)
Ql [t + 1] = (Ql [t] + rl [t] − Rl [t])+ , ∀l,
(2.4.8)
η=(ηl )L
l=1 ∈R l=1
Queue evolution:
()
()
Ql [t]1{Q() [t]≥b
l
()
l,Kl }
()
where M is the maximum allowable input rate.
The Dual Algorithm also uses the scheduling assumption as the DRDO Algorithm
that does not schedule link l with queue length less than bl,Kl , which helps establish
its utility benefit optimality. However, removing this scheduling constraint does not
improve the convergence speed, which is validated through simulations.
We are now ready to give the convergence speed of the Dual Algorithm in terms
of utility benefit.
Proposition 2.4.9. For the Dual Algorithm with parameter > 0, the generated
departure sequence {D() [t]}t≥1 satisfies
where G()
L
()
kQ() [1]k2 − (M 2 + 3 max b2l,Kl )
U (d [T ]) ≥ U (r∗ ) −
l∈L
2
√ 2T
√
hmax L
−
kQ() [1]k + G() L , ∀T ≥ 1,
(2.4.9)
T
√
hmax
βmax
3βmax
2
, W+
and W ,
+ 2 LM +
+ 2 L max b2l,Kl .
l∈L
32
Proof. The proof first shows the boundedness of queue length at all times for each
link. Then, we establish the relationship between the utility benefit and the utility of
the running average of flow rate vector sequence. By using similar technique in [64],
we can give the lower bound on the utility of the running average of flow rate vector
sequence. Note that the scheduling component in our setup makes our analysis more
challenging than that in [64]. Please see Appendix A.8 for details.
Remark: The difference between our analysis and that in [64] lies in that we add
the scheduling component in wireless networks and consider the utility of the running
average of departure rate vector sequence rather than that of the running average of
primal vector sequence, which makes it more challenge to deal with.
()
From (2.4.9), we can see that the utility benefit U (d [T ]) converges to the optimal
L
1
2
2
∗
(M + 3 max bl,Kl ) with the speed of Ω
. When
value U (r ) within error level
l∈L
2
T
the parameter decreases, the error level will decrease in the price of the slower
convergence speed. Next, we establish the utility benefit optimality of the Dual
Algorithm.
Proposition 2.4.10. The Dual Algorithm is utility benefit optimal (cf. Definition
βmin δ 2
, the Dual Algo2.4.4), i.e., for any δ > 0, by choosing ≤ √
L(M 2 + 3 maxl∈L b2l,Kl )
rithm can achieve the upper bound in (2.4.1).
Proof. The proof directly follows from propositions 2.4.3, 2.4.9 and the definition of
utility benefit optimality.
2.5
Simulation Results
In this section, we consider a single-hop network topology with L = 5 links in both
non-fading and fading channels. In each time slot, at most one link can be active.
We take the additive utility function with Ul (y) = log(y + γ), ∀l, where γ = 10−8 ,
33
for both non-fading and fading scenarios. Recall that this function satisfies Assumption 2.4.2 on the utility function to establish the utility benefit optimality of both
DRDO Algorithm and the Dual Algorithm. For the non-fading scenario, each link
has a fixed rate and the link rate vector is p = [0.8, 0.4, 0.6, 0.5, 0.3]. For the fading
scenario, each link suffers from ON-OFF channel fading independently and the link
ON probability vector is also p. For DRDO and RDO algorithms with parameter ,
we use Newton’s method (see [5]) to generate sequence {r() [t]}t≥1 that satisfies the
condition for Proposition 2.3.9.
2.5.1
Non-fading Scenario
In this subsection, we mainly investigate the impact of parameter on the performance of the Dual Algorithm and compare it with the DRDO Algorithm with = 0.
Figures 2.6a, 2.6b and 2.6c show the impact of parameter on the convergence speed
and the average queue length per link for the Dual Algorithm. From Figure 2.6a, we
can observe that when is too large (e.g., = 20), the utility benefit cannot converge
to the optimal value. From Figure 2.6b, we can see that the convergence under rate
deviation metric requires much smaller than that under utility benefit metric. This
is probably because that the utility function is concave while the l2 norm of the rate
vector is convex. If there is some difference between rate vectors, there is smaller difference between their utility values than that between their distance especially when
the rate vectors are away from zero.
In addition, among the set of parameters guaranteeing the convergence to the
optimal value, the smaller leads to the slower convergence speed under both interest
metrics. From Figure 2.6c, we can observe that the average queue length per link
increases as decreases. This observation matches the theoretical upper bound on
the average queue length. Thus, for the Dual Algorithm, we need to choose as large
34
0.30
( )
[T]-r
-14
( )
-16
-18
Rate deviation ||d
Utility benefit U(
d
( )
[T])
||
-12
-20
Dual with
-22
Dual with
-24
Dual with
-26
Dual with
-28
DRDO with
-30
10
100
1000
0.25
Dual with
0.20
Dual with
Dual with
0.15
Dual with
0.10
DRDO with
0.05
0.00
10000
100
1000
Time step T
10000
Time step T
(a) Utility Benefit metric
(b) Rate deviation metric
Average queue length
120
100
Dual with
80
Dual with
Dual with
60
Dual with
40
DRDO with
20
0
0
10000
20000
30000
40000
50000
60000
70000
Time step T
(c) Average queue length
Figure 2.6: Dual Algorithm performance with varying as possible among the set of parameters guaranteeing convergence to the optimal
value, which not only leads to faster convergence but also enjoys smaller average
delay.
In Figures 2.6a, 2.6b and 2.6c, we also compare the performance between the
Dual Algorithm and the DRDO Algorithm with = 0. We can observe that the
35
Dual Algorithm with proper parameter (e.g., = 5) converges slightly faster than
the DRDO Algorithm under the utility benefit metric. This does not contradict our
result in Section 2.4 that both Dual and DRDO algorithms are utility benefit optimal
and thus their convergence speed may differ at most a constant factor. However, the
DRDO Algorithm converges faster than the Dual Algorithm under rate deviation
metric. Finally, we can see from Figure 2.6c that the DRDO Algorithm has quite
small average queue length.
2.5.2
Fading Scenario
In this subsection, we mainly consider the performance of variants of the RDO Algorithm over wireless fading channels. We consider a variant of the RDO Algorithm
that does not require maintaining a virtual queue and performs MWS directly among
data queues, and another variant of the RDO Algorithm that has independent random arrivals and performs MWS directly among data queues. Recall that the purpose
of introducing virtual queues in RDO Algorithm is to show the boundedness of the
average data queue length by avoiding the difficulty in using Lyapunov Drift argument. From Figures 2.7a, 2.7b and 2.7c, we can observe that for both rate deviation
and utility benefit metrics, the original RDO Algorithm converges faster than and
has smaller average queue length than a variant of the RDO Algorithm with independent random arrivals, but converges slower than and has larger average queue
length than a variant of the RDO Algorithm without virtual queues. The reason
why the RDO Algorithm without virtual queue has the smallest queue-length levels
is that the RDO Algorithm with independent arrivals creates the randomness to the
queues, potentially increasing the mean queue-length levels, and the original RDO
Algorithm maintaining the virtual queue length requires some overhead. According
to the equation (A.5.1), the algorithm leading to the smaller queue-lengths has the
36
faster convergence speed. Thus, the variant of the RDO Algorithm that directly
||
performs MWS among data queues has the best performance.
)
r
(
[T])]
-7
RDO Algorithm
0.28
RDO Algorithm
0.26
0.24
without virtual queue
0.22
Rate Deviation E||
d
(
)
-9
-10
RDO Algorithm
-11
RDO Algorithm
-12
without virtual queue
RDO Algorithm
-13
with independent arrival
-14
and without virtual queue
-15
10
100
1000
10000
RDO Algorithm
0.20
0.18
with independent arrival
0.16
and without virtual queue
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
10
100
Time step T
20
(b) Rate deviation metric
RDO Algorithm
18
RDO Algorithm
16
without virtual queue
14
RDO Algorithm
12
with independent arrival
10
and without virtual queue
8
6
4
2
0
0
1000
Time step T
(a) Utility Benefit metric
Average queue length
Utility benefit E[U(
d
(
)
[T]-
-8
0.30
10000
20000
30000
40000
50000
60000
70000
Time step T
(c) Average queue length
Figure 2.7: Variants of the RDO Algorithm
37
10000
2.6
Summary
In this chapter, we considered the convergence speed of joint flow control and scheduling algorithms in Network Utility Maximization (NUM) problem in multi-hop wireless
networks. We realized that the discreteness of scheduling constraints and transmission
rates are two of the most important features in wireless networks. We incorporated
these important characteristics into the analysis and design of algorithms in terms of
their convergence speed by defining two metrics of interest: rate deviation and utility
benefit.
We showed that the convergence speed of any algorithm cannot be faster than
1
for both rate deviation and utility benefit metrics due to the discrete choices
Ω
T
of transmission rates at each link. This interesting and fundamental finding reveals
that designing faster (e.g., Interior-Point or Newton based) algorithms for the flow
1
in wireless networks caused by the
rate control cannot break the barrier of Ω
T
scheduling component. Then, we provided an algorithm that can achieve optimal
convergence speed under both rate deviation and utility benefit metrics. Moreover,
we showed that the well-known dual algorithm also has optimal convergence speed
in terms of utility benefit, which is a somewhat surprising outcome in view of the
first-order nature of its iteration.
38
CHAPTER 3
ALGORITHM DESIGN WITH OPTIMAL
CONVERGENCE SPEED IN OVERLOADED SYSTEMS
Having understood the algorithm design with optimal convergence speed in stable
wireless networks, we shift our focus on the convergence speed of efficient algorithms
in overloaded systems that demand fast convergence (cf. Chapter 1). Noting the
complexity of analyzing the overloaded systems, we consider the classical problem
of routing random arrivals to parallel queues with random services in overloaded
regimes, i.e., when the arrival rate is greater than the total sum of service rates.
There are several interesting works (e.g., [22, 8, 86, 50]) dealing with overloaded
queueing systems. Under various conditions, these works study the performance
and optimization of the metrics of queue overflow rates (i.e., the rates at which the
queues grow in the overloaded regime), and the related metric of the departure rates
of served packets. For example, in [8, 86], the authors focus on the design and analysis
of maximum-weight type scheduling policies to minimize the queue overflow rates or
to maximize the total departure rate from the system.
One caveat with the performance metrics of overflow rate and departure rate is
that, being long-term time averages, they may not be able to differentiate between
policies in terms of their convergence speeds to the same limit. In fact, as will be noted
in Section 3.1, even suitably selected randomized decisions can achieve optimal overflow or departure rate levels in overloaded queueing systems. With this motivation, in
39
this paper we propose and analyze the metric of cumulative unused service over time
to analyze the performance of routing policies in overloaded systems. This metric not
only measures the amount of under-utilization in the multi-server system over time,
but also captures the speed with which the running-average of the departure rates
converges to their limiting value (cf. Section 3.1).
The proposed cumulative unused service process is difficult to analyze in both
stable and unstable systems due to its non-stationary nature. To tackle this challenge,
we establish a novel “queue reversal” result (cf. Theorem 3.2.1) that equates the
expected cumulative unused service in the unstable system to the expected queuelength of a related (in fact, reversed) stable system. With this connection, we can
obtain the mean cumulative unused service metric by studying the mean queue-length
of a stable Markov chain, for which a rich set of tools and results exists. Based on this
fundamental result, we are able to obtain a nontrivial lower bound (cf. Section 3.2)
on the expected cumulative unused service for any feasible routing policy serving N
parallel queues under overloaded conditions.
This lower bound motivates us to study the performance of two well-known policies, namely Randomized Routing (RR) and Join-the-Shortest Queue (JSQ) policies,
with respect to this fundamental limit (cf. Section 3.3). It is easy to observe that
both RR and JSQ policies are departure-rate-optimal, in that, they both achieve
the maximum total departure rate. After utilizing the queue-reversal theorem once
again, we establish tight upper and lower bounds on the total mean cumulative unused service under the RR policy (cf. Proposition 3.3.1). This result reveals that the
cumulative unused service performance under the RR policy deviates significantly
from the lower-bound, and suggests that the RR policy is sub-optimal. We also note
40
that the JSQ policy minimizes the total cumulative unused service over all feasible policies under symmetric servers through a standard path-coupling argument (cf.
Proposition 3.3.2).
To compare the unused service performance of the JSQ policy to the lower bound
and the RR policy, we then perform numerical studies for both symmetric and asymmetric conditions (cf. Section 3.4). These investigations show that for both conditions, the expected cumulative unused service performance of the JSQ policy approaches the lower bound, suggesting its optimality both in the scaling of the network
size and the critically overloaded regime (where the total service rate approaches the
arrival rate from above). Moreover, these numerical results demonstrate that the
lower bound we derive through the queue-reversal theorem is indeed tight in these
two scaling regimes.
3.1
Problem Formulation
We consider the discrete-time parallel queueing system depicted in Figure 3.1. Packets
arrive according to an i.i.d. Bernoulli process1 {A[t]}t≥0 with mean rate of λ. Arriving
packets are routed to one of L infinite-size queues, which provides service according
to an i.i.d. Bernoulli process {Sl [t]}t≥0 with mean rate of µl , ∀l = 1, 2, ..., L.
(p)
In each slot t, a routing policy p routes Al [t] of the incoming packets to the lth
L
X
(p)
(p)
queue such that A[t] =
Al [t]. We let Ql [t] be the length of queue l in slot t
l=1
under policy p, whose evolution is given by
(p)
Ql [t
+ 1] = max
(p)
(p)
0, Ql [t]
(p)
+
(p)
Al [t]
− Sl [t]
(3.1.1)
(p)
= Ql [t] + Al [t] − Sl [t] + Ul [t],
1
In this work, we focus on the case of Bernoulli processes to simplify the exposition and analysis.
However, many of our results can be extended to more general processes.
41
Q1(p) [t]
A1(p) [t]
Ql(p) [t]
Al(p) [t]
A[t]
Router
QL(p) [t]
AL(p) [t]
S1[t]
Sl[t]
SL[t]
Figure 3.1: Routing to parallel queues.
(p)
(p)
(p)
where Ul [t] , max(0, Sl [t] − Ql [t] − Al [t]) denotes the amount of unused service
(p)
at the lth server in slot t. Accordingly, Sl [t]−Ul [t] denotes the number of departures
from queue l in slot t under policy p.
In this work, we are interested in the operation of the system in overloaded conL
X
µl > 0. An important metric of
ditions, i.e., when the overload rate , λ −
l=1
performance in such a scenario is the expected total cumulative unused services until
time T starting from zero initial state 2 :
"T −1
#
X (p)
E
UΣ [t] ,
(3.1.2)
t=0
under routing policy p, where
(p)
UΣ [t]
,
L
X
(p)
Ul [t].
l=1
2
Throughout this work, we assume (unless stated otherwise) the initial condition of the system to
be zero for all queues in order to capture the worst case cumulative unused service performance.
Yet, the results are extendable to non-zero initial conditions.
42
This metric can characterize the convergence speed of the running-average of the
expected departure rate by noting that
T −1
i
h (p) i
1X h
(p)
E Sl [t] − Ul [t]
E d [T ] ,
T t=0
"T −1
#
L
X
X (p)
1
=
UΣ [t] .
µl − E
T
t=0
l=1
(3.1.3)
This expression clearly shows that any policy p satisfying
"T −1
#
X
1
(p)
lim E
UΣ [t] = 0
T →∞ T
t=0
achieves the maximum departure rate of
L
X
µl that the system can provide. Yet,
l=1
many policies, including randomized policy (see discussion
"T −1 following
# Definition 3.1.2),
X (p)
can possess this limiting behavior. The study of E
UΣ [t] is important in ext=0
tracting additional critical information about the convergence speed of the runningaverage of the expected departure rate to its limit. This motivates us to investigate
the expected cumulative unused service performance of routing over parallel queues.
In addition to fundamental bounds for all feasible policies, in this work, we study
the performance of two well-known routing policies: Join-the-Shortest-Queue (JSQ)
policy ([19]), and Randomized Routing (RR) policy, described next.
Algorithm 3.1.1 (JSQ policy). In each time slot, the Join-the-Shortest-Queue (JSQ)
policy forwards all incoming packets to the queue with the shortest queue-length in
that time-slot. In case of ties, it selects a queue uniformly at random among the
queues with the shortest queue-length.
The JSQ policy has been shown to possess many desirable properties under stable
L
X
conditions (i.e., λ <
µl ), such as throughput-optimality, and mean delay optil=1
mality in heavy-traffic regimes, i.e., it minimizes the mean delay as the arrival rate
43
approaches to the total sum of the service rates ([15]). We are interested in investigating whether the JSQ policy also performs well in terms of the metric in (3.1.2) in
the over-loaded regime.
Another simple routing policy is the Randomized Routing (RR) policy, which is
defined as follows:
Algorithm 3.1.2 (RR policy). In each time slot, the Randomized Routing (RR)
µl
policy forwards all incoming packets to the queue l with probability ql , PL
,
i=1 µi
∀l = 1, 2, ..., L, i.e., in proportion to its service rate.
We note that the RR policy requires statistical information about the service rates,
while the JSQ policy needs queue-length information in each slot. It is easy to see
that both the JSQ and RR policies can stabilize the system under stable conditions.
Under overloaded conditions, the system is unstable under both policies.Yet, it can
also be seen that all queues will overflow under both policies, and the departure rate
X
µl under each policy.
expression in (3.1.3) will converge to the optimal level of
l
Thus, in the departure rate sense, both JSQ and RR policies are optimal. Yet, their
performance in the new metric (3.1.2) will be significantly different, as we will observe.
Since the system is necessarily unstable in overloaded conditions, the traditional
steady-state analysis does not apply, making it difficult to analyze the unused service
metric. In the next section, we tackle this challenge by proposing a queue reversal
theorem. This fundamental theorem seamlessly relates the unstable system to a
related stable system, and helps us develop a lower bound on the expected cumulative
unused service for any feasible routing policy.
44
3.2
Lower Bound Analysis
In this section, we establish a fundamental lower bound on the expected cumulative
unused service (3.1.2) in the over-loaded system of Figure 3.1. To derive the lower
bound, we first establish an interesting and surprising relationship between a singleserver queue with i.i.d. arrivals and i.i.d. services, and a reverse queue in which the
roles of the arrival and service processes are interchanged.
3.2.1
Queue Reversal Theorem
We assume that the arrival and service processes to a single-server queue are two
independent sequences of i.i.d. nonnegative-valued random variables3 {α[t]}t≥0 and
{β[t]}t≥0 . Let Φ[t] be its queue-length in slot t, which evolves as
Φ[t + 1] = Φ[t] + α[t] − β[t] + γ[t],
t ≥ 0,
(3.2.1)
where γ[t] , max{0, β[t] − Φ[t] − α[t]} denotes the amount of unused service in slot
t.
Consider a hypothetical reversal of the single-server queue by exchanging the
arrival and service processes. Let Φ(r) [t] be the queue-length of the reverse queue in
slot t. Then, the evolution of Φ(r) [t] can be described as
Φ(r) [t + 1] = β[t] − α[t] + γ (r) [t],
t ≥ 0,
(3.2.2)
where γ (r) [t] , max{0, α[t] − Φ(r) [t] − β[t]} denotes the amount of unused service of
the reverse queue in slot t. The single-server queue (also named the forward queue)
and its reverse queue are shown in Figure 3.2.
We now provide a key relationship between the forward and reverse queues.
3
We note that these random variables need not be Bernoulli distributed as in the original system. This generality is necessary to apply the result to the multi-server system in the following
subsection.
45
Į[t]
ĭ[t]
ȕ[t]
ĭ(r)[t]
ȕ[t]
(a)
Į[t]
(b)
Figure 3.2: (a) Forward queue; (b) Reverse queue
Theorem 3.2.1 (Queue Reversal Theorem). Suppose the forward and reverse
queues introduced above (cf. Fig. 2) start from zero, i.e., Φ[0] = Φ(r) [0] = 0. Then,
for any t ≥ 0, the total expected unused service until time t in the forward queue is
equal to the expected queue-length at time t of the reverse queue. Similarly, for any
t ≥ 0, the expected queue-length at time t of the forward queue is equal to the total
expected unused service until time t in the reverse queue.
In other words, given that Φ[0] = Φ(r) [0] = 0, we have
E[Φ[t]] =
t−1
X
E[γ (r) [τ ]],
τ =0
t−1
X
E[Φ(r) [t]] =
E[γ[τ ]],
∀t ≥ 1
(3.2.3)
∀t ≥ 1.
(3.2.4)
τ =0
Proof. Let X[τ ] , α[τ ] − β[τ ], ∀τ ≥ 0. Then, we have
"
( t−1
)#
X
(a)
E[Φ[t]] = E max
X[τ ], 0
0≤m≤t−1
(b)
=
(c)
=
E
"
t
X
1
k=1
max
0≤m≤t−1
k
"
(τ =m
m
X
E max
)#
X[τ ], 0
τ =0
( k−1
X
τ =0
)#
X[τ ], 0
,
(3.2.5)
where step: (a) follows from the Lindley’s equation; (b) uses the fact that (X[0], X[1], ..., X[t−
46
1]) and (X[t − 1], ..., X[1], X[0]) have the same distribution since {X[τ ]}τ ≥0 are i.i.d.;
(c) follows from the Spitzer’s Identity (see [82]). Similarly, we can show that
"
( k−1
)#
t
X
X
1
E[Φ(r) [t]] =
E max −
X[τ ], 0
.
(3.2.6)
k
τ =0
k=1
By using the identity max{x, y} = x + y − min{x, y}, we have
" k−1
( k−1
)#
t
X
X
X
1
E
X[τ ] − min
X[τ ], 0
E[Φ[t]] =
k
τ =0
τ =0
k=1
"
( k−1
)#
t−1
t
X
X
1
(a) X
=
E[X[τ ]] +
E max −
X[τ ], 0
k
τ =0
τ
=0
k=1
(b)
=
t−1
X
E[X[τ ]] + E[Φ(r) [t]],
(3.2.7)
τ =0
where (a) follows from the fact that X[τ ], τ ≥ 0 are i.i.d.; and (b) utilizes equation
(3.2.6). By summing (3.2.1) over τ = 0, 1, ..., t − 1 and taking the expectation on
both sides, we have
E[Φ[t]] =
t−1
X
E[X[τ ]] +
τ =0
t−1
X
E[γ[τ ]].
(3.2.8)
τ =0
By comparing the equations (3.2.7) and (3.2.8), we have
E[Φ(r) [t]] =
t−1
X
E[γ[τ ]],
(3.2.9)
τ =0
which proves (3.2.3). The proof of (3.2.4) follows the same steps, but with the roles
of forward and reverse queues switched.
We note that this result is a bit surprising, since the cumulative unused service
is non-decreasing in each sample path while the queue-length in the reverse queue
may increase, decrease, or stay the same over time. Yet, their means are remains
equal for all t. The key contribution of the Queue Reversal Theorem is that it relates
the metric of interest in an overloaded (forward) queue (i.e., the mean arrival rate
is strictly greater than the mean service rate) to the queue-length in a under-loaded
47
(reverse) queue, for which we have mature and rich analytical tools. In fact, this
theorem helps us establish a lower bound on the expected cumulative unused service
under any feasible routing policy.
3.2.2
A Lower Bound on the Cumulative Unused Service
We are ready to establish the necessary lower bound on the unused service in the
over-loaded system of Figure 3.1. We construct a hypothetical single-server queue
illustrated in Figure 3.3 with an infinite buffer, the same arrival process {A[t]}t≥0 as
L
X
before, and the service process {SΣ [t]}t≥0 , where SΣ [t] ,
Sl [t] is the total services
l=1
of the original multi-server system. Thus, the single-server queueing system stores all
A[t]
Figure 3.3: Lower bounding system.
arrivals in a single queue for service at the combined service amount of its multi-server
counterpart. Then, the queue-length process, {Ψ[t]}t≥0 , of the new system evolves
as:
Ψ[t + 1] = max (0, Ψ[t] + A[t] − SΣ [t])
= Ψ[t] + A[t] − SΣ [t] + W [t],
(3.2.10)
where W [t] , max(0, SΣ [t] − A[t] − Ψ[t]) denotes the amount of unused service in slot
t offered by the combined (also called resource pooled ) server. The following lemma
48
establishes the stochastic dominance relationship4 between the original multi-server
system and this single-server system.
Lemma 3.2.2. For any initial queue-length vector Q[0] = (Ql [0])Ll=1 , set Ψ[0] =
L
X
Ql [0]. Then, under any feasible (possibly non-causal) routing policy p, the queuel=1
length process {Q(p) [t]}t≥0 and the unused service process {U(p) [t]}t≥0 satisfies (i)
L
t
t
X
X
X
(p)
(p)
(p)
Ψ[t] 4st QΣ [t] ,
Ql [t], for all t ≥ 0, and (ii)
W [τ ] 4st
UΣ [τ ], for all
t ≥ 0.
τ =0
l=1
That is, Ψ[t] and
t
X
τ =0
W [τ ] are stochastically dominated by the total queue-length
τ =0
and the total unused services of the L-queue system of Figure 3.1.
Proof. This follows from coupling the sample paths of queue-length process in the
original system with that of the lower bounding system.
Next, we utilize the result (ii) of Lemma 3.2.2 that the total unused service of the
original multi-server system of Figure 3.1 is lower-bounded by the cumulative unused
service in the single-server system of Figure 3.3, and then apply the Queue Reversal
Theorem to get a lower bound on the expected cumulative unused service. Note that
our lower bound holds for more general arrival and service processes.
Assumption 3.2.3 (Basic Assumptions). We assume that: (i) The service processes
{(Sn [t])n }t≥0 are i.i.d. sequences of non-negative integer-valued and bounded random
variables with P(Sl [1] ≤ Smax ) = 1, for each l. We use the notations: µl , E[Sl [1]],
L
X
2
σl , var(Sl [1]), and µ ,
µl . (ii) The arrival process {A[t]}t≥0 is an i.i.d. sequence
l=1
of non-negative integer-valued and bounded random variables with: E[A[1]] = µ + ,
and P(A[1] ≤ Amax ) = 1, where Amax is independent of . We use the notations:
λ , E[A[1]], and (σ () )2 , var(A[1]).
4
A random variable X is said to be stochastically dominated by another random variable Y, denoted
as X 4st Y , if FY (z) ≤ FX (z) for all z ∈ R.
49
Proposition 3.2.4. Suppose Assumption 3.2.3 hold, and that the system starts from
zero initial state Q[0] = 0. Then, the unused service process {U(π) [t]}t≥0 achieved by
any feasible routing policy π satisfies
"∞
#
X (π)
ζ () Amax
()
−
,
E
UΣ [t] ≥ b1 ,
2
2
t=0
where ζ () , σ ()
↓ 0, then,
2
+
L
X
(3.2.11)
σl2 + 2 . Further, if (σ () )2 converges to a constant σ 2 as
l=1
lim E
↓0
"
∞
X
t=0
(π)
#
UΣ [t] ≥
1
2
σ2 +
L
X
l=1
σl2
!
.
(3.2.12)
Proof. The proof directly follows from Queue Reversal Theorem and Lemma 4 in
[15].
Based on the Proposition 3.2.4, we can easily calculate the lower bound for the
system with Bernoulli arrivals and symmetric Bernoulli services:
#
"∞
X (p)
λ2
1
UΣ [t] ≥ λ −
lim E
1+
.
↓0
2
L
t=0
(3.2.13)
We remark that this proposition not only shows that the total cumulative unused
service over time for the unstable system is lower-bounded, but also explicitly relates
the bound to the degree of over-load and the number of servers L.
3.3
Overload Analysis of RR and JSQ policies
In this section, we analyze the performance of the two well-known policies: the RR
policy and the JSQ policy under the Bernoulli arrivals and services. We calculate
lower and upper bounds on the expected cumulative unused service under the RR
policy. The upper and lower bounds match as scales down to zero. Then, we show
that the JSQ policy minimizes the cumulative unused service in the stochastic order
sense under symmetric servers.
50
3.3.1
Lower and Upper Bounds under the RR Policy
In this subsection, we provide lower and upper bounds on the expected cumulative
unused services under the RR policy in the system with Bernoulli arrivals and services.
Recall that λ is the arrival rate, µl is the service rate of the lth queue, and λ =
L
X
µl + , where > 0. Then, we have the following result.
l=1
Proposition 3.3.1. Assume the system starts from the zero initial state Q[0] = 0.
Then, the unused service process {U(RR) [t]}t≥0 under the RR policy satisfies
"∞
#
X
(RR)
(RR)
(RR)
bLB ≤ E
UΣ [t] ≤ bU B ,
(3.3.1)
t=0
ζ1
ζ1 L
(RR)
− , and bU B , , and also ζ1 , λ (L − λ)+(λ − ) (L − λ + )+
2 2
2
2
. This implies
#
"∞
X (RR)
(3.3.2)
lim E
UΣ [t] = λ(L − λ).
(RR)
where bLB ,
↓0
t=0
Proof. Under the RR policy, the lth queue is equivalent to having the Bernoulli arrivals
µl
with the mean of λ PL
and the Bernoulli services with the mean of µl . Then, we
i=1 µi
apply the Queue Reversal Theorem to each individual queue and utilize Lemma 4 in
[15] to get desired results.
"∞
#
X (RR)
Since E
UΣ [t] < ∞ for any > 0, the running-average departure rate
t=0
"∞
#
X (RR)
can converge to its service rate. Yet, for each > 0, E
UΣ [t] linearly
t=0
increases with the number of queues L, which implies that the speed at which the
running-average departure rate converges under the RR policy scales down linearly
with the number of queues, which is undesirable in large-scale networks. In contrast,
the limiting behavior of the lower bound in (3.2.13) is inversely related to L, which
suggests that there is a potential for significant performance improvement over the
RR policy in large scale systems. This motivates us to study the expected cumulative
51
unused services under the JSQ policy, which possesses many good properties in underloaded regimes.
3.3.2
Optimality of the JSQ Policy in Symmetric Conditions
In this subsection, we study the problem of optimal routing policy with respect to
cumulative unused service under symmetric Bernoulli service processes. We show
that the JSQ policy minimizes the process of the total cumulative unused service in
the stochastic ordering sense.
Proposition 3.3.2. For the parallel queueing system with Bernoulli arrivals and
symmetric Bernoulli services in Figure 3.1, let U(JSQ) [t] t≥0 and U(p) [t] t≥0 be
the unused service processes under the JSQ policy and any feasible routing policy p,
respectively. Then,
t
X
(JSQ)
UΣ
[τ ] 4st
t
X
(p)
UΣ [τ ].
(3.3.3)
τ =0
τ =0
Proof. This follows from coupling the queue-length realizations under JSQ and policy
p appropriately. The proof is almost the same as in [92], and thus is omitted here for
brevity.
Even though the JSQ policy is optimal in terms of the cumulative unused service,
its closeness to the lower bound we derived in Section 3.2 is unclear. In the next
section, we demonstrate with numerical investigations that the expected cumulative
unused service under the JSQ policy, for both symmetric and asymmetric service
processes, matches the lower bound as scales down to zero, and more importantly,
is independent of the number of queues when is sufficiently small.
52
3.4
Simulation Results
In this section, we provide simulations to compare the expected cumulative unused
service performance of the JSQ policy with the RR policy. In the simulation, we take
λ = 0.8. We consider both symmetric and non-symmetric Bernoulli service processes.
In the symmetric setup, the service rate for the lth queue is µl = (λ − )/L, while in
the non-symmetric case, the service rate for lth queue is µl = 2l(λ − )/(L(L + 1)).
X
We study the impact of the overload level = λ −
µl > 0 and the number of
l
queues L on the expected cumulative unused service.
3.4.1
The Impact of Overload Level on Mean Unused Services
Figure 3.4 shows the impact of on the expected cumulative unused service of JSQ
and RR policies when there are L = 5 queues. From Figure 3.4, we can observe
that the expected cumulative unused service under the JSQ policy converges to the
theoretical lower bound as scales down to zero, while the RR policy always keeps
away from the theoretical lower bound. Thus, we conjecture that the JSQ policy
is overload-optimal, i.e., it minimizes the expected cumulative unused service as diminishes, while the RR policy is sub-optimal. We leave the proof of this conjecture
to future investigation.
3.4.2
The Impact of Server Number L on Mean Unused Service
In this subsection, we study the impact of the number of queues on the expected
cumulative unused service under the JSQ policy and the RR policy. Here, we fix to
0.005, and vary the number of queues from 5 to 25. From Figure 3.5, we can observe
that the expected cumulative unused service under the JSQ policy stays close to the
theoretical lower bound as the number of queues L increases, while it scales linearly
with L under the RR policy, as derived in Proposition 3.3.1. This indicates that the
53
3.5
3.5
3
JSQ
Theoretical lower bound of RR
Theoretical upper bound of RR
Theoretical lower bound
2.5
t=0 U Σ [t]]
2.5
2
P∞
2
P∞
t=0 U Σ [t]]
3
JSQ
Theoretical lower bound of RR
Theoretical upper bound of RR
Theoretical lower bound
ǫE [
1.5
ǫE [
1.5
1
1
0.5
0.5
0
−1
0
−2
10
ǫ
10
−1
−2
10
(a) Symmetric services
ǫ
10
(b) Asymmetric services
Figure 3.4: The impact of on the expected cumulative unused service.
20
20
18
t=0 U Σ [t]]
14
16
12
8
ǫE [
ǫE [
14
12
10
P∞
10
P∞
t=0 U Σ [t]]
16
8
6
6
4
4
2
2
0
5
JSQ
Theoretical lower bound of RR
Theoretical upper bound of RR
Theoretical lower bound
18
JSQ
Theoretical lower bound of RR
Theoretical upper bound of RR
Theoretical lower bound
10
15
20
0
25
N
5
10
15
20
N
(a) Symmetric Services
(b) Asymmetric Services
Figure 3.5: The impact of L on the expected cumulative unused service.
54
25
performance of the JSQ policy is insensitive to the network size, which is desirable in
large-scale networks.
3.5
Summary
We considered a queueing system in which Bernoulli arrivals are routed to parallel
servers with independent Bernoulli service processes. We studied the system performance in the overloaded regime, i.e., the total arrival rate is greater than the sum of
the service rates of the queues. We proposed the use of cumulative unused service as
a key metric in overloaded conditions, and provided a fundamental lower bound on it
for any feasible routing policy. In the process of deriving the lower bound, we found a
surprising result, which may be interesting in its own right: the expected cumulative
unused service in a single-server queue with i.i.d. arrivals and i.i.d. services is equal
to the expected queue-length in a queueing system where the roles of arrival and service processes are exchanged. Then, we compared the derived lower bound with the
performance of two well-known routing policies, namely randomized and join-theshortest queue, to observe their optimality and sub-optimality characteristics with
respect to the new metric.
55
CHAPTER 4
RESOURCE ALLOCATION ALGORITHM FOR
ACHIEVING MAXIMUM THROUGHPUT,
HEAVY-TRAFFIC OPTIMALITY, AND SERVICE
REGULARITY GUARANTEE
After understanding the convergence speed of efficient algorithms in both underloaded
and overloaded networked systems, we turn our attention to the algorithm design
for achieving maximum throughput, minimum delay and best service regularity –
critical for real-time applications. As we discussed in Chapter 1, service regularity
is extremely important in real-time applications. For example, the QoS of users in
video application is highly related to the average Perceived Video Quality (PVQ)
across the sequence of scenes forming the video, where the PVQ traditionally is a
local quality measure associated with a particular scene or a short period of time.
In [96, 36], the authors point out that the variance in PVQ leads to the worse QoS
than the constant quality video with even smaller average PVQ. Intuitively, both the
time-varying nature of wireless channels and the scheduling policy significantly affect
the variance of the PVQ. The traditional throughput-based schedulers can guarantee
the rate requirements (e.g., [91, 16, 54, 69, 56]). However, the service received by the
user under these schemes may suffer from large variations in the inter-service times,
and hence disrupt the regularity of service they demand.
56
This motivates us to develop scheduling policies that not only maximize system
throughput and minimize mean delay but also provide regulated inter-service times for
links with heterogeneous arrival processes over wireless fading channels. However, the
inter-service time characteristics are difficult to analyze directly due to: its complex
dependence on the high-order statistics of the arrival and service processes, and its
non-Markovian evolution.
In [52], we introduce a new quantity, namely the time-since-last-service, that has
a tight relationship with the service regularity performance, and hence enables novel
design strategies. Then, we develop a novel maximum-weight type scheduling policy
that combines the time-since-last-service parameter and the queue-length in its weight
measure, and show that the proposed scheduling policy not only achieves maximum
throughput but also provides regular service guarantees in single-hop networks, where
at most one link can be scheduled in each time slot.
In this chapter, we extend our earlier work [52] in several key aspects: (i) we conduct novel analyses that extend both throughput optimality and service regularity
guarantee results to general multi-hop fading networks; (ii) we show the existence of
all moments of the system state under our proposed algorithm, which establishes the
foundation to utilize the Lyapunov-drift based analysis of the steady-state behavior
of stochastic processes; (iii) we also establish the mean delay optimality of our proposed algorithm in heavy-traffic regimes, where the delay is most pronounced. Our
contributions in this chapter can be summarized as follows:
• We show that the proposed scheduling policy possesses the desirable throughput
optimality property by using a novel Lyapunov function (cf. Section 4.2).
• We derive lower and upper bounds on the service regularity performance (cf. Section 4.3) by utilizing a novel Lyapunov-drift-based argument, inspired by the approach
in [15]. We further show that, by properly scaling the design parameter in our policy,
57
we can guarantee a degree of service regularity within a factor of our fundamental
lower bound. This factor is a function of the system statistics and design parameters
and can be as low as two under symmetric arrival rates in some special networks.
• We reveal that our proposed algorithm explores the tradeoff between the service
regularity performance and mean delay (cf. Section 4.4).
• We show that the proposed algorithm can minimize the total mean queue-length to
establish mean delay optimality under heavily-loaded conditions as long as the design
parameter scales slowly with the network load (cf. Section 4.5).
• We support our analytical results with extensive numerical investigations (cf. Section 4.6), which show significant performance gains in the service regularity over the
traditional queue-length-based policies. Furthermore, the numerical investigations
indicate that the service regularity performance of our policy actually approaches the
lower bounds as the weight of the time-since-last-service increases in some special
networks.
4.1
Problem Formulation
In this chapter, we are interested in providing regular service for each link, which
relates to the statistics of the inter-service time. We use Il [m] to denote the time
between the (m − 1)th and the mth service for link l. If the system is stable, the
steady-state distribution of the underlying Markov Chain exists (see [66]) and thus
we use Q = (Ql )Ll=1 , S = (S l )Ll=1 and I = (I l )Ll=1 to denote the random vector
with the same steady-state distribution of the queue-length, service processes and
inter-service time, respectively. We use the normalized second moment of the inter2
service time under the steady-state distribution, i.e., E[I l ]/(E[I l ])2 , as a measure
2
of the “regularity” of the service that link l receives. Noting that E[I l ]/(E[I l ])2 =
Var(I l )/(E[I l ])2 + 1, the normalized second moment of the inter-service time reflects
58
its normalized variance. Hence, the smaller the normalized second moment of the
inter-service time, the smaller its normalized variance and thus the received service
is more regular.
We would like to develop throughput-optimal policies that achieve low values of a
2
linear increasing function of (E[I l ]/(E[I l ])2 )Ll=1 in steady-state, implying more regular
service. However, unlike queue-lengths with Markovian evolution, the dynamics of
inter-service times do not lend themselves to commonly used Markovian analysis
methods. To overcome this obstacle, we introduce the following related quantity,
namely the time-since-last-service, which has much more tractable form of evolution,
and whose mean has a close relationship to the normalized second moment of the
inter-service time.
For each link l, we introduce a counter Tl , namely Time-Since-Last-Service (TSLS),
to keep track of the time since it was lastly served. In particular, each counter Tl
increases by 1 in each time slot when link l has zero transmission rate, either because
it is not scheduled, or because its channel is unavailable, i.e., Cl [t] = 0, and drops to
0, otherwise. More precisely, the evolution of the counter Tl can be written as


 0
if Sl [t]Cl [t] > 0;
(4.1.1)
Tl [t + 1] =

 Tl [t] + 1 if Sl [t]Cl [t] = 0.
It can be seen from (4.1.1) that the evolution of Tl [t] differs significantly from
that of a traditional queue. In particular, unlike the slowly evolving nature of queuelengths, the Tl [t] is incremented until link l receives service at which time it drops
to zero. In our design, we will consider policies that not only use queue-lengths to
achieve throughput-optimality, but also include TSLS to improve service regularity.
In [52], we show the following lemma relating the inter-service time and TSLS in
steady-state.
59
Lemma 4.1.1. For any policy under which the steady-state distribution of the underlying Markov Chain exists, we have
1
E[T l ] =
2
1
2
E[I l ] − 1 ,
E[I l ]
(4.1.2)
where T l and I l denote the steady-state TLSL and inter-service time at link l, respectively.
Lemma 4.1.1 reveals the connection between the second moment of the interservice time I l and the mean of TLSL T l in steady-state. In this chapter, we are
interested in designing algorithms that reduce the total weighted-sum of the norL
X
2
2
βl ρl E[I l ]/ E[I l ] , where
malized second moment of the inter-service time, i.e.,
l=1
µl , 1/E[Il ], ρl , λl /µl , and βl is some parameter related to the link preference.
According to Lemma 4.1.1, we have
L
X
l=1
Since
L
X
2
X
X
E[I l ]
βl λl .
βl ρ l
βl λl E[T l ] +
2 = 2
E[I l ]
l=1
l=1
L
L
βl λl only depends on the system parameters, we will use
(4.1.3)
L
X
βl λl E[T l ]
l=1
l=1
as our measure for the service regularity. In this work, we aim to design a scheduling
policy that achieves maximum throughput, minimum delay and best service regularity.
We achieve this triple objective by developing a parametric class of throughputoptimal schedulers (cf. Section 4.2) that utilize a combination of queue-lengths and
TSLS in its decisions. Our policy is shown to guarantee a ratio (as a function of the
system statistics) in its service regularity with respect to a fundamental lower bound
(cf. Section 4.3.1).
60
4.2
Regular Service Scheduler
As discusssed above, the introduced TLSL counter has a direct impact on service
regularity: the smaller the mean TLSL value, the more regular the service. This
interesting connection motivates the following parametrized policy which is later revealed to possess the characteristics of throughput optimality and service regularity.
Algorithm 4.2.1 (Regular Service Guarantee (RSG) Algorithm). In each time slot
t, select a schedule S∗ [t] such that
∗
S [t] ∈ argmax
S∈S
L
X
(αl Ql [t] + γβl Tl [t])Cl [t]Sl ,
(4.2.1)
l=1
where αl > 0 and γ ≥ 0 are fixed control parameters.
We note that there are two sets of control parameters in the RSG Algorithm and
they affect different behaviors of the algorithm. Yet, it will be revealed later that none
of them affects its throughput optimality. The parameters αl are weighing factors for
the queue-lengths, where a larger αl will result in a smaller average queue-length. The
parameters βl weigh Tl [t] differently for each link l, with γ being a common scaling
factor for all links. Noting that βl ≥ 0, ∀l, we can set βl > 0 if link l prefers regular
service and βl = 0 otherwise. It will be revealed in Section 4.3.2 that the design
parameter γ can improve the service regularity as it increases. Also note that when
γ = 0, our policy coincides with the Maximum Weight Scheduling (MWS) Algorithm.
When γ > 0, with the addition of Tl [t] terms in the weight of each link, our algorithm
operates completely different from the MWS and its approximate algorithms, which,
to the best of our knowledge, are the only known policies possessing the throughputoptimality characteristic in general multi-hop network topologies. Despite of this, we
can still show that our algorithm is throughput-optimal.
61
Proposition 4.2.2. The RSG Algorithm with any αl > 0, βl ≥ 0 and γ ≥ 0, is
throughput-optimal, i.e., for any arrival rate vector λ ∈ Int(R), the RSG Algorithm
stabilizes the system, with
lim sup
K→∞
K−1 L
1 XX
B(α, β, γ)
,
αl E[Ql [t]] ≤
K t=0 l=1
2
where Int(A) denotes the interior points of the region A, B(α, β, γ) , 4γCmax
(4.2.2)
L
X
βl +
l=1
L
X
l=1
αl E A2l [t] + Cl2 [t] , is some positive constant satisfying λ + 1 ∈ R, and 1 is a
vector of ones.
Proof. Consider the Lyapunov function
W (Q[t], T[t]) ,
L
X
αl Q2l [t]
+ 4γCmax
l=1
L
X
βl Tl [t].
(4.2.3)
l=1
It is shown in Appendix B.1 that there exists a positive constant > 0 such that
∆W , E [W (Q[t + 1], T[t + 1]) − W (Q[t], T[t])|Q[t], T[t]]
L
X
≤ −2
αl Ql [t] + B(α, β, γ).
(4.2.4)
l=1
Taking the expectation on the both sides of (4.2.4) and summing over t = 0, 1, ..., K −
1, we have the desired result.
Proposition 4.2.2 establishes the throughput optimality of the RSG Algorithm,
∗
∗
thus Ql [t] and Tl [t] will converge in distribution to Ql and T l , which attain the steadystate distribution under our policy. Proposition 4.2.2 also gives an upper bound for
the expected total queue-length under the steady-state, which increases linearly with
the design parameter γ. It will be revealed later that γ controls the tradeoff between
the average total queue-length, and the service regularity performance, especially in
the heterogenous networks.
62
Next, we will show that all moments of steady-state system variables, such as
queue-lengths and TLSL, are bounded under the RSG Algorithm, which enables us
to analyze the service regularity performance by using the Lyapunov-type approach
developed in [15]. In [26], the sufficient condition for all moments of state variables
of a Markov Chain to exist in steady state is given as finding a Lyapunov function
that satisfies: (1) it has a negative Lyapunov drift when the system variable is large
enough; (2) the absolute value of the Lyapunov drift is bounded or has the exponential
tail. Yet, the second condition is hard to hold due to the unique evolution of TLSL
counters, which have bounded increment but unbounded decrement. We tackle this
challenge by properly partitioning the system space.
Proposition 4.2.3. For any arrival rate λ ∈ Int(R), all moments of steady-state
queue length and TLSL exist under the RSG Algorithm with any αl > 0, βl ≥ 0 and
γ > 0.
Proof. We show the boundedness of E eηkY[t]k2 for some η > 0 by intelligently
√
√
p
αQ[t], 4γCmax βT[t] , x denotes
partitioning the system space, where Y[t] ,
the component-wise square root of the vector x, and xy denotes the component-wise
product of the vectors x and y. Please see Appendix B.2 for details.
Having established the throughput optimality and the moment existence of the
system states of the RSG Algorithm, we are ready to analyze the service regularity
L
X
performance, i.e.,
βl λl E[T l ].
l=1
4.3
Service Regularity Performance Analysis
In this section, we study the service regularity performance of our proposed RSG
Algorithm analytically. We first establish a fundamental lower bound on the service
regularity for any feasible scheduling algorithm. Then, we derive an upper bound on
63
the service regularity under the RSG Algorithm. These investigations reveal that the
service regularity performance of the RSG Algorithm can be guaranteed to remain
within a factor of the lower bound, which is expressed as a function of the system
statistics and the design parameters, and can be as low as 2 in some special networks.
We assume the parameter γ > 0 throughout this section.
4.3.1
Lower Bound Analysis
In this subsection, we derive a lower bound based on a Lyapunov drift argument
inspired by the technique used in [15]. To study the lower bound of the service
regularity by the Lyapunov drift argument, we consider a class of policies, called P,
that not only stabilize the system but also yield the bounded second moment of the
steady-state TSLS1 Note that our proposed algorithm, as well as the MWS algorithm,
falls into this class by Propositions 4.2.2 and 4.2.3.
(p)
Let T l
(p)
and S l
be the steady-state TLSL and scheduling variable for link l
under policy p, respectively. The following lemma gives key identities for the first
and second moment of the steady-state TSLS, which are useful in deriving a lower
bound on the service regularity.
Lemma 4.3.1. For any policy p ∈ P, we have




L
X
X
X
(p)
E
βl λl T l  =
βl λl − E 
βl λl  ,
l=1
(p)
l∈H
2
L
X
l=1
where H
(p)
(4.3.1)
(p)
l∈H



L
h (p) i
X
X
X
(p) 2
(4.3.2)
βl λl E T l
=
βl λl − E 
βl λl  + E 
βl λl T l
,
l=1
(p)
, {l : C l S l

(p)
l∈H
(p)
l∈H
> 0}, and C = (C l )Ll=1 has the same probability distribution
as C[t] = (Cl [t])Ll=1 .
1
We conjecture that the second moment of the steady-state TSLS is bounded as long as the system
is stable.
64
Proof. See Appendix B.6 for the proof.
We are ready to give a lower bound on the service regularity for any feasible policy
p ∈ P.
Proposition 4.3.2. For any policy p ∈ P, we have
L
X
l=1
h (p) i 1
βl λl E T l ≥
2
PL
βλ
l=1
Pl l
maxS∈S
l∈S
βl λl
!
−1
L
X
βl λl .
l=1
Proof. In the rest of proof, we will omit superscript p for conciseness. For any sample
path, by Cauchy-Schwarz inequality, we have
2
2 



Xp
X
X
X
p
2

βl λl · βl λl T l  ≤ 
βl λl T l  = 
βl λl 
βl λl T l , (4.3.3)
l∈H
l∈H
l∈H
l∈H
where we recall that H , {l : C l S l > 0}. This implies
X
2
βl λl T l
l∈H
Hence, we have
2
P
l∈H βl λl T l
P
.
≥
l∈H βl λl


" P
2 #
X
β
λ
T
2
l
l
l
l∈H
P
E
βl λl T l  ≥ E
l∈H βl λl
l∈H
P
2
(a)
E
l∈H βl λl T l
P
≥
E
l∈H βl λl
P
P
2
L
β
λ
β
λ
−
E
l=1 l l
l∈H l l
(b)
P
=
,
E
β
λ
l
l
l∈H
(4.3.4)
(4.3.5)
x2
is convex and Jensen’s inequality
y
for a multi-variable function; step (b) follows from (4.3.1). By substituting (4.3.5)
where the step (a) uses the fact that f (x, y) =
into (4.3.2), we have
L
X
l=1
1
βl λl E T l ≥
2
! L
PL
X
β
λ
l
l
Pl=1
−1
βl λl .
E
l∈H βl λl
l=1
65
(4.3.6)
Note that


" L
#
X
X
βl λl  = E
E
βl λl 1{C l S l >0}
l=1
l∈H
=
≤
L
X
l=1
L
X
βl λl Pr{C l S l > 0}
βl λl Pr{S l = 1} ≤ max
S∈S
l=1
X
βl λl .
(4.3.7)
l∈S
By substituting (4.3.7) into (4.3.6), we have the desired result.
Consider a single-hop non-fading network, where only one link is scheduled in
each time slot. Let βl = β and λl = λ for each link l. Then, the lower bound becomes
L
X
l=1
E
h
(p)
Tl
i
1
≥ L(L − 1).
2
(4.3.8)
This lower bound can be achieved by the Round-Robin (RR) policy, which serves
each link periodically. Thus, in the steady-state, the TSLS vector under the RR
L
h (RR) i 1
X
= L(L − 1).
E Tl
policy is a permutation of {0, 1, 2, ..., L − 1} and thus
2
l=1
Yet, we would like to point out that the RR policy is not throughput-optimal.
Thus, for an arrival rate vector λ that cannot be supported by the RR policy, we
do not expect a throughput-optimal policy to approach the above lower bound when
serving it. However, for the arrival rate vectors that can be supported by the RR
policy, we shall see in our numerical results that the performance of our policy can
approach this lower bound when we increase the scaling parameter γ.
4.3.2
Upper Bound Analysis
In this subsection, we obtain an upper bound on the service regularity under the RSG
∗
∗
∗
Algorithm. Let Ql , S l and T l be the steady-state queue-length, scheduling variable
and TSLS for link l under the RSG Algorithm, respectively.
66
Proposition 4.3.3. For the RSG Algorithm, we have



L
L
i
h
X
X
Cmax X
∗
βl 
βl λl E T l ≤
βl − E 
1
+
∗
l=1
l=1
l∈H
i
h 2
X
1
2
αl E Al + C l ,
2γ(1 + ) l=1
L
+
∗
(4.3.9)
∗
where > 0 satisfies λ(1 + ) ∈ R, H , {l : C l S l > 0}, and A = (Al )Ll=1 has the
same distribution as A[t] = (Al [t])Ll=1 .
Proof. See Appendix B.7 for the details.
Note that the second term of the right hand side of (4.3.9) captures various random effects in the network: the burstiness of the arrival processes and the channel
variations. Under our policy these effects diminish as the scaling factor γ goes to
infinity. Hence, together with Proposition 4.2.2, Proposition 4.3.3 reveals a tradeoff:
when increasing γ, the upper bound on the total queue-length increases linearly with
γ, but the upper bound for the service regularity decreases.
Consider the single-hop non-fading network as in Section 4.3.1. Let βl = β and
1
for each link l. Then, as γ goes to infinity, (4.3.9) becomes
λl = λ =
L(1 + )
L
X
l=1
h ∗i
E T l ≤ L(L − 1),
(4.3.10)
which is always within twice the value of the lower bound (4.3.8). In the more general
case, the upper bound converges to a constant that is determined by the system
statistics and design parameters as γ goes to infinity. Moreover, we shall see in the
numerical results presented in Section 4.6.2 that as γ increases, the service regularity
performance under the RSG Algorithm actually converges to the lower bound (4.3.8)
in the single-hop non-fading network with the symmetric parameters.
67
4.4
Tradeoff Between Mean Delay and Service Regularity
We have seen that the large values of γ improve the service regularity. Yet, it may
also deteriorate the mean delay performance. To see this, we consider a single-hop
non-fading network with 4 links, where the number of packets arriving at each link
follows a Bernoulli distribution with the arrival rate of 0.225. Let αl = βl = 1 for
each link l. Figure 4.1 shows the mean delay and service regularity performance of
the RSG Algorithm with varying γ.
12
Total mean queue length
Total mean TSLS
12
11
10
10
8
9
mean delay increases
6
8
service regularity improves
4
2
0
1
2
3
4
5
6
7
Total mean TSLS
Total mean queue length
14
7
8
9
6
10
γ: parameter of the RSG algorithm
Figure 4.1: Delay and service regularity performance of the RSG Algorithm
Figure 4.1 reveals that the improved service regularity of the RSG Algorithm with
increasing γ comes at the cost of larger mean delays. We can show that the mean of
the total TSLS value is minimized as γ goes to ∞ (see Section 4.3.3). On the other
hand, it is known (e.g. [88, 15]) that the mean queue-lengths are minimized under
68
heavily-loaded conditions (cf. Section 4.5 for more detail) when γ = 0. In view of the
tradeoff observed in the above figure, our objective is to understand whether both the
regularity and the mean-delay optimality characteristics of the RSG Algorithm can be
preserved, especially under heavily-loaded conditions, by carefully selecting γ.
In the next section, we answer this question in the affirmative by explicitly characterizing how γ should scale with respect to the traffic load in order to achieve the
heavy-traffic optimality while also optimizing the service regularity performance of
the RSG Algorithm.
4.5
Heavy-Traffic Optimality Analysis
In this section, we present our main result for the RSG Algorithm in terms of its
mean delay optimality under the heavy-traffic limit, where the arrival rate vector
approaches the boundary of the capacity region.
In the rest of this chapter, we consider the RSG Algorithm2 with αl = 1, ∀l.
We first note that the capacity region R is a polyhedron due to the discreteness and
2
√
√
√
0
0 (c)
Let A0l [t] ,
, { α · c · S : S ∈ S}
X αl Al [t] and(c)Cl [t] , αl Cl [t], ∀t ≥ 0, ∀l. Also let S
and R0 ,
ψc · CH{S 0 }. Construct a hypothetical system with the arrival process {A0 [t] =
c
0
0
L
(A0l [t])L
l=1 }t≥0 and the channel fading process {C [t] = (Cl [t])l=1 }t≥0 under the following RSG
Algorithm in the hypothetical system:
L X
βl 0
0
S [t] ∈ argmax
Ql [t] + γ √ Tl [t] Cl0 [t]Sl ,
αl
S∈S
0∗
l=1
where
Tl0
is the TSLS counter for link l in the hypothetical system and evolves as follows:
0
if Sl0 [t]Cl0 [t] > 0;
Tl0 [t + 1] =
0
Tl [t] + 1 if Sl0 [t]Cl0 [t] = 0.
0
0
L
Let {Q[t] = (Ql [t])L
l=1 }t≥0 and {Q [t] = (Ql [t])l=1 }t≥0 be the queue-length process under the
RSG Algorithm in the original system and the RSG Algorithm with
√ αl = 1 for each link l in the
hypothetical system, respectively. Then, it is easy to show that { α · Q[t]}t≥0 is stochastically
equal to {Q0 [t]}t≥0 . Thus, we can study the queue length behavior under the RSG Algorithm
with αl = 1 for each link l in the hypothetical system.
69
finiteness of the service rate choices, and thus has a finite number of faces. We consider
the exogenous arrival vector process {A() [t]}t≥0 with mean rate vector λ() ∈ Int(R),
where measures the Euclidean distance of λ() to the boundary of R (see Figure 4.2).
In heavy-traffic analysis, we study the system performance as decreases to zero,
i.e., as the arrival rate vector approaches λ(0) belonging to the relative interior of
a face, referred to as the dominant hyperplane H(d) . We denote H(d) , {r ∈ RL :
hr, di = b}, where b ∈ R, and d ∈ RL is the normal vector of the hyperplane H(d)
satisfying kdk = 1 and d 0.
r2
Line of attraction
λ(0)
ε
d
λ
(e )
H (d )
r1
Figure 4.2: Geometric structure of capacity region
We are interested in understanding the steady-state queue-length values with
vanishing . To that end, we first provide a generic lower bound for all feasible
schedulers by constructing a hypothetical single-server queue with the arrival process
hd, A() [t]i, and the i.i.d service process β[t] with the probability distribution
Pr {Ψ[t] = bc } = ψc ,
for each channel state c ∈ C,
70
(4.5.1)
where bc , max hd, c·si is the maximum d-weighted service rate achievable in channel
s∈S (c)
state c ∈ C. By the construction of capacity region R, we have E[Ψ[t]] = b. Also, it is
easy to show that the constructed single-server queue-length {Φ[t]}t≥0 is stochastically
smaller than the queue-length process {hd, Q() [t]i}t≥0 under any feasible scheduling
policy. Hence, by using Lemma 4 in [15], we have the following lower bound on the
expected limiting queue-length vector under any feasible scheduling policy.
Proposition 4.5.1. Let Q
()
be a random vector with the same distribution as the
steady-state distribution of the queue length processes under any feasible scheduling
2
policy. Consider the heavy-traffic limit ↓ 0, suppose that the variance vector σ ()
of the arrival process {A() [t]}t≥0 converges to a constant vector σ 2 . Then,
h
i
ζ
lim E hd, Q i ≥ ,
↓0
2
()
(4.5.2)
where ζ , hd2 , σ 2 i + Var(Ψ).
This fundamental lower bound of all feasible scheduling policies motivates the
following definition of heavy-traffic optimality of a scheduler.
Definition 4.5.2. (Heavy-Traffic Optimality) A scheduler is called heavy-traffic optimal, if its limiting queue length vector Q
()
satisfies
h
i ζ
()
lim E hd, Q i ≤ ,
↓0
2
(4.5.3)
where ζ is defined in Proposition 4.5.1.
It is well-known that the MWS Algorithm, which corresponds to the RSG Algorithm with γ = 0, is heavy-traffic optimal (e.g., [88, 15]). This is shown by first
establishing a state-space collapse, i.e., the deviations of queue lengths from the direction d are bounded, independent of heavy-traffic parameter . Since the lower
1
bound of mean queue length is of order of , the deviations from the direction d are
71
negligible compared to the large queue length for a sufficiently small , and thus the
queue lengths concentrate along the normal vector d. Because of this, we also call
the normal vector d the line of attraction.
However, as discussed in Section 4.3, we are interested in large values of γ to
provide satisfactory service regularity. Yet, it is unknown whether the RSG Algorithm
can remain heavy-traffic optimal when γ is non-zero, since larger values of γ leads
to higher mean queue-lengths (cf. Figure 4.1). Also, the state-space collapse result
is not applicable since the deviations from the line of attraction depend on γ. This
raises the question of how γ() should scale with in order to achieve heavy-traffic
optimality while allowing γ() to take large values (providing more regular services).
We answer this interesting and challenging question by providing the following main
result, proved in Appendix B.9.
Proposition 4.5.3. Let Q
()
be a random vector with the same distribution as the
steady-state distribution of the queue length processes under the RSG Algorithm. Con2
sider the heavy-traffic limit ↓ 0, suppose that the variance vector σ () of the
arrival process {A() [t]}t≥0 converges to a constant vector σ 2 . Suppose the channel
fading satisfies the mild assumption3 Pr{Cl [t] = 0} > 0, for all l ∈ L. Then,
i ζ ()
h
()
()
+B ,
E hd, Q i ≤
2
(4.5.4)
2
()
where ζ () , hd2 , σ () i + Var(Ψ) + 2 and B is defined in (B.9.15).
1
()
Further, if γ() = O( √
), then lim B = 0 and thus the RSG Algorithm is
5
↓0
heavy-traffic optimal.
This result is interesting in that it provides an explicit scaling regime in which the
design parameter γ() can be increased to utilize the service regulating nature of the
3
We note that our result holds in single-hop network topologies without this assumption, and its
extension to more general settings is part of our future work.
72
RSG Algorithm without sacrificing the heavy-traffic optimality. Intuitively, if γ()
scales slowly as vanishes, each link weight is dominated by its own queue length
in the heavy-traffic regime and thus the heavy-traffic optimality may be maintained;
otherwise, the heavy-traffic optimality result may not hold, as will be demonstrated
in the next section.
4.6
Simulation Results
In this section, we provide simulation results for our proposed RSG Algorithm and
compare its performance to the MWS Algorithm and bounds. In addition to investigating the throughput (cf. Section 4.6.1) and service regularity (cf. Section 4.6.2)
performances of our policy in both single-hop network with L = 4 links and 3 × 3
switch, we also look at heavy-traffic behavior (cf. Section 4.6.3) of the RSG Algorithm. In all simulations, we assume Bernoulli arrivals to each link and αl = βl = 1
for each link l.
4.6.1
Throughput Performance
In this subsection, we illustrate the throughput performance of the RSG Algorithm
in three different network setups with symmetric arrivals: (i) single-hop non-fading
network, (ii) single-hop network with symmetric ON-OFF fading channels with probability q = 0.8 that the channel is available, and (iii) 3 × 3 switch. The capacity
regions for these three networks, respectively, are
R1 ,
R2 ,
R3 ,
λ=
(λl )4l=1
λ = (λl )4l=1
λ = (λl )9l=1
1
,
: λ1 = λ2 = ... = λ4 <
4
1 − (1 − q)4
: λ1 = λ2 = ... = λ4 <
,
4
1
: λ1 = λ2 = ... = λ9 <
.
3
73
2000
1800
1600
1000
MWS Algorithm
RSG Algorithm with γ=10
RSG Algorithm with γ=100
900
800
l=1 E[Ql ]
700
1200
600
500
800
PL
1000
PL
l=1 E[Ql ]
1400
600
300
400
200
200
100
0
0.248
MWS Algorithm
RSG Algorithm with γ=10
RSG Algorithm with γ=100
400
0
0.2485
0.249
0.2495
0.25
0.248
0.2485
Arrival Rate
0.249
(a) Single-hop non-fading network
(b) Single-hop fading network
5000
4500
4000
MWS Algorithm
RSG Algorithm with γ=10
RSG Algorithm with γ=100
3500
3000
2500
PL
l=1 E[Ql ]
0.2495
Arrival Rate
2000
1500
1000
500
0
0.331
0.3315
0.332
0.3325
0.333
0.3335
Arrival Rate
(c) 3 × 3 switch
Figure 4.3: Throughput performance of the RSG Algorithm
74
In Fig. 4.3, we compare the total mean queue-length under the MWS Algorithm,
as well as the RSG Algorithm with different γ values. It can be observed in Fig. 4.3
that the RSG Algorithm can stabilize the system in the above network setups. It also
can be seen that the total mean queue-length of the RSG Algorithm increases with
the parameter γ. This is expected since as γ increases, it becomes more likely for the
RSG Algorithm to choose a queue with less packet to serve, potentially wasting some
service while improving the service regularity, as we shall see next.
4.6.2
Service Regularity Performance
In this subsection, we investigate the service regularity performance of our RSG Algorithm, as well as illustrate the tradeoff between the total mean queue-length and the
service regularity. We present our results in three different networks: single-hop nonfading network, single-hop fading network and 3×3 switch. In both single-hop nonfading and fading networks, we consider the symmetric setup with the arrival rate vector
λ , [0.225, 0.225, 0.225, 0.225], and the asymmetric setup with the arrival rate vector
λ , [0.4, 0.3, 0.15, 0.05]. For a single-hop ON-OFF fading network, the probability
vectors that the channels are available are q = [0.8, 0.8, 0.8, 0.8] in symmetric setup
and q = [0.6, 0.5, 0.4, 0.3] in asymmetric setup. For a 3 × 3 switch, we consider the
symmetric setup with the arrival rate vector λ = [0.3, 0.3, 0.3; 0.3, 0.3, 0.3; 0.3, 0.3, 0.3]
and the asymmetric setup with the arrival rate vector λ = [0.5, 0.3, 0.1; 0.2, 0.4, 0.3; 0.1,
0.2, 0.5]. In all simulations, we choose the scaling parameter γ to be the powers of 2,
ranging from 2−7 to 27 .
Fig. 4.4 shows the relationship between the total mean queue-length and the service regularity in different network setups. The tradeoff between the service regularity
and the total mean queue-length can be clearly seen: as γ increases, the service regularity improves while the total mean queue-length also increases. It can be observed
75
3
Symmetric setup
Asymmetric setup
Lower bound in symmetric setup
Lower bound in asymmetric setup
3
λl E[T l ]
2
2.5
2
γ = 2−7
γ = 128
PL
1.5
1.5
1
1
1
2
10
0.5
3
10
10
1
l=1
λl E[T l ]
3.5
3
10
3
10
Total Queue Length
(a) Single-hop non-fading network
4
2
10
Total Queue Length
(b) Single-hop fading network
MWS Algorithm
(γ = 0)
γ = 2−7
γ = 128
2.5
PL
0.5
MWS
Algorithm
(γ = 0)
l=1
γ = 128
PL
l=1
λl E[T l ]
2.5
MWS Algorithm
(γ = 0)
Symmetric setup
Asymmetric setup
Lower bound in symmetric setup
γ = 2−7
Lower bound in asymmetric setup
2
1.5
Symmetric setup
Asymmetric setup
Lower bound in symmetric setup
Lower bound in asymmetric setup
1
0.5
1
10
2
10
3
10
Total Queue Length
(c) 3 × 3 switch
Figure 4.4: Tradeoff between mean queue length and service regularity
76
that the simulated values converge to the fundamental lower bound in non-fading
networks with symmetric setup (Figs 4.4(a) and (c)), while they stay away from the
lower bound in asymmetric setups. This motivates us to refine the lower bound analysis in asymmetric setups, which is left for future investigation. Here, it is worth
mentioning that even with very small γ values (e.g., 2−6 ), our RSG Algorithm significantly improves the service regularity, while introducing negligible increase in the
total mean queue-length.
4.6.3
Heavy-Traffic Performance
In this section, we provide simulation results to compare the mean delay and service
regularity performance of the RSG Algorithm with the MWS Algorithm. In the
simulation, we consider a single-hop non-fading network with 4 links. We consider
1 1 1 1
()
the symmetric case λ = (1 − ) × , , , , and the asymmetric case λ() =
2
4 4 4 4
1 1 1 1
1 1 1 1
, , ,
.
[ , , , ] + (1 − ) ×
2 4 8 16
32
64 64 64 64
22
−0.1
MWS (γ = 0)
γ=1
10
20
ε E[<d,Q>]
−0.3
10
MWS (γ = 0)
γ=1
18
γ = ε−1/5
16
γ = ε−1
theoretical lower bound
−0.4
10
γ = ε−1/5
γ = ε−1
Σ4l=1E[Tl]
−0.2
10
−0.5
10
14
12
−0.6
10
10
−0.7
10
8
−0.8
10
1
10
2
3
10
10
6
4
10
2
10
1/ε
3
4
10
10
1/ε
(a) Mean queue length
(b) Service regularity
Figure 4.5: Heavy-traffic performance in the symmetric case
77
5
10
24
22
Σ4l=1E[Tl]
ε E[<d,Q>]
20
MWS (γ = 0)
γ=1
1
10
−1/5
γ=ε
γ = ε−1
theoretical lower bound
0
10
MWS (γ = 0)
γ=1
18
γ = ε−1/5
16
−1
γ=ε
14
12
10
−1
10
2
10
3
4
10
10
8
5
10
2
10
1/ε
3
4
10
10
5
10
1/ε
(a) Mean queue length
(b) Service regularity
Figure 4.6: Asymmetric arrivals in the asymmetric case
From Figure 4.5a and 4.6a, we can observe that the RSG Algorithm with both
1
, and the MWS Algorithm converge to the theoretical lower bound
γ = 1 and γ = √
5
and thus is heavy-traffic optimal, which confirms our theoretical results. Yet, the
1
RSG Algorithm with γ = has large mean queue length, which does not match with
the theoretical lower bound and thus is not heavy-traffic optimal. Hence, γ should
1
) to preserve heavy-traffic optimality.
scale as slowly as O( √
5
From Figure 4.5b and 4.6b, we can see that the RSG Algorithm with even γ = 1
significantly outperforms the MWS Algorithm in terms of service regularity. More
1
remarkably, the RSG Algorithm with γ = √
can achieve the lower bound achieved
5
by the round robin policy under symmetric arrivals.
4.7
Summary
In this chapter, we investigated the problem of designing a scheduling policy that is
throughput-optimal, heavy-traffic optimal, and possesses favorable service regularity
78
characteristics. We proposed a parametric class of maximum-weight type scheduling
policies, where each link weight consists of its own queue-length and a counter that
tracks the time since the last service. After establishing the throughput optimality
of our policy, we showed that it also has provable service regularity performance. In
particular, the service regularity of our policy can be guaranteed to remain within a
factor distance of a fundamental lower bound for any feasible scheduling policy. We
explicitly expressed this factor as a function of the system statistics and the design
parameters. We further showed that the proposed algorithm is heavy-traffic optimal
as long as its design parameter scales slowly with the network load. Finally, We
performed extensive numerical studies to illustrate the significant gains achieved by
our policy over the traditional queue-length-based policies. This is the first work
that simultaneously addresses the throughput, delay and service regularity of the
scheduling design.
79
Part II
Distributed Resource Allocation
Algorithm Design
80
CHAPTER 5
LIMITATIONS OF RANDOMIZATION FOR
DISTRIBUTED RESOURCE ALLOCATION
We have discussed the efficient algorithm design for time-sensitive and dynamic applications. Yet, all these algorithms call for computationally heavy and typically centralized operations, which is impractical. Such restrictions motivate us to develop more
practical algorithms with reduced complexity. One possible thread is through the
development of a class of evolutionary randomized algorithms (also named pick and
compare algorithms) with throughput-optimality characteristics (e.g., [90, 13, 84]).
Another thread may utilize distributed but suboptimal randomized/greedy strategies
(e.g., [55, 35, 6]). More recently, another exciting thread have emerged that can
achieve excellent network performance by cleverly utilizing queue-length information
in the context of carrier sense multiple access (CSMA) (e.g., [60, 33, 80, 81, 72]). Yet,
to the best of our knowledge, there does not exist a general framework in which a variety of randomized schedulers can be studied in terms of their throughput-optimality
characteristics.
Thus, in this chapter, we aim to fill this gap by developing a common framework
for the modeling and analysis of queue-length-based randomized schedulers, and then
by establishing necessary and sufficient conditions on the throughput-optimality of a
large functional class of such schedulers under the time-scale separation assumption.
Our framework is built upon the observation that a common characteristic to most of
81
the developed schedulers is their randomized selection of transmission schedules from
the set of all feasible schedules. Specifically, given the existing queue-lengths of the
links, each scheduling strategy can be viewed as a particular probability distribution
over the set of feasible schedules. While the means with which this random assignment
may vary in its distributiveness or complexity, this perspective allows us to model a
large set of existing and an even wider set of potential randomized schedulers within
a common framework.
This work builds on this original point-of-view to explore the boundaries of randomization in the throughput-optimal operation of wireless networks. Such an investigation is crucial in revealing the necessary and sufficient characteristics of randomized schedulers and the network topologies in which throughput-optimality can be
achieved. Next, we list our main contributions along with references on where they
appear in this chapter.
• In Section 5.1, we introduce three functional forms of randomized queue-lengthbased scheduling strategies that include many existing strategies as special cases (cf.
Definitions 5.1.1, 5.1.2 and 5.1.3). These strategies differ in the manner in which
they measure the weight of schedules, and hence are used to model fundamentally
different scheduling implementations.
• We categorize the set of all functions used by these strategies into functions
of exponential form and of sub-exponential form (cf. Definition 5.1.4), collectively
covering almost all functions of interest. These two categories capture the steepness
of the functions used in the schedulers, and help reveal a critical degree of steepness
necessary for throughput-optimality in large networks.
• Then, we find some sufficient (in Section 5.3) and some necessary (in Section 5.4) conditions on the topological characteristics of the wireless network for the
throughput-optimality of these schedulers as a function of the class of functions used
82
in their operation. Our results, graphically summarized in Section 5.2, reveal the
significance of the network’s scheduling diversity that is measured by the number of
schedules each link belongs to.
5.1
Problem Formulation
We consider a non-fading wireless network with L links. Recall that we use S = (Sl )l∈L
to denote a feasible schedule, where Sl = 1 if link l is active and Sl = 0 is link l is
inactive in the schedule. We also treat S as a set of active links and write l ∈ S if
Sl = 1. We use |S| to denote the cardinality of the set S. We further call a feasible
schedule as maximal if no more links can be added without violating the interference
constraint. We define the scheduling diversity of link l ∈ L as the number of different
maximal schedules ml that link l belongs to. Since each link l ∈ L belongs to at
least one maximal schedule, ml should be the integer greater than or equal to 1.
Consider a single-hop wireless network where all links interfere with each other, we
have ml = 1 for all l. Less trivially, a 2 × 2 switch has two maximal schedules with
2 links, and ml = 1 for each l. Roughly speaking, the scheduling diversity increases
as the network diameter 1 increases. Such a behavior can be observed directly in a
linear network with L links under the primary interference model: for L ≤ 3, ml = 1
for all l; for L ≥ 6, ml ≥ 2 for all l.
We define F , set of non-negative, nondecreasing and differentiable functions
f (·) : R+ → R+ with lim f (x) = ∞. Recall that Ql [t] denotes queue length at link l
x→∞
in time slot t. We say that the queue l is f -stable for a function f ∈ F if it satisfies
lim sup
T →∞
1
T −1
1X
E[f (Ql [t])] < ∞.
T t=0
(5.1.1)
Network diameter is the maximum of the shortest hop-count between any two nodes in the graph.
83
We note that this is an extended form of the more traditional strong stability condition
(cf. Chapter 1) that coincides when f (x) = x. Moreover, it is easy to show that f stability implies strong stability when f is also a convex function. We say that
the network is f -stable if all its queues are f -stable. Accordingly, we say that a
scheduler is f -throughput-optimal if it achieves f -stability of the network for any
arrival rate vector λ = (λl )l∈L that lies strictly inside the capacity region Λ. Again,
in the special case of f (x) = x, the notion of f -throughput-optimality reduces to
traditional throughput-optimality, and when f is convex, f -throughput-optimality
implies throughput-optimality.
Next, we consider three classes of randomized schedulers which not only model
many existing probabilistic schedulers as special cases but also contain a much wider
classes of potential schedulers that have not been analyzed. They differ in the operation of the functional forms used in them.
Definition 5.1.1 (RSOF Scheduler). For a given f ∈ F and queue-length vector Q
at the beginning of a slot, the Ratio-of-Sum-of-Functions (RSOF) Scheduler picks a
schedule S ∈ S in that slot such that
P
l∈S f (Ql )
P
, for all S ∈ S.
PS (Q) , P
{S0 :S0 ∈S}
j∈S0 f (Qj )
(5.1.2)
Definition 5.1.2 (RMOF Scheduler). For a given f ∈ F and queue-length vector Q
at the beginning of a slot, the Ratio-of-Multiplication-of-Functions (RMOF) Scheduler
picks a schedule S ∈ S in that slot such that
Q
l∈S f (Ql )
Q
υS (Q) , P
, for all S ∈ S.
{S0 :S0 ∈S}
j∈S0 f (Qj )
(5.1.3)
Definition 5.1.3 (RFOS Scheduler). For a given f ∈ F and queue-length vector Q
at the beginning of a slot, the Ratio-of-Function-of-Sums (RFOS) Scheduler picks a
schedule S ∈ S in that slot such that
P
f ( l∈S Ql )
P
πS (Q) , P
, for all S ∈ S.
{S0 :S0 ∈S} f (
j∈S0 Qj )
84
(5.1.4)
Note that all the RSOF, RMOF and RFOS Schedulers are more likely to pick a
schedule with the larger queue length, but with different distributions based on their
form and the form of f ∈ F. In particular, the steepness of the function f determines
the weight given to the heavily loaded link in both RSOF and RMOF Schedulers and
the heavily loaded schedule in the RFOS Scheduler. Also, note that the schedulers coincide in single-hop network topologies because each maximal schedule only includes
one link in such networks, and for the following choices of f : when f (x) = x, the
RSOF and RFOS Schedulers coincide; when f (x) = ex , the RMOF and RFOS Schedulers coincide. These three classes cover a wide variety of schedulers including many
of existing throughput-optimal schedulers. For example, when f (x) = ex , the RMOS
and RFOS Schedulers correspond to the stationary distribution of the throughputoptimal CSMA policy that attracted a lot of attention lately (see [33, 80, 72]). Yet,
they also contain a much wider set of schedulers, one for each f .
The aim of this work is to identify the limitations of randomization for a wide
class of randomized dynamic schedulers that utilize functions of queue-lengths to
schedule transmissions. Even though randomization has significant advantage in lowcomplexity or distributed implementation, it causes inaccurate operation and may
be hurtful if not performed within limitations. In this chapter, we find that the
performance of the randomized schedulers may especially be sensitive to the topology
of the conflict graph and the functional form used in the weighting. To see this,
consider one maximal schedule S1 including three active links l1 , l2 and l3 in a 3 × 3
switch topology. We assume that arrivals only happen to those 3 links at rates λl1 ,
λl2 and λl3 with the constraints that λli ∈ [0, 1) for all i = 1, 2, 3, which clearly can
be supported by a simple policy that always serves the schedule S1 . Thus, by setting
λli arbitrarily close to one for each i, this simple policy can achieve a sum rate of
3
X
λli < 3. However, for a RFOS Scheduler with f (x) = x, we can easily calculate
i=1
85
that
3
X
θli = 2, where θli (i = 1, 2, 3) is the probability of serving link li . Thus, the
i=1
RFOS Scheduler with f (x) = x cannot achieve full capacity region in a 3 × 3 switch.
Yet, in the same set up, if we use f (x) = ex instead of f (x) = x in the RFOS
Scheduler, the mapping has the same probabilistic form as the CSMA policy, and thus
would be throughput-optimal. This shows the significant impact of the functional
form on the throughput performance of randomized schedulers. In addition, the
RFOS Scheduler with f (x) = x is shown to be f -throughput-optimal in a 2 × 2
switch (c.f. Section. 5.2), which indicates that the network topology may also affect
the throughput performance of randomized schedulers.
,
(Į•1, ȕ•0)
(0<Į<1, ȕ•0)
(Į>0)
Figure 5.1: The relationship between classes A, B and C.
Next, we identify the three classes of functions with varying forms that turn out
to be crucial to our investigation.
Definition 5.1.4. We consider the following subsets of F:
86
f (x)
= 0}.
x→∞ f ((1 + )x)
(a) A , {f ∈ F : ∀ > 0, lim
(b) B , {f ∈ F: lim
x→∞
f (x + a)
= 1, for any a ∈ R}.
f (x)
(c) C , {f ∈ B: there exist K1 and K2 satisfying 0 < K1 ≤ K2 < ∞ such that
K1 (f (x1 ) + f (x2 )) ≤ f (x1 + x2 ) ≤ K2 (f (x1 ) + f (x2 )), for all x1 , x2 ≥ 0}.
We call A as the class of exponential functions and C as the class of sub-exponential
functions. The key examples of functions with sets A, B, C and their interrelationship
are extensively studied in Appendix C.1.
Figure 5.1 concisely demonstrates the most critical facts: that A and C are nonoverlapping classes; while B has an intersection with A. Furthermore, the example
functions are provided with a variety of forms that justify the names assigned to A
and C : A contains rapidly increasing functions generally with exponential forms;
while C contains sub-exponentially increasing polynomial and logarithmic functional
forms. In the study of necessary and sufficient conditions for throughput-optimality,
we shall find that most of the results depend on which of these three functional classes
the functions belong to.
5.2
Overview of Main Results
In this section, we present our main findings and resulting insights on the throughputoptimality of the RSOF, RMOF and RFOS Schedulers (c.f. Definitions 5.1.1, 5.1.2
and 5.1.3) with different functional forms under different network topologies. These
results are rigorously proven in Sections 5.3 and 5.4. To facilitate an accessible figurative presentation, in the horizontal dimension, we conceptually order the functions
in F in increasing level of steepness starting from f (x) = (log(x + 1))α and f (x) = xα
1 α
for any α > 0 that belong to C, followed by f (x) = β ex for any 0 < α < 1 and any
x
87
\
1 xα
e for any α ≥ 1 and any
xβ
β ≥ 0 that belongs to A. In the vertical dimension, we use the scheduling diversity
β ≥ 0 that belongs to B
A, and finishing with f (x) =
(ml )l∈L introduced in Section 5.1 to distinguish different topological and interference
scenarios. Recall that since ml denotes the number of different maximal schedules
that link l belongs to, it may be viewed as a rough measure of the network diameter. Then, the main results for the RSOF and RFOS Schedulers are presented in
Figures 5.2 and 5.3, respectively. In these figures, we also include several conjectures
that are validated through simulations in Section 5.5.
ml
ml•2,
l
f-throughput-optimal
unknown
unknown
ml=1,
l
network
with high
scheduling
diversity
non-throughput-optimal
unknown
f-throughputoptimal
conjecture:
f-throughput-optimal
throughput-optimal
(single-hop network)
(log(x+1))Į xĮ x xĮ
(Į>0) (0<Į<1) (Į>1)
class of sub-exponential
functions C
exp(xĮ)/xȕ
(0<Į<1,ȕ•0)
exp(xĮ)/xȕ
(Į•1,ȕ•0)
network
with low
scheduling
diversity
ͺΟΔΣΖΒΤΚΟΘ͑
ΤΥΖΖΡΟΖΤΤ͑ΠΗ͑f
class of exponential
functions A
Figure 5.2: Throughput performance of the RSOF Scheduler.
From Figure 5.2, we see that the RSOF Scheduler with the function f ∈ B is
f -throughput-optimal when ml = 1, ∀l ∈ L. Also, the RSOF Scheduler with the
function f ∈ A \ B is throughput-optimal in single-hop network topologies since the
RSOF and RFOS Schedulers have the same probability distribution over schedules
88
ml
unknown
ml=1,
l
network
with high
scheduling
diversity
non-throughput-optimal
unknown
ml•2,
l
Throughput-optimal
network
with low
scheduling
diversity
conjecture:
f-throughput-optimal
f-throughput-optimal
(single-hop network)
(log(x+1))Į xĮ x xĮ
(Į>0) (0<Į<1) (Į>1)
class of sub-exponential
functions C
exp(xĮ)/xȕ
exp(xĮ)/xȕ
(0<Į<1,ȕ•0) (Į•1,ȕ•0)
class of exponential
functions A
ͺΟΔΣΖΒΤΚΟΘ͑
ΤΥΖΖΡΟΖΤΤ͑ΠΗ͑f
Figure 5.3: Throughput performance of the RFOS Scheduler.
in such networks and the RFOS Scheduler with the function f ∈ A is throughputoptimal (see Figure 5.3). However, if min ml ≥ 2, the RSOF Scheduler with any
l∈L
function f ∈ F cannot be throughput-optimal. Thus, roughly speaking, the RSOF
Scheduler is non-throughput-optimal for the network with high scheduling diversity,
while the RSOF Scheduler with the function f ∈ B is f -throughput-optimal for low
scheduling diversity. We note that although the throughput performance of the RSOF
1 α
Scheduler with some exponential functions f ∈ A \ B (i.e. f (x) = β ex , α ≥ 1 and
x
β ≥ 0) is not yet explored in general topologies with ml = 1, ∀l ∈ L, we conjecture
that it is f -throughput-optimal in this region, since the RSOF Scheduler with such
functions reacts much more quickly to the queue length difference between schedules
than that with sub-exponential functions, especially under asymmetric arrival patterns. We validate this conjecture through simulations in Section 5.5. Overall, the
RSOF Scheduler is more sensitive to the network topology than the functional form
used in it.
The horizontal unknown region corresponds to network topologies where some
89
links have scheduling diversity 1 and other links have scheduling diversity at least
2. The vertical unknown region corresponds to randomized schedulers with functions
that are not in the functional classes A, B and C. In Figure 5.3, we observe that the
RFOS Scheduler with the function f ∈ A is throughput-optimal under any network
topology. Also, the RFOS Scheduler with the function f ∈ C is f -throughput-optimal
in single-hop network topologies, which follows from the fact that the RFOS and
RSOF Schedulers have the same probability probabilistic forms in such networks, the
result that the RSOF Scheduler with the function f ∈ B is f -throughput-optimal
(see Figure 5.2) and the fact that C ⊆ B. Also, when the function f is linear,
the RFOS Scheduler has the same probability form with the RSOF Scheduler and
thus is f -throughput-optimal when ml = 1, ∀l ∈ L. However, the RFOS Scheduler
with the function f ∈ C is not throughput-optimal when min ml ≥ 2. Roughly
l∈L
speaking, the network with higher scheduling diversity requires much steeper functions (e.g., exponential functions) for the throughput-optimality of the RFOS Scheduler. While the throughput performance of the RFOS Scheduler with the function
f ∈ C \{linear functions} for general network topologies with ml = 1, ∀l ∈ L is part of
our ongoing work, we conjecture that it is f -throughput-optimal in those topologies
since both RFOS and RSOF Schedulers with sub-exponential functions have almost
the same reaction speed to the queue length difference between schedules. We also
validate this conjecture via simulations in Section 5.5. Overall, the RFOS Scheduler
is more sensitive to the functional form used in it than the network topology.
The RMOF Scheduler with the function f satisfying log f ∈ B and f (0) ≥ 1 is
(log f )-throughput-optimal under any network topology. This result together with
the RFOS Scheduler with the function f ∈ A extends the throughput-optimality of
CSMA schedulers (e.g. [33, 72]) to a wider class of functional forms. While this
result proves a weaker form of throughput-optimality than f -throughput-optimality
90
for the RMOF Scheduler, we note that the RMOF Scheduler generally outperforms
the RFOS and RSOF Schedulers in numerical investigations. Hence, we leave it to
future research to strengthen this result.
Collectively these results not only highlight the strengths and weaknesses of the
three functional randomized schedulers, they also reveal the interrelation between
the steepness of the functions and the scheduling diversity of the underlying wireless
networks. This extensive understanding of the limitations of randomization may
motivate the network designers to use or avoid certain types of probabilistic scheduling
strategies depending on the topological characteristics of the network.
5.3
Sufficient Conditions
In this section, we study the sufficient conditions on the network’s topological characteristics and the functions used in the RSOF, RMOF and RFOS Schedulers to
achieve throughput-optimality.
5.3.1
f -Throughput-Optimality of the RSOF Scheduler
We study the throughput performance of the RSOF Scheduler for a network topology
with ml = 1, ∀l ∈ L. In such a network, each link belongs to only one maximal
schedule.
Lemma 5.3.1. If
N
X
λi < 1, λi > 0, and ai ≥ 0, for i = 1, ..., N , then there exists a
i=1
δ > 0 such that
N
X
a2
i
i=1
λi
≥
N
X
ai
i=1
!2
(1 + δ).
(5.3.1)
Proof. See Appendix C.2 for the proof.
Proposition 5.3.2. In a network topology with the scheduling diversity of each link
91
equal to 1, i.e., ml = 1, ∀l ∈ L, the RSOF Scheduler with the function f ∈ B is
f -throughput-optimal.
Proof. We assume that there are only N available maximal schedules. Let Si (i =
1, ..., N ) denote the ith maximal schedule. In each maximal schedule Si , there are
|Si | active links. We use (Sil , l = 1, ..., |Si |) to denote the sequence of active links
in the maximal schedule Si . Note that we use i to index maximal schedule and l to
index link. Since the schedule diversity of each link is equal to 1, each link belongs to
only one maximal schedule. Thus, we can denote the queues, arrivals, and scheduling
statistics in terms of maximal schedules for easier exposition. To that end, we let
Qil , λil and Pli (i = 1, ..., N, l = 1, ..., |Si |) denote the queue-length of link l ∈ Si ,
the average arrival rate for the link l ∈ Si and the probability of serving the link
l ∈ Si , respectively. In addition, Ail [t], Sli [t] and Uli [t] denote the number of arrivals
to link l ∈ Si at time slot t, the number of potential departures of link l ∈ Si in
slot t and the unused service for link l ∈ Si at time slot t, respectively. Recall that
each link can only belong to one maximal schedule and note that links in different
maximal schedules cannot be active at the same time. Thus, the capacity region for
such network is
CN ,
(
λ:
N
X
)
λili < 1, ∀li = 1, ..., |Si | .
i=1
(5.3.2)
Under the above notation, the RSOF Scheduler becomes :
P|Si |
f (Qil )
,
PSi = PN l=1
P|Sk |
k
f
(Q
)
l=1
l
k=1
i = 1, ..., N.
(5.3.3)
Note that Pli = PSi , for l = 1, ..., |Si |. If λil = 0 for some i and l, then no arrivals
occur in the link l ∈ Si . Thus, we don’t need to consider such links. In the rest of
proof, we assume λil > 0 (i = 1, ..., N , l = 1, ..., |Si |). Consider the Lyapunov function
92
i
V (Q) ,
|S |
N X
X
h(Qi )
l
i=1 l=1
λil
, where h0 (x) = f (x). By using Lemma 1, it is shown in the
Appendix C.3 that there exist positive constants γ and G such that
∆V
, E [V (Q[t + 1]) − V (Q[t])|Q[t] = Q]
i
≤ −γ
|S |
N X
X
f (Qil ) + G.
(5.3.4)
i=1 l=1
By using the Theorem 4.1 in [66], inequality (5.3.4) implies the desired result.
5.3.2
Throughput-Optimality of RMOF and RFOS Schedulers
In this subsection, we investigate the sufficient condition for the throughput-optimality
of RMOF and RFOS Schedulers.
Proposition 5.3.3. (i) The RMOF Scheduler with the function f ∈ F satisfying
log f ∈ B and f (0) ≥ 1 is (log f )-throughput-optimal under any network topology;
(ii) The RFOS Scheduler with the function f ∈ A is throughput-optimal under
any network topology.
Proof. To prove this, we use a similar approach as in [72] that uses the following
result from [17]: for a scheduling algorithm, given any 0 ≤ , δ < 1, there exists an
M > 0 for which the scheduling algorithm satisfies the following condition: in any
time slot t, with probability greater than 1 − δ, the scheduling algorithm chooses a
X
X
schedule x[t] ∈ S that satisfies:
w(Ql [t]) ≥ (1 − ) max
w(Ql [t]), whenever
x∈S
l∈x[t]
l∈x[t]
k Q[t] k> M , where Q[t] , (Ql [t])l∈L , and w ∈ B. Then the scheduling algorithm is
w-throughput-optimal.
(i) Given any 1 and δ1 such that 0 ≤ 1 , δ1 < 1. Let
X1 , {x ∈ S :
X
log f (Ql [t]) < (1 − 1 )W1∗ [t]},
l∈x
93
(5.3.5)
where W1∗ [t] , max
x∈S
X
log f (Ql [t]). Then, we have
l∈x
υ(X1 ) =
X
Since
X
exp
x∈S
"
X
#
Q
f (Ql [t])
x0 ∈S
l∈x0 f (Ql [t])
x∈X1
P
P
exp
log
f
(Q
[t])
l
P l∈x
= Px∈X1
exp
x∈S
l∈x log f (Ql [t])
|X1 | exp [(1 − 1 )W1∗ [t]]
P
.
< P
exp
log
f
(Q
[t])
l
x∈S
l∈x
υx =
x∈X1
X
P
l∈x
Q
log f (Ql [t]) ≥ exp(W1∗ [t]), then we get
l∈x
υ(X1 ) <
|X1 |
|X1 | exp [(1 − 1 )W1∗ [t]]
=
.
∗
exp(W1 [t])
exp(1 W1∗ [t])
(5.3.6)
If some queue lengths increase to infinity, then W1∗ [t] → ∞ and thus we have υ(X1 ) →
0. Hence, there exists a M1 > 0 such that k Q[t] k> M1 and the RMOF Scheduler
with the function f ∈ F satisfying log f ∈ B and f (0) ≥ 1 picks the schedule
S[t] ∈ S \ X1 with probability 1 − δ1 and thus is log f -throughput-optimal under any
topology.
(ii) Given any 2 and δ2 such that 0 ≤ 2 , δ2 < 1. Let W2∗ [t] , max
X2 , {x ∈ S :
X
x∈S
X
Ql [t], and
l∈x
Ql [t] < (1 − 2 )W2∗ [t]}. Then, by using the same technique as in
l∈x
(i), we can prove that the RFOS Scheduler with f ∈ A is throughput-optimal under
any topology.
5.4
Necessary Conditions
So far, we have shown that the RSOF Scheduler with the function f ∈ B is f throughput-optimal in the network topology with ml = 1, ∀l ∈ L and the RFOS
Scheduler with the function f ∈ A is throughput-optimal under arbitrary network
topologies. However, the next result establishes that in network topologies where each
94
link belongs to two or more schedules (i.e. when min ml ≥ 2), the RSOF Scheduler
l∈L
with any function f ∈ F and RFOS Scheduler with the function f ∈ C cannot be
throughput-optimal.
Proposition 5.4.1. If the network is such that min ml ≥ 2, then (i) RSOF Scheduler
l∈L
is not throughput-optimal for any f ∈ F; (ii) RFOS Scheduler is not throughputoptimal for any f ∈ C.
Proof. We prove these claims constructively by considering an arrival process that
is inside the capacity region, but is not supportable by the randomized schedulers
for the given functional forms. To that end, let us consider any maximal schedule
S0 ∈ S and index its links as {1, 2, ..., n} for convenience. We assume that arrivals
only happen to those n links at rates λ1 , · · · , λn with the constraint that λl ∈ [0, 1)
for all l = 1, · · · , n, which is clearly supportable by a simple scheduling policy that
always serves the schedule S0 . Thus, setting λl arbitrarily close to one for each l, this
n
X
λl < n.
simple policy can achieve a sum rate of
\ l=1
We define M = {S ∈ S : S S0 6= ∅}, K = S \ M, H = M \ {S0 } and
T = S \ {S0 }. In the rest of the proof, we use AB to denote the intersection of A
and B.
Given this construction, we next prove the following statements for the RSOF
n
X
1
and RFOS Schedulers respectively: (1) If
λl > n − , the RSOF Scheduler with
2
l=1
n
X
K10
any function f ∈ F is unstable. (2) If
λl > n −
, where K10 and K20 are
0
2K2
l=1
positive constants described in Appendix A, the RFOS Scheduler with the associated
function f ∈ C is unstable.
Since the aforementioned simple scheduler can stabilize the sum rate
n
X
l=1
λl <
n, the RSOF Scheduler with any function f ∈ F and RFOS Scheduler with the
95
associated function f ∈ C are not throughput-optimal. We next prove these claims
that complete the proof of Theorem 5.4.1.
(1) Under the above model, the RSOF Scheduler becomes
P
l∈SS0 f (Ql ) + |S \ S0 |f (0)
P
.
PS = P
0
S0 :S0 ∈S (
l∈S0 S0 f (Ql ) + |S \ S0 |f (0))
Let Pl denote the probability that link l ∈ S0 is served, then
n
X
Pl =
l=1
Since
n
X
X
f (Ql ) =
S:S∈S l∈SS0
n
X
(
X X
PS =
X
n
X
|
f (Ql )
X
{
f (Qi ) + |S \ S0 |f (0))
X
|S \ S0 |f (0)
S:S∈S
{z
}
,L2
X
1,
S∈M:l∈SS0
X
}|
f (Ql ) +
S:S∈S l∈SS0
l=1
S0 |f (0), and
X
l=1 S∈M:l∈SS0 i∈SS0
l=1 S∈M:l∈SS0
X X
,L1
z
n
X
X
|S\S0 |f (0) =
X X X
f (Qi )
S:S∈M l∈SS0 i∈SS0
X
=
=
|SS0 |
S:S∈M
n
X
X
f (Qi )
i∈SS0
f (Ql )
l=1
X
|SS0 |,
S∈M:l∈SS0
we can extend L1 and L2 as follows:
L1 =
=
n
X
l=1
n
X
f (Ql )
X
|SS0 | +
l=1
|SS0 ||S \ S0 |f (0)
S:S∈S
S∈M:l∈SS0
f (Ql )(n +
X
X
|HS0 |) +
X
1+
X
|TS0 ||T \ S0 |f (0),
T:T∈T
H∈H:l∈HS0
and
L2 =
=
n
X
l=1
n
X
l=1
f (Ql )
S∈M:l∈SS0
f (Ql )(1 +
X
|S \ S0 |f (0)
S:S∈S
H∈H:l∈HS0
96
X
1) +
X
T:T∈T
X
S:S∈S
l=1 S∈M:l∈SS0
f (Qi ) =
l=1 S∈M:l∈SS0 i∈SS0
n
X
.
|T \ S0 |f (0).
|SS0 ||S\
Thus, we have
n
X
Pl =
l=1
where Z1 =
n
X
l=1
Z2 =
n
X
X
f (Ql )
Z1
L1
=n− ,
L2
Z2
(n − |HS0 |) +
f (Ql )(1 +
l=1
(n − |TS0 |)|T \ S0 |f (0), and
T:T∈T
H∈H:l∈HS0
X
X
(5.4.1)
1) +
X
|T \ S0 |f (0). Note that |HS0 | ≤ n − 1, for
T:T∈T
H∈H:l∈HS0
∀H ∈ H, and |TS0 | ≤ n − 1, for ∀T ∈ T . Now, since ml =
have
X
X
1 ≥ 2, ∀l ∈ S0 , we
S∈S:l∈S
1 ≥ 1, ∀l ∈ S0 . Then, we get
H∈H:l∈HS0
P
Pn
P
1
Z1
l=1 f (Ql )
H∈H:l∈HS0 1 +
T:T∈T |T \ S0 |f (0)
P
Pn
P
= .
≥
Z2
2 l=1 f (Ql ) H∈H:l∈HS0 1 + 2 T:T∈T |T \ S0 |f (0)
2
Thus, we have
n
X
l=1
n
X
1
Pl ≤ n − . Hence, for topologies where min ml ≥ 2, if
λl >
l∈L
2
l=1
1
n − , in which case the total arrival rate is greater than the total service rate, then,
2
the RSOF Scheduler is unstable by following the Theorem 2.8 and Theorem 2.5 in
[66].
(2) With the same model, the RFOS Scheduler becomes
P
f ( l∈SS0 Ql )
P
P
.
πS = P
l∈S0 S0 Ql ) +
S00 :S00 ∈K f (0)
S0 :S0 ∈M f (
Then,
n
X
l=1
Pl =
n
X
X
l=1 S∈M:l∈SS0
Since
n
X
X
l=1 S∈M:l∈SS0
Pn P
P
l=1
S∈M:l∈SS0 f (
i∈SS0 Qi )
P
P
πs = P
.
S:S∈M f (
l∈SS0 Ql ) +
S:S∈K f (0)
f(
X
i∈SS0
Qi ) =
X
S:S∈M
97
|SS0 |f (
X
i∈SS0
Qi ),
(5.4.2)
we have
n
X
P
P
|SS0 |f ( l∈SS0 Ql )
P
P
= P
S:S∈K f (0)
S:S∈M f (
l∈SS0 Ql ) +
Pn
P
P
nf ( l=1 Ql ) + H:H∈H |HS0 |f ( l∈HS0 Ql )
P
P
P
P
=
f ( nl=1 Ql ) + H:H∈H f ( l∈HS0 Ql ) + S:S∈K f (0)
P
P
P
f (0)
H:H∈H (n − |HS0 |)f (
l∈HS0 Ql ) + n
PS:S∈K
P
P
.
= n − Pn
f ( l=1 Ql ) + H:H∈H f ( l∈HS0 Ql ) + S:S∈K f (0)
S:S∈M
Pl
l=1
The fact that f ∈ C implies that there exist K10 and K20 satisfying 0 < K10 ≤ K20 < ∞
m
m
m
X
X
X
0
0
such that K1
f (Qi ) ≤ f (
Qi ) ≤ K2
f (Qi ), for ∀m = 1, ..., n, where Qi ≥
i=1
i=1
i=1
0, i = 1, ..., m, which follows from induction. Then, we have
P
P
P
n
X
f (0)
K10
l∈HS0 f (Ql ) + n
H:H∈H (n − |HS0 |)
P
P
PS:S∈K
Pl ≤ n − 0 · Pn
K2
H:H∈H
l∈HS0 f (Ql ) +
S:S∈K f (0)
l=1 f (Ql ) +
l=1
P
P
P
n
f (Ql ) H∈H:l∈HS0 (n − |HS0 |) + n S:S∈K f (0)
K0
P
P
Pn
= n − 10 · Pnl=1
.
K2
H∈H:l∈HS0 1 +
S:S∈K f (0)
l=1 f (Ql )
l=1 f (Ql ) +
Note that |HS0 | ≤ n − 1, for ∀H ∈ H, and that ml =
that
X
X
1 ≥ 2, ∀l ∈ S0 , implies
S∈S:l∈S
1 ≥ 1, ∀l ∈ S0 . Then, we get
H∈H:l∈HS0
n
X
l=1
P
Pn
P
K10
K0
l=1 f (Ql )
H∈H:l∈HS0 1 +
S:S∈K f (0)
P
P
Pl ≤ n − 0 · Pn
≤ n − 10 . (5.4.3)
K2 2 l=1 f (Ql ) H∈H:l∈HS0 1 + 2 S:S∈K f (0)
2K2
Thus, by following the same argument as in the proof for statement (1), we know
n
X
K10
that when min ml ≥ 2 and
λl > n −
, the RFOS Scheduler is unstable.
l∈L
2K20
l=1
5.5
Simulation Results
In this section, we first perform numerical studies to validate the throughput performance of the proposed randomized schedulers with different functions in 2 × 2 and
3 × 3 switch topologies. Then, we evaluate the impact of functional forms on the
delay performance of proposed randomized schedulers in 2 × 2 switch topologies.
98
5.5.1
Throughput Performance
In a 2 × 2 switch, the scheduling diversity of each link is 1 and thus all proposed
randomized schedulers are proven to be throughput-optimal. In a 3 × 3 switch, the
scheduling diversity of each link is 2, for which the RFOS Scheduler needs to carefully
choose the functional form to preserve the throughput optimality while the RSOF
Scheduler is not f -throughput-optimal with any function f ∈ F
In a 2 × 2 switch, we consider arrival rate vector λ = ρH, where H = [Hij ] is a
doubly-stochastic matrix with Hij denoting the fraction of the total rate from input
port i that is destined to output port j. Then, ρ ∈ (0, 1) represents the average arrival
intensity, where the larger the ρ, the more heavily loaded the switch is. We present
two cases: symmetric arrival process (H1 = [0.5 0.5; 0.5 0.5]) and asymmetric arrival
process (H2 = [0.1 0.9; 0.9 0.1]) under high arrival intensity ρ = 0.99.
From Fig. 5.4a and 5.4b, we can observe that all randomized schedulers can
stabilize the system under symmetric and asymmetric arrival traffics. So, there is
a wide class of choices under which the randomized scheduling can guarantee the
throughput performance in the 2 × 2 switch. In addition, we can see that the RSOF
Scheduler with the exponential function and the RFOS Scheduler with the square
function are also stable in both symmetric and asymmetric arrival processes, which
support our conjecture in Section 5.2 that the RSOF Scheduler with the function
f ∈ A and the RFOS Scheduler with the function f ∈ B are f -throughput optimal
in network topologies with ml = 1, ∀l ∈ L.
In a 3 × 3 switch, we consider arrival rate vector λ = [0.95 0 0; 0 0.95 0; 0 0 0.95],
where the RSOF Scheduler with any function f ∈ F and the RFOS Scheduler with
any function f ∈ C cannot stabilize. The evolution of average queue length per link
over time for different schedulers with different functions is shown in Fig. 5.4c. From
Fig. 5.4c, we can observe that the average queue lengths of the RSOF Schedulers
99
45
30
RSOF or RFOS with f(x)=x
RSOF or RFOS with f(x)=x
RSOF with f(x)=x2
RSOF with f(x)=x2
40
RFOS with f(x)=x2
RFOS with f(x)=x2
RFOS or RMOF with f(x)=ex
RMOF with f(x)=x+1
30
Average Queue Length
Average Queue Length
35
RSOF with f(x)=ex
25
RSOF with f(x)=ex
25
20
15
RFOS or RMOF with f(x)=ex
RMOF with f(x)=x+1
20
15
10
10
5
5
0
0
2
4
6
8
0
10
Time Step
0
2
4
6
8
Time Step
4
x 10
(a) 2 × 2 switch: symmetric case
10
4
x 10
(b) 2 × 2 switch: asymmetric case
20
RSOF or RFOS with f(x)=x
18
RSOF with f(x)=x2
RSOF with f(x)=ex
Average Queue Length
16
RFOS with f(x)=x2
RMOF with f(x)=x+1
14
RMOF or RFOS with f(x)=ex
12
10
8
6
4
2
0
0
2000
4000
6000
Time Step
8000
10000
(c) 3 × 3 switch
Figure 5.4: Throughput performance validation of the randomized schedulers
100
with linear function, square function and even exponential function increase very fast,
which validates our theoretical result that the RSOF Scheduler with any function
f ∈ F cannot be throughput-optimal in network topologies with min ml ≥ 2. In
l∈L
addition, we can see that the average queue lengths of the RFOS Schedulers with
linear function and square function grow quickly while the RFOS Scheduler with
exponential function always keeps low queue length level, which demonstrates that
the steepness of functional form needs to be high enough for the RFOS Scheduler to
keep throughput optimality in general network topologies. Even though our result
indicates that the RMOF Scheduler with any function f satisfying log f ∈ B and
f (0) ≥ 1 is (log f )-throughput-optimal in general network topologies, we can see that
the RMOF Scheduler is still stable even with linear function. This validates that our
conjecture that the RMOF Scheduler with any function f ∈ F can be f -throughputoptimal in general network topologies.
5.5.2
Delay Performance
In this subsection, we perform numerical studies to evaluate the delay performance of
proposed randomized schedulers with different functions in a 2 × 2 switch topology.
From Fig. 5.5a, we can observe that, under symmetric arrival traffic, the delay
performance is highly insensitive to the choice of the randomization and the functional
form being used in it especially under high arrival load. So, there is a wide class of
choices under which the randomized scheduling can yield good delay performance.
On the other hand, Fig. 5.5b demonstrates that, under asymmetric arrival traffic,
the RMOF Scheduler is more robust to the choice of functions used in it than both the
RSOF and RFOS Schedulers. In particular, it appears that the steepness of f needs to
be high enough for each randomization to yield good delay performance. Generally,
the RMOF Scheduler outperforms the other two randomized schedulers especially
101
under asymmetric arrival traffic. In all cases, the RSOF and RFOS Schedulers have
similar performance and MWS has the best delay performance.
RSOF with e
100
x
RSOF
RSOF with x+1
RFOS
RSOF with log(x+e)
RFOS with e
RMOF
MWS
RFOS with x+1
Average Queue Length
Average Queue Length
10
x
RFOS with log(x+e)
RMOF with e
x
RMOF with x+1
RMOF with log(x+e)
MWS
1
10
log(x+e)
x+1
0.1
x
e
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.70
Load Factor
0.75
0.80
0.85
0.90
0.95
1.00
Load Factor
(a) Symmetric arrivals in a 2 × 2 switch
(b) Asymmetric arrivals in a 2 × 2 switch
Figure 5.5: Delay performance comparison of the randomized schedulers with different
functional forms
While these numerical studies indicate a number of interesting facts on the mean
delay performance of randomized schedulers, we leave a more careful delay performance comparison to future research. There is clearly a need for a deeper investigation of delay performance of throughput-optimal schedulers. This work forms
the foundation to investigate these higher-order performance metrics in our future
research.
102
5.6
Summary
We explored the limitations of randomization in the throughput-optimal scheduler
design in a generic framework under the time-scale separation assumption. We identified three important functional forms of queue-length-based schedulers that covers
a vast number of dynamic schedulers of interest. These forms differ fundamentally
in whether they work with the queue-length of individual links or whole schedules.
For all of these functional forms, we established some sufficient and some necessary
conditions on the network topology and the functional forms for their throughputoptimality. We also provided numerical results to validate our theoretical results and
conjectures.
103
CHAPTER 6
OPTIMAL DISTRIBUTED SCHEDULING DESIGN
UNDER TIME-VARYING CONDITIONS
As we discussed in Chapter 5, one of the most robust randomized schedulers is CSMAbased distributed scheduler (e.g., [33, 73, 23, 81]), whose stationary distribution of
the underlying Markov chain has a product-form. It is well-known that CSMA-based
scheduler can maximize long-term average throughput for general wireless topologies.
However, these results do not apply to time-varying environments (i.e., scheduling
deadline-constrained traffic over wireless fading channels), since their throughputoptimality relies: (i) on the convergence time of the underlying Markov Chain to its
steady-state, which grows with the size of the network; and (ii) on relatively stationary conditions in which the CSMA parameters do not change significantly over time
so that the instantaneous service rate distribution can stay close to the stationary
distribution. Both of these conditions are violated in time-varying environments. For
example, packets of deadline-constrained traffic are likely to be dropped before the
CSMA-based algorithm converges to its steady-state, and the time-varying fading
creates significant variations on the CSMA parameters, in which case the instantaneous service rate distribution cannot closely track the stationary distribution. To
the best of our knowledge, there does not exist a work that can achieve provably good
performance by using attractive CSMA principles under time-varying conditions.
While achieving low delay via distributed scheduling in general topologies is a
104
difficult task (see [85]), in a related work [58] that focuses on grid topologies, the
authors have designed an Unlocking CSMA (UCSMA) algorithm with both maximum
throughput and order optimal average delay performance, which shows promise for
low-delay distributed scheduling in special topologies. However, UCSMA also does
not directly apply to deadline-constrained traffic since its measure of delay is on
average. Moveover, it is not clear how existing CSMA or UCSMA will perform
under fading channel conditions. Thus, designing an optimal distributed scheduling
algorithm in time-varying environments remains an open question.
With this motivation, in this chapter, we address the problem of distributed
scheduling in fully connected networks (e.g., Cellular network, Wi-Fi network) for
time-varying environments. We propose a Fast-CSMA (FCSMA) algorithm that, despite its similarity of name, fundamentally differs from existing CSMA policies in its
design principle: rather than evolving over the set of schedules to reach a favorable
steady-state distribution, the FCSMA policy aims to quickly reach one of a set of
favorable schedules and stick to it for a duration related to time-varying scale of the
application. While the performance of the former strategy is tied to the mixing-time
of a Markov Chain, the performance of our strategy is tied to the hitting time, and
hence, yields significant advantage for time-varying applications.
Then, we apply FCSMA techniques in four main scenarios: scheduling with/without
channel state information (CSI) over wireless fading channels, and deadline-constrained
scheduling with/without CSI over wireless fading channels. The latter two are most
challenging and important application in practice, since wireless networks are expected to serve real-time traffic, such as video or voice applications, generated by
a large number of users over potentially fading channels. These constraints and requirements, together with the limited shared resources, generate a strong need for
105
distributed algorithms that can efficiently utilize the available resources while maintaining high quality-of-service for the real-time applications. Yet, the strict shortterm deadline constraints and long-term throughput requirements associated with
most real-time applications complicate the development of provably good distributed
solutions.
All existing works in scheduling over wireless fading channels (e.g., [92, 89]) and in
deadline-constrained scheduling (e.g., [27, 29, 31, 20]) assume centralized controllers,
and hence are not suitable for distributed operation. To the best of our knowledge,
this is the first work that proposes an optimal and distributed algorithm under timevarying conditions caused by channel fading or time-sensitive applications. Our main
contributions in this chapter are:
• In Section 6.1, we propose a FCSMA algorithm that aims to quickly reach one
of a set of favorable schedules.
• We design an optimal policy based on FCSMA techniques in scheduling with/without
CSI over wireless fading channels in Section 6.2 and Section 6.3, respectively.
• We design an optimal distributed policy based on FCSMA techniques in scheduling deadline-constrained traffic with/without CSI over wireless fading channels in
Section 6.4 and Section 6.5, respectively.
6.1
The Principle of Fast-CSMA design
We consider a fully-connected network topology where L links contend for data transmission over a single channel. Due to the interference constraints, at most one link can
transmit in each slot. Randomized schedulers (e.g., [13, 33, 55, 60, 80, 90]) are widely
studied due to their flexibilities in development of low-complexity and distributed
implementations. The most promising and interesting randomized schedulers are distributed CSMA-based algorithms. We give the definition of continuous-time CSMA
106
algorithm (e.g., [33]) for completeness. In this chapter, we adopt the same assumptions as in [33] that the sensing is instantaneous and the backoff time is continuous.
Algorithm 6.1.1 (CSMA Algorithm). Each link l independently generates an exponentially distributed random variable with rate Rl [t] and starts transmitting after
this random duration unless it senses another transmission before. If link l senses
the transmission, it suspends its backoff timer and resumes it after the completion
of this transmission. The transmission time of each link is exponentially distributed
with mean 1.
1
(1,0,0)
(1,0,0)
R1[t]
1
R3[t]
(0,0,0)
R2[t]
R1[t]
(0,1,0)
(0,0,0)
1
1
1
R2[t]
(0,0,1)
(0,1,0)
(a)
R3[t]
1
(0,0,1)
(b)
Figure 6.1: (a) Markov chain for a CSMA algorithm (b) Markov chain for a FCSMA
algorithm
Figure 6.1a shows the state transition diagram of the underlying Markov Chain
for the CSMA Algorithm when there are 3 available links at time t, where each state
107
stands for a feasible schedule. It is easy to see that the stationary distribution of this
Markov Chain is
Pl =
1+
Rl [t]
PL
l=1 Rl [t]
, ∀l.
(6.1.1)
Since R[t] = (Rl [t])Ll=1 is chosen as a function of network state information (e.g.,
queue length, channel state information, arrivals) in wireless networks, the underlying Markov Chain for the CSMA Algorithm is inhomogeneous. Intuitively, the CSMA
parameters R[t] should change slowly such that the instantaneous service rate distribution can stay close to the stationary distribution. Indeed, for the application of
scheduling over time-invariant channels (i.e., the transmission rate of each link does
not change over time), such mapping has been observed to be optimal (e.g., [23, 81])
if the CSMA parameter Rl [t] of each link l can take certain functional forms (e.g.,
log log(·)) of its queue length at time t. Note that the queue length will change slowly
when it is large enough. The purpose of choosing the slowly increasing function is
further to make the CSMA parameters as a function of queue length do not change
significantly over time.
However, for the application of scheduling over wireless fading channels, the
CSMA parameters R[t] need to be chosen as a function of channel sate information to yield good performance. In such case, no matter what function we choose for
the channel state, R[t] will change significantly as the fading state fluctuates and thus
the instantaneous service distribution is not expected to track the stationary distribution. More generally, extending CSMA solutions to stochastic network dynamics or
sophisticated application requirements (e.g., serving traffic with strict deadline constraints over wireless fading channels) is difficult for two reasons: 1) the mixing time
of the underlying CSMA Markov chain grows with the size of the network, which, for
large networks, generates unacceptable delay for deadline-constrained traffic; 2) since
the dynamic CSMA parameters R[t] are influenced by the arrival and channel state
108
process, the underlying CSMA Markov chain may not converge to its steady-state
under strict deadline constraints and wireless fading channel conditions.
Thus, designing an optimal and distributed scheduling algorithm for stochastic
networks becomes quite challenging. In this paper, we propose a Fast-CSMA strategy that provides provably good performance under time varying conditions. Our
approach fundamentally differs from existing CSMA solutions in that our FCSMA
policy exploits the fast convergence characteristics of “hitting times” instead of “mixing times”.
Algorithm 6.1.2 (Fast-CSMA (FCSMA) Algorithm). At the beginning of each time
slot t, each link l independently generates an exponentially distributed random variable
with rate Rl [t], and starts transmitting after this random duration unless it senses
another transmission before. If all links have their random duration greater than a
slot, all links will keep silent in the current slot; otherwise, the link that captures the
channel transmits its data1 until the end of the slot. The whole process is repeated in
the next time slot.
Remarks: (1) The operation of the FCSMA Algorithm resembles that of the UCSMA Algorithm (see [58]). The difference lies in that the UCSMA algorithm restarts
the CSMA Algorithm to achieve both maximum throughput and order optimal average delay in grid network topologies over time-invariant channels by carefully choosing
the running period. However, it is unclear whether the UCSMA algorithm can still
work well in time-varying applications.
(2) By choosing the running period for the FCSMA Algorithm the same as the
1
If there is no data awaiting in the link l, it transmits dummy data to occupy the channel.
109
time scale of network dynamics (i.e., the block length for block fading or maximum
allowable deadline), we can show in later sections that the FCSMA Algorithm exhibits
excellent performance in time-varying applications.
(3) In general multi-hop network topologies, the FCSMA Algorithm can still converge very fast to one feasible schedule. Yet, the probability of serving each schedule
may not have a product form and the performance of the FCSMA Algorithm is unclear. We leave it for future investigation.
Figure 6.1b gives the state transition diagram of underlying Markov Chain for
the FCSMA Algorithm when there are 3 available links, where each state represents
a feasible schedule. The convergence time of the FCSMA Algorithm is tied to the
hitting time2 , while the convergence time of the CSMA Algorithm is dominated by
the mixing time of Markov chain, which generally is large. The hitting time of
1
,
the FCSMA Algorithm at slot t is exponentially distributed with mean PL
R
[t]
l
l=1
which is generally small in practice as we will see in simulations. Due to its small
hitting time, the FCSMA Algorithm yields significant advantages over existing CSMA
policies evolving slowly to the steady-state and may work well in more challenging
environments, i.e., scheduling real-time traffic over wireless fading channels. Because
of the fast convergence property of the FCSMA Algorithm, we introduce the idealized
FCSMA algorithm for easier theoretical analysis. The simulation results in the later
sections indicate that both FCSMA and Idealized FCSMA Algorithm have the same
system performance.
Algorithm 6.1.3 (Idealized FCSMA Algorithm). Idealized FCSMA Algorithm is
2
The hitting time is an empty duration after which the Markov Chain stays in a non-zero feasible
schedule state (i.e., the channel is occupied by one of users)
110
the FCSMA Algorithm with zero hitting time, which assumes that it can reach the
favorable state instantaneously.
For the Idealized FCSMA Algorithm, the probability of serving the link l in each
slot t will be:
Rl [t]
.
πl [t] = PL
l=1 Rl [t]
(6.1.2)
Let Wl [t] = log(Rl [t]) and W ∗ [t] = max Wl [t]. The following lemma establishes the
l
fact that the Idealized FCSMA Algorithm picks a link with the weight close to the
maximum weight with high probability when the maximum weight W ∗ [t] is large
enough at each slot t.
Lemma 6.1.4. Given > 0 and ζ > 0, ∃W ∈ (0, ∞), such that if W ∗ [t] > W , then
the Idealized FCSMA Algorithm picks a link l satisfying
Pr{Wl [t] ≥ (1 − )W ∗ [t]} ≥ 1 − ζ.
(6.1.3)
The proof is similar to that in [73] and [41], and thus is omitted here for brevity.
We mainly consider two types of traffic: elastic and inelastic traffic, where the
inelastic traffic means that each arrival has a maximum delay requirement while the
elastic traffic does not have such a requirement. We apply the FCSMA technique in
four challenging scenarios: scheduling elastic/inelastic traffic with/without Channel
State Information (CSI) over wireless fading channels. In each application, we need
to carefully design the FCSMA parameters R[t] = (Rl [t])Ll=1 at each slot t to yield
optimal performance. To facilitate the flexibility in the design and implementation
of the FCSMA algorithm, we mainly consider functions within the functional class B
that is defined in Definition 5.1.4.
111
Now, we are ready to develop optimal FCSMA algorithms in four challenging
applications: scheduling elastic/inelastic traffic with/without CSI over wireless fading
channels.
6.2
6.2.1
Scenario 1: Scheduling Elastic Traffic with CSI
FCSMA Algorithm Implementation
In this section, we consider the elastic traffic scheduling with the knowledge of channel
state information (CSI) at the beginning of each time slot in single-hop networks,
where at most one link can be scheduled in each time slot. We propose an optimal
FCSMA Algorithm with CSI that can support any arrival rate vector within the
capacity region R.
Algorithm 6.2.1 (Idealized FCSMA Algorithm with CSI for Scheduling Elastic
Traffic). In each time slot t, choose the rates
Rl [t] = f (Ql [t])Cl [t] , ∀l,
(6.2.1)
where f ∈ F.
Proposition 6.2.2. If log f ∈ B and f (0) ≥ 1, the FCSMA Algorithm with CSI for
scheduling elastic traffic is (log f )-optimal over wireless fading channels, i.e., for any
arrival rate λ ∈ Int(R), it makes the system (log f )-stable.
The proof is a special case of that in Proposition 6.4.8 which deals with the
deadline-constrained traffic over wireless fading channels. The main difference lies
in that the proof in Proposition 6.2.2 considers stabilizing data queues, instead of
112
virtual queues in the proof for Theorem 6.4.8. However, the argument is almost the
same as that in Proposition 6.4.8 and thus is omitted here for brevity. Note that there
are lots of flexibilities in choosing the function f . The FCSMA algorithm with less
steep function is easier to be implemented in practice. However, the higher steepness
of function f in FCSMA algorithm leads to smaller average queue length, as we will
see in the following simulations.
6.2.2
Simulation Results
In this subsection, we perform simulations to validate the optimality of the proposed
FCSMA policy with CSI for scheduling elastic traffic over wireless fading channels.
In the simulation, there are L = 10 links. The number of arrivals in each slot
follows a common Bernoulli distribution with mean λ. All links suffer from the
ON-OFF channel fading independently with probability p = 0.8 that the channel
is available at each time slot. Under this setup, we can get the capacity region
1 − (1 − p)L
. Through numerical calculation, we can get
[92]: Λ1 (C) = λ : λ <
L
λ < 0.1. In the simulation, we also consider the FCSMA algorithm with zero hitting
time, which assumes it can reach the favorable state instantaneously. We compare
our proposed FCSMA policy with f (x) = ex with QCSMA algorithm [73] with the
weight log log(Xl [t]Cl [t] + e). To that end, we divide each time slot into M mini-slots.
In FCSMA policy, if the link contends for the channel successfully, it will occupy that
channel in the rest of time slot; while in QCSMA policy, each link contends for the
channel and transmits the data in 1 mini-slot. Here, we do not consider the overhead
that the QCSMA policy needs to contend for the channel, which will greatly degrade
its performance.
From Figure 6.2, we can observe that FCSMA algorithm with both f (x) = ex
and f (x) = x + 1 can achieve full capacity. However, the average queue length under
113
exponential function is smaller than that under linear function, which implies that
the FCSMA algorithm with steeper function yields better delay performance. Recall
that FCSMA policy waits for random duration before accessing the channel, this
random duration can be arbitrarily small when the number of links increases and the
queue length is high. We can see from simulations that FCSMA policy has almost
the same delay performance as that with zero hitting time.
30
x
Idealized FCSMA with f(x)=e
Average queue length
25
x
FCSMA with f(x)=e
Idealized FCSMA with f(x)=x+1
20
FCSMA with
f(x)=x+1
15
10
5
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
Arrival rate
Figure 6.2: Performance of FCSMA for scheduling elastic traffic with CSI
From Figure 6.3, we can observe that the average queue length grows very fast
under the QCSMA policy with M = 1 while the average queue length of FCSMA
always stays at a low level. The reason for the poor performance of QCSMA scheme
over wireless fading channels is that the underlying Markov chain is controlled by
114
the channel state processes. If the running time of QCSMA policy has the same
time scale with the fading block, this Markov chain cannot converge to the steadystate. However, FCMSA policy can quickly lock into one state and exhibits good
performance, which is shown in Theorem 6.2.2 to be optimal if we carefully choose
the parameters. In addition, as M increases, the performance of QCSMA improves.
The reason is that the underlying Markov chain has enough time to converge to the
steady-state and thus yields better performance.
50
Idealized FCSMA with f(x)=e
Average queue length
45
40
FCSMA with f(x)=e
x
x
QCSMA with M=1
QCSMA with M=10
35
QCSMA with M=10
30
QCSMA with M=10
2
4
25
20
15
10
5
0
-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
Arrival rate
Figure 6.3: Performance comparison between FCSMA and QCSMA
115
6.3
6.3.1
Scenario 2: Scheduling Elastic Traffic without CSI
FCSMA Algorithm Implementation
In this section, we study the case when each link does not know the CSI at the
beginning of each time slot. In
(
) such a network, the capacity region becomes R =
L
X λl
λ = (λl )Ll=1 :
<1 .
E[C
l [t]]
l=1
Next, we propose an optimal FCSMA algorithm without CSI that can stabilize
the system for any arrival rate within the capacity region R.
Algorithm 6.3.1 (Idealized FCSMA Algorithm without CSI for Scheduling Elastic
Traffic). In each time slot t, choose the rates
Rl [t] = f (Ql [t]), ∀l,
(6.3.1)
where f ∈ F.
Proposition 6.3.2. If log f ∈ B and f (0) ≥ 1, FCSMA algorithm without CSI for
scheduling elastic traffic is (log f )-optimal over wireless fading channels, i.e., for any
arrival rate λ ∈ Int(R), it makes the system (log f )-stable.
The proof is similar to that in Proposition 6.2.2 which considers the elastic traffic
with known CSI. We skip it for brevity. Note that CSMA algorithms (e.g., [23] and
[81]) without CSI can also achieve optimal throughput over wireless fading channels
if the weight has the form log log(q) or log(q)/g(q), where q is the queue length. However, due to the fast hitting time, FCSMA algorithm yields better delay performance
than QCSMA algorithm, as we will see in the following simulations.
116
6.3.2
Simulation Results
In this subsection, we perform simulations to validate the throughput optimality
and investigate the delay performance of the proposed FCSMA policy without CSI
for scheduling elastic traffic over wireless fading channels. The simulation setup
is the same as that in Section 6.2.2. It is easy to see that the capacity region:
Λ2 (C) = {λ : λ < p}. Thus, we can get λ < 0.8. We compare our proposed FCSMA
policy with QCSMA algorithm [73] with the weight log log(Ql [t] + e).
70
x
Idealized FCSMA with f(x)=e
Average queue length
x
60
FCSMA with f(x)=e
Idealized FCSMA with f(x)=x+1
50
FCSMA with f(x)=x+1
QCSMA with loglog(x+e)
40
30
20
10
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Arrival rate
Figure 6.4: Performance of FCSMA for scheduling elastic traffic without CSI
From Figure 6.4, we can clearly see that FCSMA algorithm with both f (x) = ex
and f (x) = x + 1 can achieve full capacity. However, the average queue length
of FCSMA algorithm with both exponential and linear functions is smaller than
117
that of QCSMA (see [73, 23]) with log log function. The reason for the poor delay
performance of QCSMA scheme is that the convergence of underlying Markov Chain
is governed by the mixing time, which is normally large. As in Section 6.2.2, we can
also observe that FCSMA policy has almost the same delay performance as that with
zero hitting time.
6.4
6.4.1
Scenario 3: Scheduling Inelastic Traffic with CSI
Basic Setup
In this and next section, we consider the inelastic traffic scheduling. We assume that
all arrivals have the same delay bound of T time slots, which means that if the data
cannot be served during T slots after it arrives, it will be dropped. For convenience,
we call a set of T consecutive time slots a frame. In the context of fully-connected
networks, we associate each real-time flow with a link, and hence use these two terms
interchangeably. We assume that all data arrives at each link at the beginning of each
frame. Let Al [kT ] denote the amount of data arriving at link l in frame k that are
independently distributed over links and identically distributed over time with mean
λl , and Al [kT ] ≤ Amax for some Amax < ∞. All the remaining data is dropped at the
end of a frame. Each link has a maximum allowable drop rate ρl λl , where ρl ∈ (0, 1)
is the maximum fraction of data that can be dropped at link l. For example, ρl = 0.1
means that at most 10% of data can be dropped at link l on average.
Our goal is to find the schedule {S[t]}t≥1 under the scheduling constraint that at
most one link can be scheduled in each time slot and dropping rate constraint that the
average drop rate of each link should not be greater than its maximum allowable drop
rate. To solve this optimal control problem, we use the intelligent technique in [66] to
introduce a virtual queue Zl [kT ] for each link l to track the amount of dropped data in
118
frame k. Specifically, the amount of data arriving at virtual
 queue l at the end of frame

−1
(k+1)T

X
k is denoted as Dl [kT ], which is equal to Al [kT ] − min
Cl [t]Sl [t], Al [kT ] .


t=kT
We use Il [kT ] to denote the service for virtual queue l at the end of the frame k with
mean ρl λl , and Il [kT ] ≤ Imax for some Imax < ∞. Further, we let Ul [kT ] denote the
unused service for queue l at the end of frame k, which is upper-bounded by Imax .
Then, the evolution of virtual queue l is described as follows:
Zl [(k + 1)T ] = Zl [kT ] + Dl [kT ] − Il [kT ] + Ul [kT ], ∀l.
(6.4.1)
In this and next section, we consider two main scenarios: known channel state
and unknown channel state. For the known channel state case, we assume that the
channel state is constant for the duration of a frame and each link knows CSI at
the beginning of each frame. For the unknown channel state case, we allow that the
channel state changes from time slot to time slot and each link does not know CSI
before each transmission, but can determine how much data has been transmitted at
each slot after we get feedback from the receiver. These assumptions are also adopted
in [32].
We consider the class of stationary policies P that select S[t] as a function
of (Z[kT ], A[kT ], C[kT ]) for the known channel state scenario and a function of
(Z[kT ], A[kT ]) for the unknown channel state scenario in frame k, which, then, form
a Markov Chain, where Z[kT ] = (Xl [kT ])Ll=1 and A[kT ] = (Al [kT ])Ll=1 . If this Markov
Chain is positive recurrent, then the average drop rate will meet the required dropping
rate constraint automatically (see [12]). We define the maximal satisfiable region as
a maximum set of arrival processes for which this Markov Chain is positive recurrent
under any policy. We call an algorithm optimal if it makes Markov Chain positive
recurrent for any arrival process within the maximal satisfiable region.
119
6.4.2
FCSMA Algorithm Implementation
In this subsection, we first characterize the maximal satisfiable region and then propose an optimal FCSMA algorithm with CSI for scheduling inelastic traffic over fading
channels.
Consider the class P of stationary policies that base their scheduling decision on
the observed vector (Z[kT ], A[kT ], C[kT ]) in frame k. The next lemma establishes a
necessary condition for stabilizing the system.
Lemma 6.4.1. If there is a policy P0 ∈ P that can stabilize the virtual queue X,
then there exist non-negative numbers α(a, c; s0 , s1 , ..., sT −1 ) such that
X
s0 ,s1 ,...,sT −1 ∈S
λl (1 − ρl ) <
α(a, c; s0 , s1 , ..., sT −1 ) = 1, ∀a, c,
X
a
X
PA (a)
X
(6.4.2)
PC (c)
c
α(a, c; s0 , s1 , ..., sT −1 ) min
s0 ,s1 ,...,sT −1 ∈S
(T −1
X
i=0
cl sil , al
)
, ∀l,
(6.4.3)
where si = (sil )Ll=1 , PA (a) = Pr{A[t] = a} and PC (c) = Pr{C[t] = c}.
The proof is almost the same as in [89]. The main difference lies in that our
proof deals with the necessary condition for stabilizing virtual queues instead of data
queues as in [89]. We omit it for conciseness. Note that the right hand side of the
inequality (6.4.3) is the average service provided for link l during one frame; while
λl (1 − ρl ) is the average amount of data at link l that needs to be served. Thus, to
meet the maximum allowable drop rate requirement, (6.4.3) should be satisfied. We
define the maximal satisfiable region Λ1 (ρ, C) as follows:
Λ1 (ρ, C) , {A : ∃α(a, c; s0 , s1 , ..., sT −1 ) ≥ 0, such that both (6.4.2) and (6.4.3) satisfy}.
We are now ready to develop an optimal centralized algorithm with CSI for
scheduling inelastic traffic over wireless fading channels.
120
Algorithm 6.4.2 (Centralized Algorithm with CSI). In each frame k, given (Z[kT ],
A[kT ],C[kT ]),
(k+1)T −1
{S∗ [t]}t=kT
∈
argmax
(k+1)T −1
{S[t]}t=kT
X
f (Zl [kT ]) min
l
where f ∈ F.



(k+1)T −1
Cl [kT ]
X
Sl [t], Al [kT ]
t=kT



, (6.4.4)
Remark: In [32], the authors proposed a centralized algorithm with f (x) = x.
Our proposed centralized algorithm is more general, which allows more flexibilities in
distributed implementations.
Next, we establish the optimality of the centralized algorithm with CSI under
certain conditions for function f .
Proposition 6.4.3. If f ∈ B, the Centralized Algorithm with CSI for scheduling
inelastic traffic is optimal over wireless fading channels, i.e., for any arrival process
A ∈ Λ1 (ρ, C), it makes the underlying Markov Chain positive recurrent.
The proof is a generalization of that in [32] and is a special case of that in Proposition 6.4.8, where we use FCSMA techniques to mimic the Centralized Algorithm.
Thus, we omit it for brevity. Even though the above centralized algorithm is optimal,
it cannot directly be applied in practice due to the need of centralized coordination.
Next, we propose a greedy algorithm that is well suited for distributed implementation. To that end, we first give the key identity that facilitates the development of
greedy solutions.
121
Lemma 6.4.4. Let a ≥ 0 and c[t] ≥ 0, ∀t = 0, 1, ..., T − 1. If s[t] ∈ {0, 1}, ∀t, then

(T −1
) T −1
!+ 
t−1


X
X
X
min
c[t]s[t], a =
min c[t], a −
c[j]s[j]
s[t],
(6.4.5)


t=0
t=0
j=0
where (x)+ = max{x, 0}.
Proof. Please see Appendix D.1 for the proof.
Based on Lemma 6.4.4, the objective function in (6.4.4) can be rewritten as


(k+1)T −1


X
X
f (Zl [kT ]) min Cl [kT ]
Sl [t], Al [kT ]


l
t=kT

!+ 
(k+1)T −1
t−1


X
X X
Sl [j]
Sl [t].
f (Zl [kT ]) min Cl [kT ], Al [kT ] − Cl [kT ]
=


t=kT
j=kT
l
(6.4.6)
We can observe that the equation (6.4.6) decouples the scheduling decisions over a
frame and help develop the greedy solutions that are easy to be implemented distributively.
Algorithm 6.4.5 (Greedy Algorithm with CSI). At each time slot t ∈ {kT, kT +
1, ..., (k + 1)T − 1} in frame k, select link lG [t] such that

!+ 
t−1


X
lG [t] ∈ argmax f (Zl [kT ]) min Cl [kT ], Al [kT ] − Cl [kT ]
Sl [j]
, (6.4.7)


l
j=kT
where f ∈ F.
Proposition 6.4.6. The Greedy Algorithm with CSI is an optimal solution to the
problem (6.4.4) and thus is optimal for scheduling inelastic traffic over wireless fading
channels if f ∈ B.
122
The proof is a special case of that in Proposition 6.5.5: the channel state is
constant over a frame in the proof for Proposition 6.4.6, while the channel state
changes from slot to slot in a frame in that for Proposition 6.5.5, which makes it
more challenging to deal with. Next, we expand on the distributed implementation
of the greedy solution by using the FCSMA technique.
Algorithm 6.4.7 (Idealized FCSMA Algorithm with CSI for Scheduling Inelastic
Traffic). At each time slot t ∈ {kT, kT + 1, ..., (k + 1)T − 1} in frame k, choose the
rates
n
o
P
+
min Cl [kT ],(Al [kT ]−Cl [kT ] t−1
j=kT Sl [j])
Rl [t] = g(Zl [kT ])
, ∀l,
(6.4.8)
where g ∈ F.
Note that the Idealized FCSMA Algorithm with CSI does not take the convergence
time into consideration. For the FCSMA Algorithm with CSI, the rates can be
selected as
Rl [t] = g(Zl [kT ])min{Cl [kT ],Jl [t]} ,
for any link l and any t ∈ {kT, kT +1, ..., (k+1)T −1}, where Jl [t] is the remaining data
at link l at the beginning of each time slot t. Next, we will show that the Idealized
FCSMA Algorithm yields the optimal performance. Simulation results show that
both FCSMA and Idealized FCSMA Algorithm have the same performance.
Proposition 6.4.8. If f (x) = log g(x) ∈ B and g(0) ≥ 1, the Idealized FCSMA
Algorithm with CSI for scheduling inelastic traffic is optimal over wireless fading
channels, i.e., for any arrival process A ∈ Λ1 (ρ, C), it makes the underlying Markov
Chain positive recurrent.
123
Proof. The proof follows from the Lyapunov drift argument. However, it is quite
challenging to argue that the Idealized FCSMA Algorithm with CSI mimics the Centralized/Greedy Algorithm with CSI over a frame, which is an obvious case when
T = 1 (see [40]). By properly partitioning the space of weights chosen by the Greedy
Algorithm with CSI within a frame, we tackle this difficulty and refer the reader to
see Appendix D.2 for the details.
6.4.3
Simulation Results
In this subsection, we perform simulations to validate the optimality of the proposed
FCSMA policy with CSI for scheduling inelastic traffic with deadline constraint of
T slots over wireless fading channels. In the simulation, there are L = 10 links and
each frame has T = 5 slots. All links require the maximum fraction of dropped
data to not exceed ρ = 0.3. The amount of arrivals in each frame follows common
Bernoulli distribution that the amount of arrivals equal to T with probability λ.
All links suffer from the ON-OFF channel fading independently with probability
p = 0.8 that the channel is available in each frame. The service for virtual queue also
follows Bernoulli distribution that the maximum available service equals to T with
probability ρλ. Under this setup, we can use the same technique in paper [92] to get
the maximal satisfiable region: Λ1 (ρ, C) = {λ : L(1 − ρ)λ < 1 − (1 − pλ)L }. Through
numerical calculations, we can get the maximal satisfiable region: {λ : λ < 0.038}.
In the simulations, we also compare our proposed FCSMA policy with the QCSMA
algorithm (see [73]) with the log log function.
From Figure 6.5, we can observe that the FCSMA Algorithm with both g(x) = ex
and g(x) = x+1 can achieve maximal satisfiable region. Also, we see that the average
virtual queue length of the FCSMA Algorithm with exponential function is smaller
than that with linear function. However, the meaning of smaller virtual queue length
124
is unclear in this setup. We will explore it in our future research. In addition, we
can observe that the FCSMA Algorithm has almost the same performance as that
with the Idealized FCSMA Algorithm, which indicates that the hitting time should
be negligibly small. Furthermore, the QCSMA algorithm with log log function cannot
even support the arrival rate of λ = 0.001 (i.e., its corresponding virtual queues are
unstable for the arrival rate of 0.001). The reason for the poor performance of the
QCSMA algorithm is that it does not have enough time to converge to the steady
state under fast dynamics of the arrival and channel processes.
1000
Average virtual queue length
900
800
x
Idealized FCSMA with g(x)=e
x
FCSMA with g(x)=e
Idealzied FCSMA with g(x)=x+1
700
600
FCSMA with g(x)=x+1
QCSMA with loglog function
500
400
300
200
100
0
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
Arrival rate
Figure 6.5: Performance of FCSMA for scheduling inelastic traffic with CSI
125
6.5
Scenario 4: Scheduling Inelastic Traffic without CSI
In this section, we consider the inelastic traffic scheduling without CSI over wireless
fading channels. We assume that each link knows how much data has been transmitted at the end of each slot by using per-slot feedback information. The per-slot
feedback complicates the design of distributed scheduling algorithm. But, we still can
find a similar greedy solution as in Section 6.4 and design its distributed algorithm
by using FCSMA techniques.
6.5.1
FCSMA Algorithm Implementation
Consider the class P of stationary policies that base their scheduling decision on the
observed vector (Z[kT ], A[kT ]) in frame k. The next lemma establishes a condition
that is necessary for stabilizing the system.
Lemma 6.5.1. If there is a policy P0 ∈ P that can stabilize the virtual queue Z, then
there exist non-negative numbers α0 (a; s0 ), α1 (a, s0 ; s1 ), ..., αT −1 (a, s0 , ..., sT −2 ; sT −1 ),
such that
X
α0 (a; s0 ) = 1, ∀a,
(6.5.1)
αi (a, s0 , ..., si−1 ; si ) = 1, ∀a, i = 1, 2, ..., T − 1,
(6.5.2)
s0 ∈S
X
si ∈S
λl (1 − ρl ) <
X
PA (a)
a
X
α0 (a; s0 )α1 (a, s0 ; s1 )...
s0 ,s1 ,...,sT −1 ∈S
"
αT −1 (a, s0 , ..., sT −2 ; sT −1 )E min
(T −1
X
i=0
cl sil , al
)#
, ∀l.
(6.5.3)
The proof is almost the same as [89] and hence is omitted here. We define maximal
satisfiable region Λ2 (ρ, C) as follows:
Λ2 (ρ, C) , {A : ∃α0 (a; s0 ), α1 (a, s0 ; s1 ), ...,
αT −1 (a, s0 , ..., sT −2 ; sT −1 ) ≥ 0, such that both (6.5.1), (6.5.2), and (6.5.3) satisfy}.
126
Next, we develop an optimal centralized algorithm without CSI for scheduling inelastic traffic over fading channels.
Algorithm 6.5.2 (Centralized Algorithm without CSI). In each frame k, given
(Z[kT ], A[kT ]), solve the following optimization problem:



(k+1)T

X
X−1
max
E
f (Zl [kT ]) min
Cl [t]Sl [t], Al [kT ]  ,
(k+1)T −1


{S[t]}t=kT
l
(6.5.4)
t=kT
where f ∈ F, and the schedule at each slot is determined after knowing how much
data has been transmitted in the previous slots in each frame.
Remark: In [32], the authors designed a centralized algorithm with f (x) = x. Our
proposed centralized algorithm generalizes this to a large space of functions f , and
allows for more flexibilities in distributed implementations.
Next, we establish the optimality of the centralized algorithm without CSI under
certain conditions for function f .
Proposition 6.5.3. If f ∈ B, the Centralized Algorithm without CSI for scheduling
inelastic traffic is optimal over wireless fading channels, i.e., for any arrival process
A ∈ Λ2 (ρ, C), it makes the underlying Markov Chain positive recurrent.
The proof is a generalization of that in [32] and follows the same argument as that
in Proposition 6.4.8. Thus, we omit it here for brevity. The centralized algorithm
without CSI is quite complicated, since it couples the scheduling decisions in each
frame. Under the per-slot feedback assumption, the optimization problem (6.5.4)
can be solved by using dynamic programming. Based on Lemma 6.4.4, we have the
following key identity:
127
X
f (Zl [kT ]) min
l
(k+1)T −1
X
=
t=kT
X

(k+1)T
X−1

t=kT
f (Zl [kT ]) min
l





Cl [t]Sl [t], Al [kT ]

Cl [t], Al [kT ] −
t−1
X
Cl [j]Sl [j]
j=kT
!+ 


Sl [t]. (6.5.5)
By using (6.5.5), we can get the following backward equation (see [2]) for the
optimization problem (6.5.4).
Backward Equation for (6.5.4): At each slot t ∈ {kT, kT + 1, ..., (k + 1)T − 1}
∗
in frame k, given (Z[kT ], A[kT ]) and {(C[j], S[j])}t−1
j=kT , select link l [t] such that

l∗ [t] ∈ argmax f (Zl [kT ])E min
l
+
max
(k+1)T −1
{S[r]}r=t+1

E min
where f ∈ F.



(k+1)T −1 L
X X
i=t+1



Cl [t], Al [kT ] −
t−1
X
j=kT
f (Zl0 [kT ])
l0 =1
Cl0 [i], Al0 [kT ] −
i−1
X
j=kT
!+ 
!

 Sl0 [i] ,
Sl0 [j]Cl0 [j]

! + 


Sl [j]Cl [j]

(6.5.6)
At first glance, the optimal solution to problem (6.5.6) at each time slot depends
on the future slots and thus is difficult to be implemented distributively. However,
it may still be possible to decouple the scheduling decisions over a frame, since the
channel states are i.i.d. across over time slots. Next, we will show that this is the
case in our setup.
Algorithm 6.5.4 (Greedy Algorithm without CSI). In each time slot t ∈ {kT, kT +
128
1, ..., (k + 1)T − 1} in frame k, given (Z[kT ], A[kT ]) and {(C[j], S[j])}t−1
j=kT , select link
lG [t] such that

!+ 
t−1


X
G

 , (6.5.7)
l [t] ∈ arg max f (Zl [kT ])E min Cl [t], Al [kT ] −
Cl [j]Sl [j]
l



j=kT
where f ∈ F.
Proposition 6.5.5. The Greedy Algorithm without CSI is optimal for problem (6.5.6)
and thus is optimal for scheduling inelastic traffic over wireless fading channels if
f ∈ B.
Proof. Without loss of generality, we consider the frame k = 0. We will show that if
lG [t] satisfies (6.5.7) at time t ∈ {0, 1, ..., T − 1}, then lG [t] is an optimal solution to
the backward equation (6.5.6), that is,

f (ZlG [t] [0])E min



ClG [t] [t], AlG [t] [0] −
t−1
X
j=0
! + 


SlG [t] [j]ClG [t] [j]

!+ 

 Sl [i]
+ max
f (Zl [0])E min Cl [i], Al [0] −
Sl [j]Cl [j]
−1


{S[r]}T
r=t+1 i=t+1
j=0
l



!+ 
t−1


X


≥f (Zm [0])E min Cm [t], Am [0] −
Sm [j]Cm [j]


j=0


!+ 
T
−1
i−1


X X
X
 Sl [i],
f (Zl [0])E min Cl [i], Al [0] −
Sl [j]Cl [j]
+ max
−1


{S[r]}T
r=t+1
T −1 X
X
i=t+1



i−1
X
j=0
l
(6.5.8)
holds for any m 6= lG [t]. Recall that at most one link can be scheduled at each
slot. For ease of exposition, let d , (d[t], d[t + 1], ..., d[T − 1]) generically denote
129
the sequence of feasible links chosen from time slot t to the end of the frame by any
algorithm, where the element d[i] denotes the link that is scheduled at slot i. Note
that the elements in d can be any possible links. The purpose of introducing d is to
simplify the expression of (6.5.8). Let D be the collection of the sequence of selected
links from time slot t to the end of the frame. Let Wd for a given d ∈ D be defined
as
Wd

!+ 
i−1


X

,
f (Zd[i] [0])E min Cd[i] [i], Ad[i] [0] −
Sd[i] [j]Cd[i] [j]


i=t
j=0


!+ 
t−1


X


= f (Zd[t] [0])E min Cd[t] [t], Ad[t] [0] −
Sd[t] [j]Cd[t] [j]


j=0


!+ 
T
−1
i−1


X X
X

 Sl [i],
+
f (Zl [0])E min Cl [i], Al [0] −
Sl [j]Cl [j]



T −1
X
i=t+1
j=0
l
where Sd[i] [i] = 1, and Sl [i] = 0, ∀l 6= d[i], for i = t, t + 1, ..., T − 1.
Let Fl = {d ∈ D : d[t] = l}. Then, (6.5.8) can be rewritten as
max Wd ≥ max Wd , ∀m 6= lG [t].
d∈FlG [t]
d∈Fm
(6.5.9)
Given any m 6= lG [t], we have the following two cases:
(1) If d ∈ Fm includes the element lG [t], then a permutation of d with the first
element being lG [t] should be in FlG [t] . Since the channel states are i.i.d. over
time slots, any permutation of d does not change the value Wd and thus Wd ≤
max We .
e∈FlG [t]
(2) If d ∈ Fm does not include the element lG [t], then it is easy to see that Wd ≤
Wc , where c = (lG [t], d[t + 1], d[t + 2], ..., d[T − 1]). Since c ∈ FlG [t] , we have
Wd ≤ max We .
e∈FlG [t]
Thus, we have max We ≥ Wd , ∀d ∈ Fm , and hence we have the desired result
e∈FlG [t]
(6.5.9).
130
Next, we illustrate the distributed implementation of greedy solutions by using
FCSMA techniques.
Algorithm 6.5.6 (Idealized FCSMA Algorithm without CSI for Scheduling Inelastic Traffic). In each time slot t ∈ {kT, kT + 1, ..., (k + 1)T − 1} in frame k, given
(Z[kT ], A[kT ]) and {(C[j], S[j])}t−1
j=kT , choose the rates
h
n
oi
P
+
E min Cl [t],(Al [kT ]− t−1
j=kT Cl [j]Sl [j])
Rl [t] = g(Zl [kT ])
, ∀l,
(6.5.10)
where g ∈ F.
Note that the Idealized FCSMA Algorithm without CSI does not consider the
impact of the convergence time. For the FCSMA Algorithm without CSI, the rates
can be chosen as
Rl [t] = g(Zl [kT ])E[min{Cl [t],Jl [t]}] ,
(6.5.11)
for any link l and any time t ∈ {kT, kT + 1, ..., (k + 1)T − 1}, where Jl [t] is the
remaining data at link l at the beginning of each time slot t. Next, we will show
that the Idealized FCSMA Algorithm yields the optimal performance. Simulation
results indicate that both FCSMA and Idealized FCSMA Algorithm have the same
performance.
Proposition 6.5.7. If f (x) = log g(x) ∈ B and g(0) ≥ 1, the Idealized FCSMA
Algorithm without CSI for scheduling inelastic traffic is optimal over wireless fading
channels, i.e., for any arrival process A ∈ Λ2 (ρ, C), it makes the underlying Markov
Chain positive recurrent.
The proof is similar to that in Theorem 6.4.8 which considers the inelastic traffic
with CSI over wireless fading channels. We skip it for conciseness.
131
6.5.2
Simulation Results
In this subsection, we perform simulations to validate the optimality of the proposed
FCSMA policy without CSI for scheduling inelastic traffic with deadline constraint
T slots over wireless fading channels. The simulation setup is the same as that in
Section 6.4.3. The main difference is that the fading channels change from slot to
slot. The maximal satisfiable region under this setup is Λ2 (ρ, C) = {λ : L(1 −
ρ)λ < p 1 − (1 − λ)L }. Through numerical calculations, we can get the maximal
satisfiable region: {λ : λ < 0.031}. As in Section 6.4.3, we also compare the Idealized
FCSMA Algorithm with the FCSMA Algorithm and the QCSMA algorithm with
log log function. From Figure 6.6, we can observe the same phenomenon as in 6.4.3.
1000
Average virtual queue length
900
800
700
x
Idealized FCSMA with g(x)=e
x
FCSMA with g(x)=e
Idealzied FCSMA with g(x)=x+1
FCSMA with g(x)=x+1
QCSMA with loglog function
600
500
400
300
200
100
0
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
Arrival rate
Figure 6.6: Performance of FCSMA for scheduling inelastic traffic without CSI
132
6.6
Practical Implementation Suggestions
In the previous sections, we assume that the sensing is instantaneous and the backoff
time is continuous, which excludes the possible collisions. These key assumptions
are important in allowing us to concentrate on the challenging distributed scheduling
problem in time-varying environments without considering the contention resolution
procedure. Yet, in practice, the sensing time is non-zero and the backoff time is
typically a multiple of mini-slots, where a mini-slot is equal to the time required to
detect the data transmission from another link (e.g., in IEEE 802.11b, a mini-slot
should be at least 8µs). In such cases, collisions happen, which reduces the system
throughput.
In this section, we explicitly consider these practical challenges and propose an
easily implementable and efficient algorithm that is similar to the one in [33]. The
basic idea is to quantize the continuous rate Rl [t] into a set of discrete values, where
each discrete value is assigned to a different contention window (CW) size. The
smaller the quantized value is, the larger the corresponding CW size is. Thus, this
can be easily mapped to the “service classes” in IEEE 802.11e. The suggested rate
quantization procedure is as follows: (i) if Rl [t] ≥ Rmax , then let Rl0 [t] = Rmax .
1
1
This corresponds to the first class; (ii) if i−1 Rmax ≤ Rl [t] < i−2 Rmax for some
2
2
1
0
i = 2, 3, ..., N , then let Rl [t] = i−1 Rmax , where N is the number of classes; (iii) if
2
1
Rl [t] < Rmin , N −1 Rmax , then do not start transmissions. Thus, the probability of
2
links accessing the channel in class i is roughly twice as large as that in class i + 1,
which implies that the CW size of class i + 1 should be roughly twice that of class i.
Algorithm 6.6.1 (Discrete-time version of the FCSMA Algorithm). At the beginning
of each slot t, each link l generates a uniformly distributed random variable rl over
133
{0, 1, ..., CW[t] − 1}, where CW[t] is chosen according to the quantized value of rate
Rl [t] as described above. Each link l keeps sensing the channel for rl mini-slots. If
the channel is busy in any one of the first rl mini-slots, then link l suspends its
transmission; otherwise, link l starts its transmission3 from the rlth mini-slot to the
end of this slot. If two or more links have the same backoff time, then a collision
happens in the current slot. The whole process restarts in the next slot.
Without loss of generality, we mainly consider the inelastic traffic scheduling in
this section. The simulation setups for scheduling inelastic traffic with and without
CSI are the same as that in Section 6.4.3 and Section 6.5.2, respectively. We assume
that the coherence time for scheduling inelastic traffic with and without CSI are
500ms and 100ms, respectively. Since a mini-slot is typically 10µs, without loss of
generality, we assume that a time slot contains 10000 mini-slots in both cases. In the
simulations, we let Rmax = e5 , N = 6, and CWi = 32 × 2i−1 , i = 1, 2, ..., N , where
CWi is the CW of class i. We also compare the discrete-time version of the FCSMA
Algorithm with IEEE 802.11 Distributed Coordination Function (DCF). In IEEE
802.11 DCF, the contention window (CW) size depends on whether the transmission
is successful or not, rather than the current system state information. In particular,
the CW for all links are initialized to 32; if the transmission of link l is unsuccessful,
then its CW is doubled until it reaches to the maximum value of 1024; otherwise, its
CW drops to the initial value.
From Figure 6.7a and 6.7b, we can observe that the performance of the discretetime version of the FCSMA Algorithm remains close to that of the FCSMA Algorithm,
3
If the number of links is large, each link uses short packets, such as Request-To-Send (RTS) and
Clear-To-Send (CTS) in IEEE 802.11b, to contend for the wireless channel, which will significantly
reduce the cost of a collision.
134
500
Average virtual queue length
Average virtual queue length
500
400
300
200
FCSMA with g(x)=e
x
Discrete-time version of FCSMA
100
QCSMA with loglog function
IEEE 802.11 DCF
0
1E-5
400
300
200
FCSMA with g(x)=e
x
Discrete-time version of FCSMA
100
QCSMA with loglog function
IEEE 802.11 DCF
0
1E-4
1E-3
0.01
1E-5
Arrival rate
1E-4
1E-3
0.01
Arrival rate
(a) Scheduling inelastic traffic with CSI
(b) Scheduling inelastic traffic without CSI
Figure 6.7: Performance comparison between FCSMA and its discrete-time version
and continue to perform much better than the QCSMA algorithm with log log function and IEEE 802.11 DCF in both scheduling inelastic traffic with and without CSI.
However, we note that if the coherent time is comparable with the maximum CW
size, then, the discrete-time version of the FCSMA Algorithm can perform poorly,
since a non-negligible amount of resources is consumed by the backoff process instead
of the data transmission.
6.7
Summary
In this chapter, we first proposed a Fast-CSMA (FCSMA) Algorithm that quickly
reaches the favorable state in fully connected network topologies. Due to the fast
convergence time, the FCSMA Algorithm exhibits significant advantages over existing
CSMA algorithms for time-varying applications, which are important and popular in
wireless networks. Then, we apply the FCSMA Algorithm to design optimal policies
for scheduling elastic/inelastic traffic with/without CSI over wireless fading channels.
135
CHAPTER 7
EFFICIENT DISTRIBUTED CHANNEL PROBING AND
SCHEDULING DESIGN
Having the knowledge of the channel state information (CSI) at the outset of each
transmission decision can significantly improve the system performance in wireless
fading systems. However, in the presence of many contending users that utilize the
time-varying channel, acquiring CSI per user is not only energy-consuming, but,
more importantly, operationally difficult since it typically requires non-overlapping
pilot training phases to obtain reliable channel quality estimates. Moreover, such persistent probing is likely unnecessary given that only few of them may be allowed to
transmit due to the interference constraints. Yet, opportunistic gains from multi-user
diversity cannot be realized if sufficient CSI is not present. This implies a natural
tradeoff between exploring the multi-user diversity and energy consumption for channel acquisition, and raises a fundamental question on the design of opportunistic
scheduling towards the determination of which subset of users to probe the channel
given limited average probing rates.
The seminal works of Tassiulas and Ephremides (e.g., [91, 92, 89]) have showed
the throughput-optimality of the opportunistic scheduling, which prioritizes activation of links with the largest product of backlog awaiting service and corresponding
channel rate given the full knowledge of CSI, also called Maximum Weight Scheduling
(MWS). Recently, there has been an increasing understanding on efficient scheduling
136
with limited CSI (e.g., [24, 51, 9, 74, 77, 78, 75, 76, 79]). In [24], the authors propose
a two-stage throughput-optimal MWS-type algorithm given partial CSI under the
assumption that only users with known channel states can contend for the channel.
However, they do not answer how to select a subset of users to probe the channel.
In [77, 78], the authors consider the efficient scheduling design without knowing CSI
before each transmission over Markovian fading channels. In [51], the authors also
develop a similar MWS-type algorithm that minimizes the energy consumption. However, the resulting decision space being exponentially increasing with the number of
users appears to limit its applicability in multi-user environments. In fact, existing
works in the design of joint probing and transmission strategies assume centralized
controllers that utilize all state information, and hence are not suitable for distributed
operation in large-scale networks. However, as we shall point out, the design for distributed probing strategies generates difficult challenges that require novel techniques
beyond existing approaches discussed next.
In an exciting thread of work, it has been shown that Carrier Sense Multiple
Access (CSMA) based distributed scheduling strategies (e.g., [33, 73, 23, 81]) can
maximize long-term average throughput for general non-fading wireless topologies.
Yet, the design of distributed schedulers in a fading environment has been observed
to be much more difficult. Nevertheless, when CSI is available, a distributed FastCSMA (FCSMA) algorithm has also been developed in Chapter 6 that guarantees
throughput-optimal scheduling over wireless fading channels in a fully-connected network topology. Yet, to the best of our knowledge, there does not exist a distributed
solution that also accounts for the energy and operational limitations in the CSI
acquisition.
With this motivation, in this chapter, we address the problem of distributed joint
probing and transmission scheduling when users have heterogeneous loads, probing
137
rate constraints, and channel statistics. The following items list our main contributions along with references on where they appear in this chapter:
• In Section 7.2, we study an important basic setup with many users sharing a
common resource that motivates the rest of the work by illustrating that a small
probing rate is sufficient to achieve almost the same performance as the case when all
users continuously probe their channels. Yet, it is also observed that simplistic randomized solutions will under-perform, thus motivating more sophisticated distributed
solutions.
• In Section 7.3, we first characterize the capacity region given the allowable probing rate for general fading channels. Then, we develop a throughput-optimal joint
probing and transmission algorithm assuming a centralized controller. This algorithm, while impractical as is, forms the basis for the subsequent design of algorithms
that are suitable for distributed operation.
• In Section 7.4, based on the maximum-minimums identity [83], we first develop
a novel Sequential Greedy Probing (SGP) algorithm where users probe the channel
sequentially. Then, we show that the SGP algorithm can get the optimal probing
schedule, leading to throughput-optimal performance over symmetric and independent ON-OFF fading channels.
• In Section 7.5, we introduce and analyze a Modified SGP (MSGP) algorithm
that is adapted to general fading channels, and explicitly characterize the efficiency
ratio that it achieves as an explicit function of the channel statistics and rates. The
efficiency ratio is tight for symmetric and independent ON-OFF channels.
• In Section 7.6, we utilize the FCSMA strategy [40] to develop distributed implementations of proposed greedy algorithms, and analyze the performance of the
resulting algorithm.
138
7.1
Problem Formulation
We consider a single-hop fading network with L links, where at most one link is allowed to transmit in each time slot. We assume that the channel for each link has
M +1 possible rates c0 , c1 , c2 , ..., cM , where c0 < c1 < c2 < ... < cM and c0 = 0. We assume that C[t] = (Cl [t])Ll=1 are independently and identically distributed (i.i.d.) over
time, with plj , Pr{Cl [t] = cj }, ∀l = 1, ..., L; j = 0, 1, ..., M . We reasonably assume
that the channel for each link is unavailable with a strictly positive probability1 , that
is, pl0 > 0, ∀l. In the rest of chapter, we interchangeably use “link” and “user”.
In order to get CSI, each user needs to probe the channel by transmitting small
control packets. We denote the probing schedule as X = (Xl )Ll=1 , where Xl = 1 if
user l probes the channel and Xl = 0 otherwise. We also treat X as a set of probing
users. Let X be the collection of probing schedules. Recall that we use S = (Sl )Ll=1
to denote a feasible schedule, where Sl = 1 if link l is active at slot t and Sl = 0
otherwise.
If the user does not probe the channel at the beginning of each time slot, it may
underestimate the channel rate or may even fail to transmit due to a bad channel
condition. Thus, it is reasonable to assume (as in [24]) that each user will not start
a transmission if it does not observe the channel state at the beginning of each time
slot. We denote the allowable probing rate for each user l as ml ∈ (0, 1], ∀l, which
puts an upper bound on the average number of probing operations that each user is
allowed to make. This bound, as noted in the introduction, may be due to energy or
operational constraints associated with the channel estimation operation.
1
In practice, the probing packets and data packets are transmitted in low-rate (e.g., 1Mbps in
IEEE 802.11b) and high-rate (e.g., 2/5.5/11Mbps in IEEE 802.11b) respectively, which implies
that the transmission of probing packets requires lower signal-to-noise-ratio than that of data
packets. Thus, it is reasonable to assume that when the channel is very poor, the user can still
probe the channel but cannot transmit the data packets.
139
Recall that Ql [t] denoting the queue length at link l in time slot t. Then, the
evolution of data queue l is described as follows.
Ql [t + 1] = (Ql [t] + Al [t] − Xl [t]Sl [t]Cl [t])+ , ∀l.
(7.1.1)
Our goal is to find an efficient joint probing and transmission schedule {X[t], S[t]}t≥1
under the scheduling constraint that at most one user can be scheduled at each time
slot and probing constraint that the average probing rate of each user should not
be greater than its allowable probing rate. A key difficulty in the solution of this
problem is that the information available at the transmission scheduling decision S[t]
critically depends on the previously made probing decision X[t], which in turn must
be performed distributively with only local information. We will address the problem
of optimal centralized control, and then return to the distributiveness challenge.
We define the capacity region as a maximum set of arrival rate vectors λ = (λl )Ll=1
for which the system is stable and the average probing rate of each user is no greater
than its allowable probing rate under any policy. We call an algorithm optimal if
it can make the system stable for any arrival rate vector that lies strictly inside the
capacity region. An algorithm can achieve the efficiency ratio ρ if it can stabilize the
system for any λ strictly within a fraction ρ of the capacity region. Next, we study
a basic setup that motivates further investigations.
7.2
A Motivating Scenario
Here, we consider symmetric and independent ON-OFF fading channels with probability p of each channel being ON to support a unit rate in each time slot. Assume that
each user has a uniform arrival rate λ and uniform allowable probing rate m ∈ (0, 1].
Thus, all users should be expected to have the same maximum achievable rate, which
140
is denoted by λmax (m). The next proposition explicitly characterizes λmax (m) under
any strategy with a long-term average as a piece-wise linear function of m.
Proposition 7.2.1. For the above setup, the maximum supportable arrival rate under
any policy with a well-defined long term average is characterized as follows:
λmax (m) = mp, if 0 ≤ m ≤
λmax (m) =
1
;
L
1
l
1
+ (m − )p(1 − p)l − (1 − p)l ,
L
L
N
l
l+1
if ≤ m ≤
, l = 1, ..., L − 1.
L
L
Proof. See Appendix E.1 for the proof.
Maximum Throughput λmax(m)
0.5
L=2
L=5
L=10
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
Probing rate m
Figure 7.1: Maximum throughput under different number of users
Figure 7.1 illustrates λmax (m) as a function of the allowable probing rate m for
a range of the number of users, L, when p = 0.8. An interesting observation is that
141
when the number of users increases a small probing rate appears enough to achieve
almost the same maximum achievable rate as the case when all users always probe
their channels, i.e., when m = 1. This observation can be accurately captured in the
following corollary.
Corollary 7.2.2. The maximum achievable throughput λmax (m) approaches the upper
limit λmax (1) asymptotically as L increases as long as the scaled probing rate mL
diverges, however slowly. More explicitly, we have
)
λmax ( bh(L)c
L
= 1,
lim
L→∞
λmax (1)
(7.2.1)
where h is any non-negative and non-decreasing function with h(x) ≤ x, ∀x, and
lim h(x) = ∞, and byc is the maximum integer that cannot be greater than y.
x→∞
Proof. From Proposition 7.2.1, we get λmax (
1 − (1 − p)bh(L)c
bh(L)c
)=
. Then, we
L
L
λmax ( bh(L)c
)
1 − (1 − p)bh(L)c
L
= lim
have lim
= 1.
L→∞
L→∞
λmax (1)
1 − (1 − p)L
Note that h(x) can be log x or log log x. Thus, when the number of users is
bh(L)c
, however small, is enough to guarantee the good
large, the probing rate
L
performance. In practice, we are interested in the design of a distributed probing
and scheduling algorithm that can support the maximum achievable rate. One may
be inclined to suggest a natural Randomized Probing (RP) policy whereby each user
independently probes the channel with probability m. From [92], the maximum
1
achievable throughput of RP policy is given by
1 − (1 − mp)L .
L
Figure 7.2 compares this rate to the maximum achievable rate by any policy to
demonstrate that the RP policy falls short of reaching the maximum achievable rate,
especially for small allowable probing rates. This motivates us in the rest of the work
to develop more sophisticated algorithms that can support the maximum achievable
rates.
142
0.1
0.09
0.08
Theoretical Maximum Throughput
Randomized Probing Policy
Throughput
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.2
0.4
0.6
0.8
1
Probing rate m
Figure 7.2: Throughput performance of RP policy
7.3
Optimal Centralized Probing and Transmission
In this section, we first study the capacity region given the allowable probing rate in
a general fading channel. Then, we propose a centralized probing and transmission
algorithm that supports any throughput in it.
7.3.1
Characterization of the Capacity Region
The next lemma gives the capacity region Λ(m, C) under the allowable probing rate
vector m = (ml )Ll=1 in a general fading channel C.
Lemma 7.3.1. The capacity region Λ(m, C) is a set of arrival rate vectors λ such
143
that there exist non-negative numbers α(x) and β(x, c; s) satisfying
λl ≤
X
X
α(x)
x
X
P (C[t] = c)
c
X
β(x, c; s)xl cl sl , ∀l,
(7.3.1)
s∈S
β(x, c; s) = 1, ∀x, c,
(7.3.2)
s∈S
X
α(x) = 1,
(7.3.3)
α(x)xl ≤ ml , ∀l.
(7.3.4)
x
X
x
Proof. See Appendix E.2 for the proof.
In (7.3.1), the right-hand-side (RHS) is the total average service provided for each
user and the left-hand-side (LHS) is just the average arrival rate. Thus, to stabilize
the data queue, (7.3.1) should be satisfied. In (7.3.4), the LHS is the average probing
rate for each user and the RHS is the allowable probing rate for each user. To meet
the constraint of allowable probing rates, (7.3.4) should be satisfied.
Next, we characterize the equivalent capacity region for ON-OFF fading channels,
which will be useful in the performance analysis of the algorithm proposed in the next
section in general fading channels.
Lemma 7.3.2. For the case of ON-OFF fading channels, the capacity region Λ(m, C)
is equivalent to the following region Γ(m, C) which is a set of arrival rate vectors
λ such that there exist non-negative numbers α(x) satisfying: for any A ⊆ L ,
{1, 2, · · · , L},
X
i∈A
X
λl ≤ 1 −
X
α(x)
x
α(x)xl ≤ ml , ∀l,
X
Pr{C[t] = c}1{xl cl =0,∀l∈A} ,
(7.3.5)
c
(7.3.6)
x
X
α(x) = 1.
(7.3.7)
x
where 1{·} is the indicator function.
144
Remark: If a random probing schedule X = (Xl )Ll=1 has the probability distribution α(x), then
X
x
α(x)
X
Pr{C[t] = c}1{xl cl =0,∀l∈A} = Pr{Xl Cl [t] = 0, ∀l ∈ A}
(7.3.8)
c
In addition, since
E[max Xl Cl [t]] = 1 − Pr{Xl Cl [t] = 0, ∀i ∈ A},
i∈A
(7.3.9)
(7.3.5) is equivalent to
X
l∈A
λl ≤ E[max Xl Cl [t]],
l∈A
∀A ⊆ L.
(7.3.10)
Proof. See Appendix E.3 for the proof.
7.3.2
An Optimal Joint Probing and Transmission Algorithm
To obtain the optimal centralized joint probing and transmission algorithm, we use
the standard technique in [66] to introduce and guarantee stability of a virtual queue
for each user that conveniently measures the degree of violation of the average probing
constraint. Specifically, we let Zl [t] denote the virtual queue length for user l at the
beginning of slot t. The number of packets entering the virtual queue l at slot t is
just Xl [t]. We use Il [t] to denote the service for virtual queue l at slot t that are
i.i.d. over time with E[Il [t]] = ml , and E[Il2 [t]] ≤ Imax for some Imax < ∞. Then, the
evolution of the virtual queue l is as follows:
Zl [t + 1] = (Zl [t] + Zl [t] − Il [t])+ , ∀l.
(7.3.11)
E[Zl [T ]]
= 0.
T →∞
T
If the virtual queue l is mean rate stable, then, by using Theorem 2.5 in [66], the
We say that virtual queue l is mean rate stable if it satisfies lim
average probing rate constraint of user l is automatically satisfied. Thus, we aim
to design a joint probing and transmission policy that provides strong stability for
145
data queues and mean rate stability for virtual queues under any arrival rate vector
strictly within the capacity region Λ(m, C).
Algorithm 7.3.3 (Joint Probing and Transmission (JPT) Algorithm). In each slot
t, given (Q[t], U[t]), perform:
(1) Probing Decision: select the probing vector X∗ [t] as
L
i X
X [t] ∈ argmax E max Ql [t]Xl Cl [t] −
Zl [t]Xl
∗
X
h
l
l=1
!
,
(7.3.12)
(2) Transmission Scheduling Decision: After the channel states of the selected users
are probed, schedule the transmission of user l∗ [t] that satisfies
l∗ [t] ∈ argmax Ql [t]Xl∗ [t]Cl [t].
(7.3.13)
l
Remark: Since at most one user can be scheduled at each time slot, we can also
interpret l∗ as the index such that Sl∗∗ [t] = 1, where
∗
S [t] ∈ argmax
S∈S
L
X
Ql [t]Xl∗ [t]Cl [t]Sl [t].
l=1
In the JPT algorithm, we first need to solve the optimization problem (7.3.12) to
get the optimal probing schedule X∗ [t] in the probing stage at slot t. Then, we need
to solve the optimization problem (7.3.13) to get the optimal transmission schedule
in the transmission stage given the optimal probing schedule X∗ [t] and the observed
channel states. Next, we will show that the JPT algorithm is optimal in the sense
that it can stabilize the system for any arrival rate vector strictly within the capacity
region.
Proposition 7.3.4. The JPT algorithm is optimal, i.e., for any arrival rate λ ∈
Int(Λ(m, C)), the JPT algorithm stabilizes the system subject to the average probing
rate constraints.
146
Proof. See Appendix E.4 for the proof.
Even though the JPT algorithm is optimal, it cannot directly be applied in practice due to the complexity of computing an optimal probing schedule and the need of
centralized coordination. In Chapter 6, we proposed a distributed FCSMA algorithm
over a wireless fading channel in a fully-connected network topology. We can use a
similar technique as in Chapter 6 to solve transmission scheduling component (7.3.13)
of the JPT algorithm distributively if we know the optimal probing schedule. However, how to reduce the complexity of computing an efficient probing schedule and
implement it in a distributed way still remains an open question. Next, we develop a
sequential greedy algorithm that is well-suited for distributed computation of (7.3.12)
and analyze its performance. From now on, we always use the well-known MWS algorithm or its distributed variants (e.g., the FCSMA algorithm) in the transmission
stage.
7.4
Sequential Greedy Probing Policy and Analysis
In this section, we propose a sequential greedy algorithm for the probing component
of the JPT algorithm, which can be implemented distributively as we will explain
in Section 7.6. Then, we show that it can get an optimal probing schedule in a
symmetric and independent ON-OFF fading channel.
7.4.1
A Sequential Greedy Probing Algorithm
We need to establish some new notations to introduce our proposed algorithm. For
any non-empty set E ⊆ L (recall that L = {1, 2, · · · , L}), we define the function
f (E, e) as follows:
f (E, e) , E[max min{Ql Cl , Qe Ce }],
l∈E
147
(7.4.1)
where e ∈
/ E. Appendix E.5 explores some properties of f (E, e) over a symmetric
and independent ON-OFF fading channel. Here, it is worth noting that, by using the
maximum-minimums identity [83], f (E, e) can be computed recursively.
Also, let φl , E[Ql Cl ] − Zl , ∀l ∈ L, and consider a set F ⊆ N of probing users and
r ∈ N \ F. By using the maximum-minimums identity, we have the key relationship:
!
X
X
E max
Ql Cl −
Zl = E[max Ql Cl ] −
Zl + φr − f (F, r). (7.4.2)
S
l∈F
{r}
l∈F
S
{r}
l∈F
l∈F
For the derivation of this identity, please see Appendix E.6 for details. Based on
the iterative equation (7.4.2), we can define a directed graph G, where each probing
X
schedule X denotes a node with an associated value of E[max Ql Cl ] −
Zl . Thus,
l∈X
l∈X
X also represents the collection of all nodes. Since each node is a binary vector of N
dimensions, we have |X | = 2L , where | · | denotes the cardinality of the set. For two
nodes X1 and X2 , there is a directed link from node X1 to node X2 if and only if
X1 is a subset of X2 with the cardinality |X2 | − 1. Let q = X2 \ X1 . We define the
weight of a link from node X1 to node X2 as φq − f (X1 , q). Let E be the collection
of edges, and let node X0 denote the all-zero probing schedule where no user probes
the channel, and thus the value of node X0 is 0. We say node X is in level |X| in the
directed graph G = (X , E). Finally, let I = {l ∈ L : φl > 0}. Figure 7.3 shows the
directed graph for L = 3.
Given the directed graph G, the optimization problem (7.3.12) is equivalent to
finding a path with the largest total weight emanating from node X0 . By noting
that the directed graph is acyclic, if we delete all edges with the negative weights
and negate the weight of remaining edges, the optimization problem (7.3.12) is also
equivalent to finding a shortest path from node X0 in the directed graph, which can
be solved polynomially by Dijkstra’s algorithm [10]. However, Dijkstra’s algorithm
always goes back and forth to find a shortest path, which is not allowed in the probing
problem since once a node probes its channel its energy is consumed. Fortunately,
148
(1,1,1)
ӿ4! f({1,2},3)
ӿ1
ӿ2
(1,1,0)
f({1,3},2)
(1,0,1)
(0,1,1)
ӿ1
ӿ1
ӿ2
f({1},2)
ӿ3
f({3},1)
f({2},1)
ӿ3
f({1},3)
(1,0,0)
f({2},3)
(0,1,0)
ӿ1
f({2,3},1)
ӿ2
ӿ2
f({3},2)
(0,0,1)
ӿ3
(0,0,0)
Figure 7.3: The directed graph G = (X , E) when L = 3
the weights of edges are highly correlated with each other through the queue lengths.
Thus, it is possible to design a sequential greedy probing algorithm as follows that
can still yield good performance.
We first divide each time slot into a control slot and a data slot. The purpose of
the control slot is to determine the probing schedule to get the channel state used
for data transmission in the data slot. To achieve this goal, we further subdivide the
control slot into N mini-slots.
Algorithm 7.4.1 (Sequential Greedy Probing (SGP) Algorithm). (1) In the first
mini-slot, select user l1 such that l1 ∈ argmax φl . User l1 probes the channel while
l∈I
also announcing its queue-length. If no users probe the channel, then all users keep
silent in the rest of current slot and restarts in the next time slot.
149
(2) In the k th (1 < k ≤ L) mini-slot, select user lk such that
lk ∈
argmax
(φl − f ({l1 , ..., lk−1 }, l)) .
(7.4.3)
l∈I\{l1 ,...,lk−1 }
If φlk > f ({l1 , ..., lk−1 }, lk ), then user lk probes the channel while also announcing its
queue length. Otherwise, all users stop probing and all probing users with non-zero
channel states are candidates for transmission scheduling as dictated in (7.3.13).
Remark: In the SGP algorithm, we require that each probing user announces its
queue-length information, which may cause the heavy message exchange overhead.
Motivating by [98] that utilizes the delayed queue length information to provide the
fair resource allocation, we may only allow the transmitting user to announce its
queue-length information, and all users utilize this delayed queue length information
to calculate the probing schedule. Our simulation results indicate that this modified
version of the SGP algorithm does not degrade the system performance.
7.4.2
Optimality of the SGP Algorithm for Symmetric Channels
In this subsection, we will show that the SGP algorithm can achieve the optimal
value of the maximization problem (7.3.12) for symmetric and independent ON-OFF
fading channels. The next lemma and subsequent corollaries pave the path to this
result by establishing a key property of the directed graph G.
Lemma 7.4.2. For symmetric and independent ON-OFF fading channels with an
ON probability p, if node A∗ is the unique node with maximum value in level |A∗ | in
graph G, then the node with maximum value in level |A∗ | − 1 should be a subset of
node A∗ .
Proof. Let A be the class of the nodes in level |A∗ |; D be the class of nodes in level
150
|A∗ | − 1; and B be the class of nodes that are a subset of node A∗ in level |A∗ | − 1.
Thus, we need to show that ∃B∗ ∈ B such that
B∗ ∈ argmax E[max Ql Cl ] −
l∈D
D∈D
X
Zl
l∈D
!
.
(7.4.4)
We prove it by contradiction. Suppose there exists a D∗ ∈ D \ B such that
!
X
D∗ ∈ argmax E[max Ql Cl ] −
Zl .
l∈D
D∈D
(7.4.5)
l∈D
Let d ∈ arg min
Ql and B , A∗ \ {d}. Since A∗ is the unique node with the
l∈A∗ \D∗
[
maximum value in level |A∗ |, node D∗ {d} ∈ A does not have the maximum value
in level |A∗ | and thus we have
E[ max
Ql Cl ] −
S
∗
l∈D
{d}
X
l∈D∗
S
Zl < E[max∗ Ql Cl ] −
l∈A
{d}
X
Zl .
l∈A∗
According to the iterative equation (7.4.2), we have
E[max∗ Ql Cl ] −
l∈D
X
Zl + φd − f (D∗ , d)
l∈D∗
< E[max Ql Cl ] −
l∈B
X
Zl + φd − f (B, d).
(7.4.6)
l∈B
Since D∗ is one of the optimal solutions to (7.4.5), we have
E[max∗ Ql Cl ] −
l∈D
X
Zl ≥ E[max Ql Cl ] −
l∈B
l∈D∗
X
Zl .
(7.4.7)
l∈B
Hence, to let (7.4.6) hold, we should have f (D∗ , d) > f (B, d). To arrive at a contradiction, we need to show that f (D∗ , d) ≤ f (B, d), which is not at all obvious and
requires a challenging investigation. Please see Appendix E.7 for details.
Corollary 7.4.3. For symmetric and independent ON-OFF fading channels, let A∗
be one of nodes with maximum value in level |A∗ | in the directed graph G, then the
node with maximum value in level |A∗ | − 1 should be in the union of subsets of nodes
with maximum value in level |A∗ |.
151
Proof. The proof is exactly the same as in the proof for Lemma 7.4.2 except that
B denotes the class of nodes in level |A∗ | − 1 that are the subset of all nodes with
maximum value in level |A∗ |.
Corollary 7.4.4. For symmetric and independent ON-OFF fading channels, if node
A∗ has the maximum value in level |A∗ |, then there exists a node with maximum
value in level |A∗ | + 1 that is the superset of node A∗ .
Proof. If there is only one node with maximum value in level |A∗ | + 1, then the result
directly follows from Lemma 7.4.2. If there are multiple nodes with maximum value
in level |A∗ | + 1, then the result follows from Corollary 7.4.3.
It is important to note that Lemma 7.4.2 and its corollaries hold regardless of
whether the edge weights are positive or negative valued. This property will be
crucial in the proof of the following main result of this subsection.
Proposition 7.4.5. The SGP algorithm can achieve the optimal value of the maximization problem (7.3.12) in symmetric and independent ON-OFF fading channels.
Proof. If there are multiple nodes with optimal value in the directed graph G, then we
just consider the nodes with optimal value in the lowest level, say level K. Thus, for
any node with the level lower than K, its value is strictly less than that of the nodes
with optimal value in level K. Next, we first assume that the SGP algorithm can
continue to work even when it picks an edge with a non-positive weight. Under this
assumption, we can show that the SGP algorithm sequentially selects users l1 , l2 , ..., lK
to get to the node A∗ = {l1 , l2 , ..., lK }, which has the optimal value in the directed
graph G. Finally, we will show that all edges in a path leading to node A∗ have a
strictly positive weight and the SGP algorithm will stop at node A∗ .
Note that the proposed SGP algorithm first picks the user l1 , where the node
{l1 } has the maximum value in level 1. By corollary 7.4.4, there exists a node with
152
maximum value in level 2 that is a superset of node {l1 }. Since the SGP algorithm
picks an edge with maximum weight φl2 − f ({l1 }, l2 ), the node {l1 , l2 } has the maximum value in level 2. By using similar argument, we can see that the SGP algorithm
sequentially selects users l1 , l2 , ..., lK to get to the node A∗ in level K, where the node
{l1 , ..., lj } has the maximum value in level j for each j = 1, ..., K. Since node A∗ has
the maximum value in level K and the node with optimal value is in level K, node
A∗ has the optimal value in the directed graph G.
Let G(A∗ ) be the subgraph of G that includes all subsets of the node A∗ and their
corresponding edges. Since node A∗ has the optimal value, we have φl −f (A∗ \{l}, l) >
0, ∀l ∈ A∗ . Indeed, if φk − f (A∗ \ {k}, k) ≤ 0 for some k ∈ A∗ , then according to the
iterative equation (7.4.2), we have
E[max∗ Qj Cj ] −
j∈A
X
j∈A∗
Zj ≤ E[ max
Qj Cj ] −
∗
j∈A \{k}
X
Zj ,
j∈A∗ \{k}
which contradicts that the value of a node with the level less than K is strictly smaller
than that of node A∗ . According to the definition of the function f (see equation
(7.4.1)), it is easy to see that if E ⊆ F, then f (E, e) ≤ f (F, e), where e ∈
/ F. Thus,
for any given l ∈ A∗ and any H ⊆ A∗ \ {l}, we have
φl − f (H, l) ≥ φl − f (A∗ \ {l}, l) > 0.
(7.4.8)
Thus, all edges in the subgraph G(A∗ ) have the strictly positive weight. Hence, there
always exists an edge with strictly positive weight from node {l1 , ..., lk } in level k to
node {l1 , ..., lk , lk+1 } in level k + 1 (k = 1, 2, ..., K − 1).
In addition, there is no edge with strictly positive weight from node A∗ in level
K. Indeed, if there is an edge with strictly positive weight from node A∗ in level
K to a node in level K + 1, say node J, then node J should have the value larger
than the optimal value, which contradicts that node A∗ has the optimal value in the
directed graph G. Thus, when the SGP algorithm reaches node A∗ , it stops.
153
In a general wireless fading channel, the SGP algorithm cannot always find the
optimal value of (7.3.12) as in the above symmetric setup, and thus its performance
is unclear. Instead, we consider a Modified SGP (MSGP) algorithm in the next
subsection to show that the MSGP algorithm combined with MWS algorithm in the
transmission stage can at least achieve a constant efficiency ratio.
7.5
The Modified SGP Policy and Analysis
In this section, we consider the more general fading channels and introduce a slightly
modified version of the SGP algorithm studied in the previous section. Then, we
explicitly characterize the efficiency ratio that this modified algorithm is guaranteed
to achieve as a function of the channel statistics and rates.
We assume that the general fading channels satisfy the following assumption.
Assumption 7.5.1. The general fading channels are i.i.d. over time and the events
that the channels have zero rate are independent, that is,
Pr{Cl [t] = 0, ∀l ∈ A} =
Y
Pr{Cl [t] = 0},
∀A ⊆ L.
(7.5.1)
l∈A
Remark: If fading channels are independently over users, then condition (7.5.1)
trivially holds.
To introduce the proposed algorithm, we first let pmin , 1 − max pj0 and pmax ,
j
1 − min pj0 to denote the non-zero rate probability of the worst and the best channel,
j
respectively. Then, we define two identical and independent ON-OFF fading channels
Cmin [t] = (Clmin [t])Ll=1 and Cmax [t] = (Clmax [t])Ll=1 satisfying:
Pr{Clmin [t] = 0} = 1 − pmin ,
Pr{Clmax [t] = 0} = 1 − pmax ,
Pr{Clmin [t] = c1 } = pmin , ∀l;
Pr{Clmax [t] = cM } = pmax , ∀l,
154
where we recall that c1 and cM are, respectively, the smallest and largest transmission
rates achievable for any user.
Algorithm 7.5.2 (Modified SGP (MSGP) Algorithm). MSGP algorithm operates
exactly the same as the SGP algorithm, except that steps are computed assuming the
identical and independent ON-OFF fading channels Cmin .
Remark: The MSGP algorithm differs from the SGP algorithm only in the assumed
channel statistics and rates.
The following lemma gives the key relationship between the general fading channels under Assumption 7.5.1 and two constructed identical and independent ON-OFF
fading channels.
Lemma 7.5.3. For general fading channels under Assumption 7.5.1, the following
relationship
E[max al Clmin [t]] ≤ E[max al Cl [t]] ≤ E[max al Clmax [t]], ∀t,
l
l
l
(7.5.2)
holds for any constants al ≥ 0, ∀l.
Proof. See Appendix E.8 for the proof.
In the following lemma, we give the relationship of the capacity region for fading
channels satisfying condition (7.5.2).
Lemma 7.5.4. Let CI [t] = (ClI [t])Ll=1 and CII [t] = (ClII [t])Ll=1 represent two fading
channels. If
E[max al ClI [t]] ≤ E[max al ClII [t]], ∀t,
l
l
155
(7.5.3)
holds for any constants al ≥ 0, ∀l, then, we have
Λ(m, CI ) ⊆ Λ(m, CII ).
(7.5.4)
Proof. See Appendix E.9 for the proof.
The following lemma reveals an interesting monotonicity property of the mean of
the maximum of a set of binary random variables, and will be used in the subsequent
main result.
Lemma 7.5.5. Let A ⊆ L and X = (X1 , ..., X|A| ) be a zero-one random vector.
Wl , ∀l = 1, ..., |A|, are independent and identical Bernoulli random variables with
1
parameter p. Then, h(p) , E[max Xl Wl ] is a non-increasing function.
p l∈A
Proof. See Appendix E.10 for the proof.
Proposition 7.5.6. The MSGP algorithm combined with the MWS algorithm in the
transmission stage (see equation (7.3.13)) can at least achieve an efficiency ratio
pmin c1
in general fading channels under Assumption 7.5.1.
ρ,
pmax cM
Proof. See Appendix E.11 for the proof.
Remarks: (1) In symmetric and independent ON-OFF channels, the MSGP algorithm
can achieve the full capacity region, which matches the result in Proposition 7.4.5.
(2) Even though the efficiency ratio is low in highly asymmetric fading channels,
the MSGP algorithm still performs well in practice as we can see in the simulation
section.
7.6
Distributed Implementation with Fast-CSMA
Here, we expand on the distributed implementation of the greedy sequential algorithms developed in the previous two sections by using the FCSMA technique developed in [40]. Since the MSGP algorithm has the same performance as the SGP
156
Algorithm in the special case of symmetric ON-OFF channels, we focus on the distributed implementation of the MSGP Algorithm in the control slot.
Algorithm 7.6.1 (Distributed MSGP (DMSGP) Algorithm). In the first mini-slot,
each user l with φl > 0 independently generates an exponentially distributed random
variable with rate exp(Gφl ) (G > 0), and starts transmitting a small probing packet
after this random duration unless it senses another transmission before. The user that
grabs the channel transmits its probing packet until the end of the mini-slot. After
probing, all other users know the queue length of the current probing user. If no users
transmit the probing packet during this mini-slot, then all users keep silent in the rest
of current slot and restarts in the next time slot.
In the k th (1 < k ≤ L) mini-slot, the remaining non-probing user l with φl −
f ({l1 , ..., lk−1 }, l) > 0 generates an exponential distributed random variable with rate
exp(G(φl − f ({l1 , ..., lk−1 }, l))) and uses the same produce as in the first mini-slot
to probe the channel. If no users probe the channel in the current mini-slot or the
control slot is over, then all the probing users with the available channel state start
to contend for data transmission.
Remark: Here, we assume that the sensing is instantaneous and the backoff time is
continuous, which excludes the possible collisions. Yet, in practice, the sensing time
is non-zero and the backoff time is typically a multiple of time units, where a time
unit is equal to the time required to detect the transmission from other links. Thus,
we should use the discrete-time version of the FCSMA algorithm, whose performance
is close to its continuous counterpart as shown in [40].
The above procedure leads to a probing schedule XDM SGP by the end of the
control slot, where each selected probing user l knows its channel state Cl . Then, to
157
determine the one that transmits the data packet each probing user l distributively
runs the FCSMA algorithm as described in [40] with parameter exp(Ql Cl ). This
is known to solve the transmission decision (7.3.13) if the queue-lengths are large
enough. In order to establish the performance of such a distributed probing and
transmission algorithm, we need an additional assumption.
Assumption 7.6.2. The channel rates and their corresponding probability for each
user, i.e., cj , ∀j = 1, ..., M and plj , ∀l = 1, ..., L, j = 0, ..., M , are rational numbers.
Proposition 7.6.3. For any ζ > 0 and arrival rate vector λ satisfying λ + ζ ∈
ρInt(Λ(m, C)), with the efficiency ratio ρ given in Proposition 7.5.6, there exists a
design parameter G > 0 such that the DMSGP algorithm, combined with the FCSMA
algorithm in the transmission stage, can support λ subject to the given probing rate
constraints m in general fading channels under Assumptions 7.5.1 and 7.6.2.
Proof. See Appendix E.12 for the proof.
7.7
Simulation Results
In this section, we first study the impact of iterative steps and using the delayed
queue length information (i.e., only the transmitting user broadcasts its queue length
information) on the performance of the SGP algorithm. Then, we compare the performance between the SGP algorithm and the MSGP algorithm in asymmetric ON-OFF
fading channels and symmetric general fading channels. In the simulation, we consider
three different fading models that are i.i.d. over time and independently distributed
over users: symmetric and independent ON-OFF channels with probability p = 0.8
that the channel is available in each time slot; asymmetric ON-OFF channels that
one user has channel availability probability of 0.1 and all others have probability of
0.9 and symmetric general fading channels available to each user with rates 0, 1, 10
158
and corresponding probability 0.1, 0.2, 0.7. All users have the same arrival rate and
require that the allowable probing rate cannot exceed m = 0.4. Without loss of generality, we use arrival process where the number of arrivals in each slot follows Bernoulli
distribution and Poisson distribution when we consider ON-OFF fading channels and
general fading channels respectively.
7.7.1
The Impact of Iterative Steps
In this subsection, we study the impact of iterative steps on the performance of
the SGP algorithm. We consider L = 20 users over a symmetric and independent
ON-OFF fading channel. Under this setup, we can use Proposition 7.2.1 to get the
capacity region Λ = {λ : λ < 0.05}. We use K to denote the maximum allowable
K
K
K
K
K
Average data queue length
100
90
80
70
60
50
Average virtual queue length
iterative steps.
=1
=2
=3
=4
=20
40
30
20
10
0
0.01
0.02
0.03
0.04
0.05
0.06
K
K
K
K
K
0.30
0.25
0.20
0.15
=2
=3
=4
=20
0.10
0.05
0.00
0.01
Arrival rate
=1
0.02
0.03
0.04
0.05
0.06
Arrival rate
(a) Average data queue length
(b) Average virtual queue length
Figure 7.4: Impact of iterative steps
From Figure 7.4a and 7.4b, we observe that the SGP algorithm with unlimited
159
iterative steps can achieve full capacity. In addition, as K increases, the performance
of the SGP algorithm improves. Especially, we can see that four iterative steps are
enough to reach almost optimal performance. This implies that while the original
algorithm may be defined over more steps, in practice, we can limit the iterative steps
to a small number virtually without hurting the throughput.
7.7.2
The Impact of Using Delayed Queue Length Information
In this subsection, we study the impact of using the delayed queue length information
(i.e., each user only have the queue length information of the transmitting user) on the
performance of the SGP algorithm. Figure 7.5a and 7.5a compare the performance
between the SGP algorithm and the SGP algorithm using the delayed queue length
information in the network of L = 5 users over symmetric ON-OFF fading channels.
We can observe that using the delayed queue length information does not affect the
system performance of the SGP algorithm. This promising property allows us to
significantly reduce the overhead of exchanging queue length information under the
SGP algorithm.
7.7.3
The Performance of Greedy Probing Algorithms
In this subsection, we compare the performance among the SGP algorithm, the MSGP
algorithm and the JPT algorithm. We consider L = 5 users. Figure 7.6 and Figure 7.7 compare the performance among the SGP algorithm, the MSGP algorithm
and the JPT algorithm under an asymmetric ON-OFF channel and a symmetric
general fading channel, respectively. From Figure 7.6 and 7.7, we can see that
these algorithms have almost the same throughput performance. Noting that the
JPT algorithm is throughput-optimal, both SGP and MSGP algorithm are probably
throughput-optimal in general fading channels. We leave it for future research.
160
90
80
Average actual queue length
Average actual queue length
100
SGP algorithm with delayed
queue length information
SGP algorithm
70
60
50
40
30
20
10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.5
SGP algorithm with delayed
queue length information
0.4
SGP algorithm
0.3
0.2
0.1
0.0
0.01
Arrival rate
0.02
0.03
0.04
0.05
0.06
Arrival rate
(a) Average data queue length
(b) Average virtual queue length
Figure 7.5: Impact of using delayed queue length information
In addition, we can observe from Figure 7.6 that the SGP algorithm is insensitive
to the channel statistics. Furthermore, from Figure 7.7, we can observe that the
MSGP algorithm has the smallest average actual queue length and virtual queue
length. Thus, while the throughput performance of the SGP algorithm is not sensitive
to the channel rates, its delay performance may be significantly affected by the channel
rates.
7.8
Summary
In this chapter, we considered the distributed channel probing for opportunistic
scheduling under heterogeneous allowable probing rate constraints. We first analyzed a basic scenario with symmetric arrivals and uniform allowable probing rate to
express the maximum achievable throughput as a function of the allowable probing
rate in symmetric and independent ON-OFF fading channels. This result not only
indicates that almost the same opportunistic gains can be achieved with significant
161
Average virtual queue length
Average actual queue length
100
SGP algorithm
90
MSGP algorithm
80
JPT algorithm
70
60
50
40
30
20
10
0
0.010
0.015
0.020
0.025
0.030
0.035
0.040
60
SGP algorithm
50
MSGP algorithm
JPT algorithm
40
30
20
10
0
0.045
0.010
0.015
Arrival rate
0.020
0.025
0.030
0.035
0.040
0.045
Arrival rate
(a) Average data queue length
(b) Average virtual queue length
100
90
80
70
Average virtual queue length
Average actual queue length
Figure 7.6: Impact of asymmetric channel statistics
SGP algorithm
MSGP algorithm
JPT algorithm
60
50
40
30
20
10
0
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
70
MSGP algorithm
JPT algorithm
50
40
30
20
10
0
1.0
Arrival rate
SGP algorithm
60
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Arrival rate
(a) Average data queue length
(b) Average virtual queue length
Figure 7.7: Impact of asymmetric channel rates
162
reductions in probing rates when the number of users is relatively large, but also
points out that a simplistic randomized policy cannot achieve the full opportunistic
gains.
Then, we characterized the capacity region under the heterogeneous probing constraints and provided the centralized throughput-optimal JPT algorithm. Realizing
the operational difficulty of centralized solution, we put effort in developing a novel
SGP algorithm based on the maximum-minimums identity, which is easy for distributed implementation. Also, we showed that the SGP algorithm is optimal in the
crucial scenario of symmetric and independent ON-OFF fading channels. In the case
of more general fading channels, we analyzed a more tractable variant of the SGP
algorithm to obtain its efficient ratio as an explicit function of the channel statistics
and rates and show that this ratio is tight in the symmetric and independent ONOFF fading scenario. Finally, we discussed the distributed implementation of these
greedy probing algorithms by using the FCSMA technique.
163
CHAPTER 8
CONCLUSIONS AND FUTURE WORKS
In this dissertation, we first considered the design of efficient resource allocation algorithms for time-sensitive and dynamic applications that are characterized by secondorder metrics (e.g., convergence speed, mean delay, and service regularity) beyond
the traditional throughput requirements. In particular, we developed efficient algorithms with fast convergence in both underloaded and overloaded systems, and
designed algorithms that achieve maximum throughput, heavy-traffic optimality and
service regularity guarantees. Then, we systematically considered the distributed algorithm design for optimizing the first-order metrics (e.g., throughput and energy
consumption). Our main contributions are summarized as follows.
• We considered the algorithm design with optimal convergence speed in wireless
networks (cf. Chapter 2). By providing universal bounds on the convergence speed
of any scheme, we established the fundamental limits on the convergence speed of
cross-layer algorithms under discrete transmission rate constraints. Using this bound,
we also developed algorithms that achieve the optimal convergence speed. Somewhat
surprisingly, we even showed that the well-known dual algorithm achieves the optimal
convergence speed.
• We conducted the convergence speed analysis in overloaded systems (cf. Chapter 3). Noting the difficulty in the analysis of overloaded systems, we proposed an
interesting “queue reversal” approach that relates the metrics in an unstable system
164
to the metrics of a stable system, for which a rich set of powerful analytical tools is
available.
• We considered the regular scheduling design (cf. Chapter 4). We developed a
maximum-weight type scheduling algorithm, where the weight of each link consists of
its own queue-length and the counter that tracks the time since the last service. We
showed that the proposed algorithm achieves the throughput optimality, heavy-traffic
optimality, and service regularity guarantees. We developed a novel analytical approach for the new dynamics of the introduced counter that is beyond the traditional
queue.
• We focused on the throughput limitation of randomized scheduling algorithms
that are easily implemented distributively (cf. Chapter 5). We revealed the critical
characteristics of the scheduling rule and the network topology in maximizing system
throughput. This result provided both the insights and the machinery for developing
provably efficient resource allocation through randomization.
• We considered the design of distributed scheduling algorithm for time-varying
applications (cf. Chapter 6). Using a combination of CSMA-based scheduling and
quick release of locked decisions, we developed optimal solutions that apply to both
elastic and inelastic traffic scheduling over wireless fading channels. These results
provide one of the first promising means of effectively dealing with time-varying
conditions in efficient resource allocation.
• We investigated the problem of optimally probing and scheduling over wireless fading channels subject to energy limitations (cf. Chapter 7). By using the
Maximum-Minimum Identity, we developed a novel low-complexity solution to the
classic exploration versus exploitation problem. This work allows the network users
explore the opportunistic gains of wireless fading channels while assuring drastic reductions in the channel estimation costs.
165
Our research has opened an interesting avenue to the design of efficient and distributed algorithms for wireless networks. Also, the resulting research will have wider
applications to other complex networks, including smart grid and cloud computing.
In addition, our investigations motivate numerous future research efforts. Next, we
list several possible future research directions.
• In Chapter 3, the fascinating queue reversal result only holds for a single-server
queue. It would be interesting to expand the power of “queue reversal” methodology that enables indirect analysis of overloaded systems through the performance of
appropriately constructed stable systems.
• In Chapter 4, the proposed algorithm inherits the same complexity issue of
the Maximum Weight Scheduling (MWS) Algorithm. The low complexity or the distributed implementations of the proposed algorithm are strongly desirable in practical
networks. Also, it would be interesting to investigate the problem of how to provide
service regularity in present of short-lived or deadline-constrained flows.
• In Chapter 6, our proposed algorithm achieves the optimal performance in fullyconnected networks. It would be interesting to investigate the performance of our
proposed algorithm or develop efficient distributed scheduling algorithms in general
network topologies under time-varying conditions.
• In Chapter 7, we only considered the distributed channel probing algorithm
design over i.i.d. fading channels. It might be interesting to develop channel probing
algorithms over Markovian fading channels.
166
APPENDIX A: PROOFS FOR CHAPTER 2
A.1
Proof of Lemma 2.3.2
T
c
1 X
r
I[t] − r ≤
is equivalent to
Note that T
T
t=1
−cr ≤
T
X
I[t] − T r ≤ cr .
(A.1.1)
t=1
(i) If I[t + 1] ≤ ai , then we have
T +1
X
I[t] − (T + 1)r ≤
T
X
I[t] − T r + ai − r
t=1
t=1
≤ c r + ai − r
(A.1.2)
< −cr ,
(A.1.3)
where (A.1.2) follows from the inequality (A.1.1), and (A.1.3) follows from
r − ai
cr <
.
2
(ii) If Il [t + 1] ≥ ai+1 , then we have
T +1
X
I[t] − (T + 1)r ≥
t=1
T
X
I[t] − T r + ai+1 − r
t=1
≥ −cr + ai+1 − r
(A.1.4)
> cr ,
(A.1.5)
where (A.1.4) follows from the inequality (A.1.1), and (A.1.5) follows from
ai+1 − r
cr <
.
2
Thus, by combining (A.1.3) and (A.1.5), we have the desired result.
167
A.2
Proof of Proposition 2.3.3
We first show the following claim:
()
∈ (Bl,i , Bl,i+1
() is not an integer for some
)!i = 1, 2, ..., Kl − 1, then,
()
()
rl − Bl,i Bl,i+1 − rl 1
there exists a c ∈ 0, min
,
,
and a positive integer2
2
2
valued increasing sequence {Tk }k≥1 such that
Claim 1. If rl
()
()
|dl [Tk ] − rl | ≥
c
,
Tk
∀k ≥ 1,
(A.2.1)
holds for any sample path of departure rate vector sequence {D() [t]}t≥1 .
This implies that if the long-term average injection rate vector r() at least contains
()
one non-integer-valued coordinate, e.g., rl ∈ (Bl,i , Bl,i+1 ) is not an integer for some
non-negative integer i, then, there exists a c and a positive integer-valued increasing
sequence {Tk }k≥1 , which are defined in claim 1, such that
()
()
()
()
kd [Tk ] − rl k ≥ |dl [Tk ] − rl | ≥
c
,
Tk
∀k ≥ 1,
holds for any sample path of departure rate vector sequence {D() [t]}t≥1 .
1
()
Next, we prove Claim 1 to complete the proof. Indeed, since c < and rl > 0,
2
we have
!
n+1
c
n
c
1 − 2c
− () −
+ () = () > 0,
(A.2.2)
()
()
rl
rl
rl
rl
rl
which implies
n
()
rl
−
c
()
rl
,
n
()
rl
+
c
()
rl
!
\
n+1
()
rl
()
−
c
()
rl
,
n+1
()
rl
+
c
()
rl
!
= ∅,
()
r − Bl,i
r
for any non-negative integer n. Since c < l
≤ l , we have
2
2
!
n
c
n
c
2c
+ () −
− () = () < 1,
()
()
rl
rl
rl
rl
rl
168
(A.2.3)
n
which implies that each interval
()
−
rl
c
()
,
rl
!
n
()
rl
+
c
()
rl
!
can at most contain one
non-negative integer.
n
c n
c
If the interval
− () , () + () contains some non-negative integer T for
()
rl
rl rl rl
c
n
() ()
some non-negative integer n, i.e., − rl ≤ , where rl ∈ (Bl,i , Bl,i+1 ) for some
T(
T
)!
()
()
rl − Bl,i Bl,i+1 − rl 1
,
,
, then, by taking
i = 1, 2, ..., Kl − 1 and c ∈ 0, min
2
2
2
n+l
c
() 0
the set I = N , {0, 1, 2, ...} in Lemma 2.3.2, we have − rl ≥
for any
T +1
! T +1
n+l
c n+l
c
positive integer l, which implies T +1 ∈
/
− () , () + () , for any positive
()
r!
rl
rl
rl
l
∞
[
n
c n
c
integer l. Thus,
− () , () + () does not cover all positive integers and
()
rl
rl rl
rl
n=0
thus there exists a sequence of positive integers {Tk }∞
k=1 such that
!
c
j
c
j
−
,
+
for any non-negative integer j.
Tk ∈
/
()
()
()
()
rl
rl rl
rl
1
1
()
Figure A.1 shows an example when rl = , Sl ∈ {0, 1} and c = . We can see that
2
8
!
∞
∞ [
[
n
c n
c
1
1
− () , () + () =
2n − , 2n +
,
()
4
4
rl
rl rl
rl
n=0
n=0
does not cover odd numbers.
0
1
2
3
4
5
6
2n 2n+1 2n+2
1
1
()
Figure A.1: An example when rl = , Sl ∈ {0, 1} and c = .
2
8
169
For any sample path of random sequence
()
{Dl [t], t
≥ 1},
Tk
X
()
Dl [t] is an integer
t=1
and thus we have
Tk ∈
/
PTk
()
t=1 Dl [t]
()
−
rl
c
()
,
rl
PTk
()
t=1 Dl [t]
()
rl
c
+
()
rl
!
,
which is equivalent to (A.2.1).
A.3
Proof of Lemma 2.3.7
()
Since r() [t] = (rl [t])Ll=1 ∈ R for all t and the maximum transmission rate for each
()
link l is Bl,Kl , we have 0 ≤ rl [t] ≤ Bl,Kl . Next, we show this lemma by using
induction.
(1) If T = 1, we have
()
()
()
()
|Gl [1] − rl [1]| ≤ max{Bl,Kl − rl [1], r1 [1]} ≤ Bl,Kl ;
(2) Assume T = n, (2.3.11) holds, that is,
−Bl,Kl
n
X
()
()
(Gl [t] − rl [t]) ≤ Bl,Kl .
≤
(A.3.1)
t=1
Then, when T = n + 1, we consider the following two cases:
n
n
X
X
()
()
()
(2.1) If
Gl [t] ≥
rl [t], then, by the RDO Algorithm, we have Gl [n+1] =
t=1
t=1
0. Thus, we have
n+1
X
()
(Gl [t]
−
()
rl [t])
t=1
n+1
X
n
X
()
()
()
=
(Gl [t] − rl [t]) − rl [n + 1]
t=1
()
≤ Bl,Kl − rl [n + 1] ≤ Bl,Kl ,
()
(Gl [t]
−
()
rl [t])
=
t=1
≥
n
X
()
()
(A.3.2)
()
(Gl [t] − rl [t]) − rl [n + 1]
t=1
()
−rl [n
170
+ 1] ≥ −Bl,Kl .
(A.3.3)
(2.2) If
n
X
n
X
()
Gl [t] <
t=1
()
()
rl [t], then, by the RDO Algorithm, we have Gl [n+1] =
t=1
Bl,Kl . Thus, we have
n+1
n
X
X
()
()
()
()
()
(Gl [t] − rl [t]) =
(Gl [t] − rl [t]) + Bl,Kl − rl [n + 1]
t=1
t=1
()
≤ Bl,Kl − rl [n + 1] ≤ Bl,Kl ,
n+1
X
()
(Gl [t]
−
()
rl [t])
=
t=1
n
X
()
()
(A.3.4)
()
(Gl [t] − rl [t]) + Bl,Kl − rl [n + 1]
t=1
()
≥ −Bl,Kl + Bl,Kl − rl [n + 1] ≥ −Bl,Kl .
n+1
X
()
()
In both cases, we have (Gl [t] − rl [t]) ≤ Bl,Kl .
(A.3.5)
t=1
A.4
Proof of Lemma 2.3.8
By using Lemma 2.3.7, we have
T −1
T −1
X
X
()
() G [t] −
rl [t] =
t=n l
t=n
T −1 n−1 X
X
()
()
()
()
Gl [t] − rl [t] −
Gl [t] − rl [t] t=1
t=1
T −1 n−1 X
X
()
()
()
()
≤ Gl [t] − rl [t] + Gl [t] − rl [t] t=1
t=1
≤ 2Bl,Kl ,
(A.4.1)
which implies that
T −1
X
()
Gl [t]
≤
t=n
T −1
X
()
∀n = 1, 2, ..., T − 1.
rl [t] + 2Bl,Kl ,
t=n
By using the Lindley’s equation, we have
(T −1
)
T −1
X
X
()
()
()
Ql [T ] =
max
Gl [t] −
Rl [t], 0
1≤n≤T −1
≤
≤
max
1≤n≤T −1
max
1≤n≤T −1
()
(
t=n
t=n
2Bl,Kl +
(T −1
X
T −1
X
t=n
)
Rl [t], 0
+ 2Bl,Kl
()
rl [t] −
t=n
()
rl [t] −
t=n
T −1
X
t=n
= Yl [T ] + 2Bl,Kl .
171
()
T −1
X
()
Rl [t], 0
)
(A.4.2)
A.5
Proof of Proposition 2.3.9
()
=
=
≤
(a)
=
(b)
≤
E[φ(d [T ], r() )]
T
1 X
()
() E
D [t] − r T
t=1
T
T
T
T
T
1 X
X
X
X
X
1
1
1
1
()
()
()
()
()
() E
D [t] −
G [t] +
G [t] −
r [t] +
r [t] − r T
T t=1
T t=1
T t=1
T t=1
t=1
T
T
T
1 X
1 X () 1 X ()
()
() E
D [t] −
G [t] + (r [t] − r )
T
T
T t=1
t=1
t=1
T
T
1 X
1 X () ()
+
G [t] −
r [t]
T t=1
T t=1
T
T
1 X
1
EkQ() [T + 1]k
X
+ G() [t] − r() [t] + (r() [t] − r() )
T t=1
T
T t=1
PL
L
T
T
()
1 X
()
1 X X ()
()
l=1 E[Ql [T + 1]]
r [t] − r() ,(A.5.1)
+
Gl [t] − rl [t] +
T t=1
T
T l=1 t=1
where the step (a) follows from the fact that
T
X
()
()
Ql [T
+ 1] −
()
Ql [1]
=
T
X
()
Gl [t] −
t=1
L
X
()
Dl [t] and Ql [1] = 0 for all l; (b) follows from the fact that kyk ≤
t=1
|yl | for
l=1
any L-dimensional vector y.
First, we will show that
()
E[Yl [T ]] ≤ M1 ,
∀T ≥ 1, ∀l ∈ L,
(A.5.2)
()
where M1 is some positive number. This will imply E[Ql [T ]] ≤ M1 + 2Bl,Kl , ∀T ≥
1, ∀l ∈ L, by using Lemma 2.3.8. By choosing the Lyapunov function V1 (Y() [t]) ,
L
1 X () 2
(Y [t]) , it is easy to show that
2 l=1 l
∆V1 , E[V1 (Y() [t + 1]) − V1 (Y() [t])|Y() [t] = Y() ]
L
X
()
Yl + M2 ,
≤ −θ()
l=1
172
(A.5.3)
where M2 and θ() are some finite positive constants. By using Theorem 14.0.1 in
()
()
()
[62], there exists Y l such that lim E[Yl [T ]] = E[Y l ], where E[Y l ] < ∞. Thus,
T →∞
()
()
give ζ > 0, ∃T0 ≥ 0 such that T > T0 implies that E[Yl [T ]] ≤ E[Y l ] + ζ.
For T ≤ T0 , we have
()
E[Yl [T ]] ≤
T
X
()
rl [t] ≤
T0
X
()
rl [t] ≤ T0 Bl,Kl .
(A.5.4)
t=1
t=1
()
Hence, by taking M1 , max{T0 Bl,Kl , E[Y l ] + ζ}, we have the desired result.
l∈L
T
X ()
()
Gl [t] − rl [t] ≤ Bl,Kl . Thus, we have
By Lemma 2.3.7, we have t=1
()
()
E[φ(d [T ], r )] ≤
()
where W1 , LM1 + 2
LM1 + 2
L
X
l=1 Bl,Kl
T
Bl,Kl +
l=1
A.6
PL
L
X
+
PL
()
Bl,Kl W1
W
+
= 1 ,
T
T
T
l=1
Bl,Kl + R1 .
l=1
Proof of Proposition 2.4.3
(1) Proof of the first part of Proposition 2.4.3: For any δ ∈ (0, max kr − r∗ k), we can
r∈R
(δ)
easily find a r
=
(δ)
(rl )Ll=1
∈ R satisfying the following conditions:
(i) kr(δ) − r∗ k = δ;
(ii)
L
X
(δ)
(δ)
Ul0 (rl )rl
∈
/ G (δ) ,
l=1
where
G (δ) ,
( L
X
)
(δ)
Ul0 (rl )Rl : R = (Rl )Ll=1 ∈ R and Rl ∈ Fl , ∀l .
l=1
Due to the discrete structure of scheduling rates R, without loss of generality, we asL
X
(δ) (δ)
(δ)
(δ) (δ)
(δ)
(δ)
(δ)
sume G has K elements. Let G = {a1 , a2 , ..., aK (δ) } and assume
Ul0 (rl )rl ∈
l=1
(δ) (δ)
(ai , ai+1 )
for some i = 1, 2, ..., K
(δ)
− 1.
173
()
Consider any policy in P with parameter . By Assumption 2.4.1, Dl [t] ∈
()
Fl , ∀t ≥ 1, ∀l, for any departure rate vector sequence {D() [t] = (Dl [t])l }t≥1 . Next,
we show that there exists a
(P
c(δ) ∈
0, min
L
l=1
(δ)
(δ)
(δ)
(δ)
Ul0 (rl )rl − ai ai+1 −
,
2
PL
(δ)
(δ)
0
l=1 Ul (rl )rl
2
)!
,
(A.6.1)
and a positive integer-valued sequence {Tk }∞
k=1 such that (2.4.1) holds. Let fy (r) ,
L
X
∇U (r)T (y−r) =
Ul0 (rl )(yl −rl ), where r, y ∈ R and T means transpose operation.
l=1
Then, we have
(b)
(a) |fy (r) − fy (r∗ )| = ∇fy (z)T (r − r∗ ) ≤ k∇fy (z)kkr − r∗ k,
(A.6.2)
where the step (a) follows from the Mean Value Theorem, where z lies between r and
∂fy
r∗ ; (b) follows from the Cauchy Schwartz’s inequality. Since
= Ul00 (rl )(yl − rl ) −
∂rl
0
Ul (rl ), we have
∂fy 00
0
∂rl ≤ |Ul (rl )|(yl + rl ) + |Ul (rl )| ≤ 2βmax bl,Kl + hmax .
Thus, we have
k∇fy (z)k ≤ H ,
√
L(2βmax max bl,Kl + hmax ).
l∈L
(A.6.3)
Hence, we have
|fy (r) − fy (r∗ )| ≤ Hkr − r∗ k,
∀y, r ∈ R.
(A.6.4)
By setting r = r(δ) , we have
which implies that
fy (r(δ) ) − fy (r∗ ) ≤ Hδ,
fy (r(δ) ) ≤ fy (r∗ ) + Hδ,
174
∀y ∈ R,
∀y ∈ R.
(A.6.5)
(A.6.6)
By the first order optimality condition, we have
fy (r∗ ) = ∇U (r∗ )T (y − r∗ ) ≤ 0,
∀y ∈ R.
Thus, we have fy (r(δ) ) ≤ Hδ, ∀y ∈ R. By setting y =
L
X
l=1
Since
L
X
(δ)
(δ)
Ul0 (rl )rl
(δ)
Ul0 (rl )
T
1 X ()
(δ)
Dl [t] − rl
T t=1
!
(A.6.7)
T
1 X ()
D [t] ∈ R, we have
T t=1
≤ Hδ.
(A.6.8)
∈
/ G (δ) , by Lemma 2.3.2, there exists a c(δ) satisfying (A.6.1) and
l=1
a positive integer-valued sequence {Tk }∞
k=1 such that
Tk X
L
L
c(δ)
1 X
X
()
0 (δ) (δ) 0 (δ)
Ul (rl )rl ≥
Ul (rl )Dl [t] −
.
Tk
Tk
t=1 l=1
l=1
Thus, if Hδ <
(A.6.9)
c(δ)
c(δ)
, that is, Tk <
, then, we have
Tk
Hδ
Tk X
L
L
X
1 X
c(δ)
()
(δ) (δ)
0 (δ)
.
Ul (rl )Dl [t] −
Ul0 (rl )rl ≤ −
Tk t=1 l=1
Tk
l=1
(A.6.10)
By using the concavity of the utility function U , we have
!
!
Tk
Tk
1 X
1 X
()
(δ)
(δ) T
()
(δ)
U
D [t]
≤ U (r ) + ∇U (r )
D [t] − r
Tk t=1
Tk t=1
≤ U (r(δ) ) −
c(δ)
.
Tk
(A.6.11)
In addition, we have
1
(a)
U (r(δ) ) = U (r∗ ) + ∇U (r∗ )T (r(δ) − r∗ ) + (r(δ) − r∗ )T ∇2 U (z)(r(δ) − r∗ )
2
(b)
√
1
≤ U (r∗ ) − βmin Lkr(δ) − r∗ k2
2
√
1
= U (r∗ ) − βmin Lδ 2 ,
(A.6.12)
2
where (a) follows the Mean Value Theorem, where z is between r∗ and r(δ) ; (b)
√
uses the first order optimality condition and k∇2 U (z)k ≥ Lβmin . By substituting
(A.6.12) into (A.6.11), we have inequality (2.4.1).
175
(2) Proof of the second part of Proposition 2.4.3: We will show the following
claim:
Claim 2. If we further have
L
X
/ H,
Ul0 (rl∗ )rl∗ ∈
l=1
i.e.,
L
X
Ul0 (rl∗ )rl∗
l=1
L
X
∈ (
Ul0 (rl∗ )yl ,
l=1
L
X
Ul0 (rl∗ )zl ), where y = (yl )Ll=1 , z = (zl )Ll=1 ∈ R
l=1
and yl , zl ∈ Fl , ∀l ∈ L, then, for any
1
δ < min
min
|y 0 − z 0 | ,
(0)
0
0
0
0
2G1 y ,z ∈G :y 6=z
!
L
L
X
X
1
Ul0 (rl∗ )yl ,
U 0 (r∗ )r∗ −
G1 + G2 l=1 l l l
l=1
!
L
L
X
X
1
Ul0 (rl∗ )rl∗
U 0 (r∗ )zl −
,
G1 + G2 l=1 l l
l=1
(A.6.13)
there exists
c1
(δ)
1
=
min
4
L
X
X
L
−
l=1
Ul0 (rl∗ )zl
l=1
Ul0 (rl∗ )rl∗
L
X
Ul0 (rl∗ )yl − (G1 + G2 )δ,
l=1
−
L
X
Ul0 (rl∗ )rl∗
l=1
− (G1 + G2 )δ
(A.6.14)
and a positive integer-valued sequence {Tk }∞
k=1 such that (2.4.1) holds, where G1 ,
√
√
Lβmax max bl,Kl and G2 , L(βmax max bl,Kl + hmax ).
l∈L
l∈L
(δ)
When δ is sufficiently small (e.g., where δ satisfies (A.6.13)), we take c(δ) = c1
and thus we have the desired result. Next, we prove this claim to complete the proof.
By using similar technique in showing inequality (A.6.4), we have
L
L
X
X
(δ) (δ)
Ul0 (rl )rl −
Ul0 (rl∗ )rl∗ ≤ G2 δ
l=1
l=1
L
L
X
X
0 (δ)
0 ∗
Ul (rl )yl −
Ul (rl )yl ≤ G1 δ
l=1
l=1
L
L
X
X
(δ)
0
0 ∗
Ul (rl )zl −
Ul (rl )zl ≤ G1 δ.
l=1
l=1
176
(A.6.15)
(A.6.16)
(A.6.17)
Since δ satisfies (A.6.13), the relationship between
L
X
(δ) (δ)
Ul0 (rl )rl
and
l=1
shown in Figure A.2.
L
¦U c(r G ) y
L
( )
l
l
l
l 1
L
¦ U c( r
l
*
l
) yl
¦ U lc( rl(G ) ) rl(G )
L
(
l
l 1
l 1
L
L
¦ U lc( rl* ) rl*
2G1į
l
¦ U c( r
l
*
l
)
) zl
) zl
l 1
2G2į
Figure A.2: The relationship between
2G1į
L
X
(δ) (δ)
Ul0 (rl )rl
and
l=1
(δ) (δ)
Ul0 (rl )rl
l=1
∈
L
X
(δ)
Ul0 (rl )yl ,
L
X
(δ)
Ul0 (rl )zl
l=1
l=1
L
X
Ul0 (rl∗ )rl∗ .
l=1
Thus, we have
L
X
Ul0 (rl∗ )rl∗ is
l=1
¦ U c( r G
l 1
l 1
L
X
!
.
(A.6.18)
By statement (1), for any departure rate vector sequence {D() [t]}t≥1 , there exists a
c(δ) satisfying
)
(P
PL
PL
PL
L
0 (δ) (δ)
0 (δ)
0 (δ)
0 (δ) (δ)
U
(r
)r
U
(r
)y
U
(r
)z
−
U
(r
)r
−
l
l
l
l
l=1 l l
l=1 l l
l=1 l l
c(δ) < min
, l=1 l l
,
2
2
and a positive integer-valued sequence {Tk }∞
k=1 such that (2.4.1) holds. By using
inequality (A.6.15), (A.6.16) and (A.6.17), we have
PL
PL
PL
PL
0 (δ)
0 (δ)
0 (δ) (δ)
0 (δ) (δ)
l=1 Ul (rl )rl −
l=1 Ul (rl )yl
l=1 Ul (rl )zl −
l=1 Ul (rl )rl
min
,
2
2
P
PL
L
0 ∗ ∗
0 ∗
l=1 Ul (rl )rl −
l=1 Ul (rl )yl − (G1 + G2 )δ
≥ min
,
2
PL
PL
0 ∗
0 ∗ ∗
U
(r
)y
−
U
(r
)r
−
(G
+
G
)δ
l
1
2
l
l
l
l
l
l=1
l=1
.
2
(δ)
Thus, we can take c1 as in (A.6.14) that satisfies (A.6.1).
177
A.7
Proof of Proposition 2.4.6
(a)
() T ()
()
()
U d [T ]
≥ U (r ) + ∇U d [T ]
r − d [T ]
() ()
(b)
≥ U (r() ) − ∇U d [T ] d [T ] − r() (c)
√
()
()
() ≥ U (r ) − Lhmax d [T ] − r ,
()
(A.7.1)
where the step (a) follows from the definition of concavity; (b) follows from CauchySchwartz’s inequality; (c) follows from Assumption 2.4.2.
By using similar line of argument in Proposition 2.3.9, it is not hard to show that
()
W ()
()
()
φ(d [T ], r() ) = d [T ] − r() ≤ 2 ,
T
where W2 is some positive constant. Thus, we have
√
()
() Lhmax W2
()
U d [T ] ≥ U (r ) −
.
T
(A.7.2)
(A.7.3)
Since kr() − r∗ k ≤ ρ() , by the Mean Value Theorem, we have
|U (r() ) − U (r∗ )| = |∇U (z)T (r() − r∗ )|
≤ k∇U (z)kkr() − r∗ k
√
≤ hmax Lρ() ,
(A.7.4)
where z is between r() and r∗ . Thus, we have
√
U (r() ) ≥ U (r∗ ) − hmax Lρ() ,
(A.7.5)
By substituting (A.7.5) into (A.7.3), we have the desired result.
A.8
Proof of Proposition 2.4.9
Before analyzing the convergence speed of the Dual Algorithm, we need to establish
the boundedness of queue length for all links.
178
Lemma A.8.1. For Dual Algorithm with parameter > 0, the queue lengths for all
links are bounded all the time, i.e.,
()
Ql [t] ≤ G() , ∀l, t,
()
where G
√
hmax
and W ,
, W+
(A.8.1)
βmax
3βmax
2
+ 2 LM +
+ 2 L max b2l,Kl .
l∈L
Proof. See Appendix A.9 for the proof.
We are ready to analyze the convergence speed of the Dual Algorithm in terms
of its utility benefit. By Assumption 2.4.1, there is no unused service in the system
and thus we have
!
T
1 X ()
R [t]
U (d [T ]) = U
T t=1
!
!T P
T
T
T
()
()
1 X ()
1 X ()
t=1 (r [t] − R [t])
U
r [t] − ∇U
R [t]
T t=1
T t=1
T
!
!
T
T
T
X
1
1 X ()
1 X ()
U
r [t] − ∇U
R [t] (r() [t] − R() [t])
T t=1
T T t=1
t=1
!
√
T
1 X ()
hmax L ()
U
r [t] −
kQ [T + 1] − Q() [1]k
T t=1
T
!
√
T
1 X ()
hmax L
U
r [t] −
kQ() [1]k + kQ() [T + 1]k
T t=1
T
!
√
T
√ 1 X ()
hmax L ()
U
r [t] −
kQ [1]k + G() L ,
(A.8.2)
T t=1
T
()
(a)
≥
(b)
≥
(c)
≥
(d)
≥
(e)
≥
where the step (a) follows from the concavity of utility function; (b) follows from
the Cauchy-Schwarz inequality; (c) uses Assumption 2.4.2 and the queue evolution
(2.4.8); (d) follows from the triangle inequality; (e) follows from the Lemma A.8.1.
179
!
T
1 X ()
r [t] .
T t=1
Next, we give the lower bound for U
1
U
(a)
≥
=
(b)
≥
=
T
1 X ()
r [t]
T t=1
!
T
11X
U r() [t]
T t=1
T
1X
T t=1
1
T
1
U (r() [t]) −
T
X
1
U (r∗ ) −
t=1
1
1
U (r∗ ) +
T
L
X
l=1
T
L
XX
()
()
Ql [t]rl [t]
l=1
L
X
!
()
Ql [t]rl∗
()
!
T
L
1 X X () ()
+
Q [t]rl [t]
T t=1 l=1 l
T
L
1 X X () ()
+
Q [t]rl [t]
T t=1 l=1 l
()
Ql [t](rl [t] − rl∗ ),
(A.8.3)
t=1 l=1
where the step (a) follows from the Jensen’s inequality; (b) follows from equation
(2.4.6). Hence, we have
U
T
1 X ()
r [t]
T t=1
Next, let’s consider
!
T X
L
X
T
L
X X ()
()
≥ U (r ) +
Ql [t](rl [t] − rl∗ ).
T t=1 l=1
∗
()
(A.8.4)
()
Ql [t](rl [t] − rl∗ ).
t=1 l=1
T X
L
X
=
t=1 l=1
T X
L
X
()
()
Ql [t](rl [t] − rl∗ )
()
()
Ql [t](rl [t]
−
()
Rl [t])
t=1 l=1
+
T X
L
X
t=1 l=1
180
()
()
Ql [t](Rl [t] − rl∗ ).
(A.8.5)
For
L
X
()
()
Ql [t](Rl [t] − rl∗ ), we have
l=1
L
X
=
(a)
≥
l=1
L
X
l=1
L
X
()
()
Ql [t](Rl [t] − rl∗ )
()
()
Ql [t](Rl [t]
()
−
rl∗ )1{Q() [t]≥b }
l,Kl
l
l
l=1
≥ −
()
()
Ql [t](Rl [t] − rl∗ )1{Q() [t]<b
l
l=1
l,Kl }
()
Ql [t](Rl [t] − rl∗ )1{Q() [t]<b
L
X
+
L
X
()
Ql [t]rl∗ 1{Q() [t]<b
l
l=1
l,Kl }
l,Kl }
≥ −L max b2l,Kl ,
(A.8.6)
l∈L
where the step (a) follows from equation (2.4.7). Thus, we have
T X
L
X
()
()
Ql [t](Rl [t] − rl∗ ) ≥ −T L max b2l,Kl .
l∈L
t=1 l=1
Next, we consider
L
X
()
()
(A.8.7)
()
Ql [t](xl [t] − Rl [t]). By using the queue length evolution
l=1
(2.4.8), we have
kQ() [t + 1]k2 = kQ() [t] + r() [t] − R() [t]k2
L
X
()
()
()
()
2
= kQ [t]k + 2
Ql [t](rl [t] − Rl [t]) + kr() [t] − R() [t]k2 .
l=1
Thus, we have
L
X
()
()
()
Ql [t](rl [t] − Rl [t])
l=1
1
kQ() [t + 1]k2 − kQ() [t]k2 −
=
2
1
≥
kQ() [t + 1]k2 − kQ() [t]k2 −
2
181
1 ()
kr [t] − R() [t]k2
2
L
(M 2 + max b2l,Kl ),
l∈L
2
(A.8.8)
()
()
2
where we use the fact that kr [t] − R [t]k =
L
X
()
()
(rl [t] − Rl [t])2
l=1
()
(Rl [t])2 )
2
≤ L(M +
max b2l,Kl ).
l∈L
T X
L
X
()
()
((rl [t])2 +
l=1
Hence, we have
()
≤
L
X
()
Ql [t](rl [t] − Rl [t])
t=1 l=1
TL
1
kQ() [T + 1]k2 − kQ() [1]k2 −
(M 2 + max b2l,Kl )
l∈L
2
2
1 ()
T
L
≥ − kQ [1]k2 −
(M 2 + max b2l,Kl ).
l∈L
2
2
≥
(A.8.9)
Thus, by substituting (A.8.7) and (A.8.9) into (A.8.5), we have
T X
L
X
t=1 l=1
1
TL
()
()
Ql [t](rl [t] − rl∗ ) ≥ − kQ() [1]k2 −
(M 2 + 3 max b2l,Kl ).
l∈L
2
2
Hence, we have
T
L
X X ()
L
()
Ql [t](rl [t] − rl∗ ) ≥ − kQ() [1]k2 − (M 2 + 3 max b2l,Kl ). (A.8.10)
l∈L
T t=1 l=1
2T
2
Thus, by combining (A.8.2), (A.8.4) and (A.8.10), we have the desired result.
A.9
Proof of Lemma A.8.1
Definition A.9.1. (Invariant Pair:) The pair (Q∗() , r∗() ) forms an invariant
pair if they satisfy the following conditions:
∗()
Ul0 (rl
∗()
) = Ql
r∗() ∈ argmax
η∈R
,
L
X
(A.9.1)
∗()
Ql
ηl .
(A.9.2)
l=1
By using similar technique as in [16], we can show the existence and uniqueness
of (Q∗() , r∗() ). In addition, r∗() is the same for all and thus r∗() = r∗ .
182
1 X ()
∗()
Choose Lyapunov function V2 (Q [t]) =
(Ql [t] − Ql )2 . By using similar
2 l=1
technique as in [16], we have
L
()
∆V2 , V2 (Q() [t + 1]) − V2 (Q() [t])
≤ −
2
V2 (Q() [t]) + W3 ,
βmax
(A.9.3)
1 X ()
W3 βmax
()
,
(rl [t] − Rl [t])2 + L max b2l,Kl < ∞. Thus, if V2 (Q() [t]) >
l∈L
2 l=1
2
L
where W3 ,
then, V2 (Q() [t + 1]) < V2 (Q() [t]); otherwise, V2 (Q() [t + 1]) may be greater than or
W3 βmax
equal to V2 (Q() [t]). Thus, if V2 (Q() [t]) =
, then
2
()
2V2 (Q [t + 1]) =
L
X
∗() 2
()
(Ql [t + 1] − Ql
)
l=1
(a)
=
=
L
X
l=1
L
X
()
()
()
(Ql [t]
−
∗() 2
∗()
Ql )2
)
L
L
X
X
()
∗()
()
()
()
()
2
(Ql [t] − Ql )(rl [t] − Rl [t])
(rl [t] − Rl [t]) + 2
+
l=1
l=1
l=1
(b)
()
(Ql [t] + rl [t] − Rl [t] − Ql
≤ kQ() [t] − Q∗() k2 + kr() [t] − R() [t]k2 + 2kQ() [t] − Q∗() kkr() [t] − R() [t]k
≤ 2kQ() [t] − Q∗() k2 + 2kr() [t] − R() [t]k2 ,
(A.9.4)
where (a) uses the fact that there is always no unused service by Assumption 2.4.1;
(b) follows from Cauchy-Schwarz inequality. Since
kQ() [t] − Q∗() k2 = 2V2 (Q() [t]) =
W3 βmax
(A.9.5)
and
kr() [t] − R() [t]k2 ≤ L(M 2 + max b2l,Kl ),
l∈L
we have
()
2V2 (Q [t + 1]) ≤ 2
W3 βmax
2
2
+ L(M + max bl,Kl ) , W4 .
l∈L
183
(A.9.6)
Thus, we have
2V2 (Q() [t]) ≤ W4 , ∀t.
(A.9.7)
Now, we can give an upper bound for queue lengths at each time. Indeed, we have
()
∗()
()
Ql [t] = Ql [t] − Ql
∗()
()
∗()
∗()
≤ |Ql [t] − Ql | + Ql
p
hmax
W4 +
, ∀l, t,
≤
q
p
()
∗()
∗()
where the last step follows from |Ql [t] − Ql | ≤ 2V2 (Q() [t]) ≤ W4 and Ql =
hmax
1
Ul0 (rl∗ )
≤
. Since W3 ≤ L(M 2 + max b2l,Kl ) + L max b2l,Kl , we have
l∈L
l∈L
2
βmax
3βmax
2
W4 ≤
+ 2 LM +
+ 2 L max b2l,Kl , W.
(A.9.8)
l∈L
+ Ql
Thus, we have
()
Ql [t] ≤
√
W+
hmax
, G() .
184
(A.9.9)
APPENDIX B: PROOFS FOR CHAPTER 4
B.1
Proof of Inequality (4.2.4)
∆W , E [W (Q[t + 1], T[t + 1]) − W (Q[t], T[t])|Q[t], T[t]]
" L
L
X
X
2
βl Tl [t + 1]
= E
αl Ql [t + 1] + 4γCmax
−
≤
l=1
l=1
L
X
αl Q2l [t] − 4γCmax
l=1
l=1
L
X
l=1
L
X
#
βl Tl [t]Q[t], T[t]
αl E (Ql [t] + Al [t] − Cl [t]Sl∗ [t])2 − Q2l [t]Q[t], T[t]
+4γCmax E
"
L
X
βl Tl [t + 1] −
L
X
l=1
l=1
#
βl Tl [t]Q[t], T[t] ,
(B.1.1)
where the last step follows from the evolution of each queue, and (max{x, 0})2 ≤ x2 .
Let H∗ , {l : Sl∗ [t]Cl [t] > 0}. According to the definition of the TSLS counter,
we have
L
X
βl Tl [t + 1] =
X
βl (Tl [t] + 1)
l∈H
/ ∗
l=1
=
≤
L
X
βl Tl [t] −
X
l=1
l∈H∗
L
X
X
βl Tl [t] −
l∈H∗
l=1
185
βl Tl [t] +
L
X
βl −
βl Tl [t] +
l=1
βl
(B.1.2)
l∈H∗
l=1
L
X
X
βl .
(B.1.3)
By substituting inequality (B.1.3) into (B.1.1), we have
∆W ≤
L
X
l=1
αl E (Ql [t] + Al [t] − Cl [t]Sl∗ [t])2 − Q2l [t]Q[t], T[t]
+4γCmax E
≤
L
X
"
L
X
βl −
l=1
#
βl Tl [t]Q[t], T[t]
∗
X
l∈H
αl E [2Ql [t](Al [t] − Cl [t]Sl∗ [t])|Q[t], T[t]]
l=1
+
L
X
l=1
αl E (Al [t] − Cl [t]Sl∗ [t])2 Q[t], T[t]
#
+4γCmax
βl − 4γCmax E
βl Tl [t]Q[t], T[t]
l=1
l∈H∗
"
#
L
L
X
X
≤ 2
αl λl Ql [t] − 2E
αl Ql [t]Cl [t]Sl∗ [t]Q[t], T[t]
l=1
l=1
#
"
X
−4γCmax E
βl Tl [t]Q[t], T[t] + B(α, β, γ),
∗
"
L
X
X
(B.1.4)
l∈H
where B(α, β, γ) is defined in Proposition 4.2.2.
L
X
(MWS)
Let S
[t] ∈ argmax
αl Ql [t]Cl [t]Sl . Then, by the definition of the RSG
S∈S
l=1
Algorithm, we have
L
L
X
X
(MWS)
∗
(αl Ql [t] + γβl Tl [t])Cl [t]Sl
[t]
(αl Ql [t] + γβl Tl [t])Cl [t]Sl [t] ≥
l=1
l=1
≥
L
X
(MWS)
αl Ql [t]Cl [t]Sl
[t],
l=1
which implies
L
X
l=1
αl Ql [t]Cl [t]Sl∗ [t] ≥
L
X
(MWS)
αl Ql [t]Cl [t]Sl
l=1
[t] − γ
L
X
l=1
186
βl Tl [t]Cl [t]Sl∗ [t]. (B.1.5)
By substituting (B.1.5) into (B.1.4), we have
" L
#
L
X
X
(MWS) ∆W ≤ 2
αl λl Ql [t] − 2E
αl Ql [t]Cl [t]Sl
[t]Q[t], T[t]
l=1
l=1
" L
#
X
+2γE
βl Tl [t]Cl [t]Sl∗ [t]Q[t], T[t]
l=1
"
#
X
−4γCmax E
βl Tl [t]Q[t], T[t] + B(α, β, γ).
(B.1.6)
∗
l∈H
Given Q[t] and T[t], we have
"
#
" L
#
X
X
Cmax E
βl Tl [t]Q[t], T[t] ≥ E
βl Tl [t]Cl [t]Sl∗ [t]Q[t], T[t] ,
∗
l∈H
(B.1.7)
l=1
where we recall that H∗ = {l : Sl∗ [t]Cl [t] > 0}. By substituting (B.1.7) into (B.1.6),
we have
#
(MWS) αl Ql [t]Cl [t]Sl
[t]Q[t], T[t]
αl λl Ql [t] − 2E
∆W ≤ 2
l=1
l=1
#
" L
X
∗ (B.1.8)
−2γE
βl Tl [t]Cl [t]Sl [t]Q[t], T[t] + B(α, β, γ).
"
L
X
L
X
l=1
Note that the capacity region R (see [89]) is also equivalent to a set of arrival rate
vectors λ such that there exist non-negative numbers θ(c; s) satisfying
X
∀c,
θ(c; s) = 1,
s∈S
λl ≤
X
Pr{C[t] = c}
c
X
(B.1.9)
θ(c; s)cl sl ,
∀l,
(B.1.10)
s∈S
where s = (sl )Ll=1 . For any λ ∈ Int(R), there exists an > 0 such that
λl ≤
X
c
Pr{C[t] = c}
X
θ(c; s)cl sl − ,
s∈S
187
∀l.
(B.1.11)
Hence, we have
L
X
l=1
αl λl Ql [t] + L
X
X
αl Ql [t] ≤
Pr{C[t] = c}
c
l=1
X
(a)
≤
θ(c; s)
s∈S
Pr{C[t] = c}
c
" L
X
= E
X
l=1
X
L
X
αl Ql [t]cl sl
l=1
θ(c; s)
s∈S
L
X
(MWS)
αl Ql [t]cl Sl
[t]
l=1
#
(MWS)
αl Ql [t]Cl [t]Sl
[t]Q[t], T[t] .
(B.1.12)
where the step (a) follows from the definition of S(MWS) . By substituting (B.1.12)
into (B.1.8), we have
∆W ≤ −2
L
X
αl Ql [t] + B(α, β, γ)
l=1
−2γE
≤ −2
"
L
X
L
X
l=1
#
βl Tl [t]Cl [t]Sl∗ [t]Q[t], T[t]
(B.1.13)
αl Ql [t] + B(α, β, γ).
(B.1.14)
l=1
B.2
Proof of Proposition 4.2.3
Consider the Lyapunov function
v
u L
L
uX
X
t
2
αl Ql [t] + 4γCmax
βl Tl [t],
V (Y[t]) , kY[t]k2 =
l=1
l=1
p
√
√
where Y[t] , ( αQ[t], 4γCmax βT[t]), x denotes the component-wise square root
of the vector x, and xy denotes the component-wise product of the vectors x and y.
It is shown in Appendix B.3 that there exist positive constants b and δ such that
if kY[t]k2 > b, then,
∆V , E [kY[t + 1]k2 − kY[t]k2 |Q[t], T[t]] < −δ.
(B.2.1)
We first consider the term E eηkY[t+1]k2 Q[t], T[t] . Let l∗ [t] ∈ argmax(αl Ql [t] +
l
188
γβl Tl [t]) and λmin ,
1
min λl . Then, we partition the space (Q[t], T[t]) into sets F1 ,
2 l
F2 and F3 , where
F1 , {kY[t]k2 ≤ b} ;
F2 , kY[t]k2 > b, Ql∗ [t] [t] > λmin Tl∗ [t] [t] ;
F3 , kY[t]k2 > b, Ql∗ [t] [t] ≤ λmin Tl∗ [t] [t] .
Then, we have
ηkY[t+1]k2 E e
Q[t], T[t] =
3
X
i=1
E eηkY[t+1]k2 ; Fi Q[t], T[t] .
(B.2.2)
Next, we consider each term in (B.2.2) individually.
(i) On event F1 , we have
v
u L
L
uX
X
t
2
kY[t]k2 =
αl Ql [t] + 4γCmax
βl Tl [t] ≤ b,
l=1
which implies
L
X
l=1
αl Q2l [t]
2
≤ b and
l=1
L
X
αl Q2l [t]
+ 4γCmax
L
X
l=1
l=1
189
βl Tl [t] ≤ b2 . For kY[t +
1]k2 , we have
kY[t + 1]k22 =
≤
=
L
X
l=1
L
X
l=1
L
X
αl Q2l [t + 1] + 4γCmax
L
X
l=1
2
αl (Ql [t] + Amax ) + 4γCmax
αl Q2l [t]
βl (Tl [t] + 1)
+ 2Amax
L
X
αl Ql [t] +
L
X
βl Tl [t] + 4γCmax
L
X
l=1
αl Q2l [t] + 4γCmax
L
X
βl Tl [t]
l=1
l=1
L
X
αl
l=1
l=1
L
X
A2max
l=1
+4γCmax
≤
L
X
l=1
l=1
(a)
βl Tl [t + 1]
βl
!
v
u L
L
L
L
uX X
X
X
2
t
2
βl
αl Ql [t] + Amax
αl + 4γCmax
αl
+2Amax
l=1
l=1
l=1
v
u L
L
L
uX
X
X
2
≤ b + 2Amax bt
βl
αl + A2max
αl + 4γCmax
l=1
,
l=1
D12 ,
l=1
l=1
(B.2.3)
where step (a) follows from Cauchy-Schwarz’s inequality. Hence, we have kY[t +
1]k2 ≤ D1 , which implies
E eηkY[t+1]k2 ; F1 Q[t], T[t] ≤ eηD1
(B.2.4)
To consider other two terms in (B.2.2), we need the following lemma, which is
shown in Appendix B.4.
Lemma B.2.1. If kY[t]k2 > b, then
P
√
4γCmax Ll=1 βl
|kY[t + 1]k2 − kY[t]k2 | ≤ 2L max{Amax , Cmax } αmax +
b
P
l∈H∗ βl Tl [t]
+ 4γCmax qP
(B.2.5)
,
PL
L
2
α
Q
[t]
+
4γC
β
T
[t]
max
l=1 l l
l=1 l l
where αmax , max αl , and H∗ , {l : Sl∗ [t]Cl [t] > 0} is defined in Appendix B.1.
l
190
(ii) On event F2 , we have
αl Ql [t] + γβl Tl [t] ≤ αl∗ [t] Ql∗ [t] [t] + γβl∗ [t] Tl∗ [t] [t]
γβmax
≤
αmax +
Ql∗ [t] [t], ∀l,
λmin
(B.2.6)
where βmax , max βl . This implies
l
γβmax
Ql∗ [t] [t], ∀l.
γβl Tl [t] ≤ αmax +
λmin
(B.2.7)
Thus, we have
P
γβmax
λmin
L αmax +
Ql∗ [t] [t]
βl Tl [t]
≤
√
PL
αmin Ql∗ [t] [t]
2
l=1 βl Tl [t]
l=1 αl Ql [t] + 4γCmax
L
γβmax
= √
αmax +
, (B.2.8)
αmin
λmin
γ
qP
L
l∈H∗
where αmin , min αl > 0. By substituting (B.2.8) into (B.2.5), we get
l
|kY[t + 1]k2 − kY[t]k2 | ≤ D2 ,
(B.2.9)
P
√
4γCmax Ll=1 βl 4Cmax L
γβmax
where D2 , 2L max{Amax , Cmax } αmax +
+√
αmax +
.
b
αmin
λmin
Noting that (B.2.1) and (B.2.9) satisfy conditions of Lemma 2.2 in [26], there exists
η1 > 0 and ρ ∈ (0, 1) such that
E eη(kY[t+1]k2 −kY[t]k2 ) ; F2 Q[t], T[t] ≤ ρ, ∀0 < η < η1 .
Thus, for any 0 < η < η1 , we have
E eηkY[t+1]k2 ; F2 Q[t], T[t] ≤ ρeηkY[t]k2 .
(B.2.10)
(iii) On event F3 , we have
αl Ql [t] + γβl Tl [t] ≤ αl∗ [t] Ql∗ [t] [t] + γβl∗ [t] Tl∗ [t] [t]
≤ (λmin αmax + γβmax ) Tl∗ [t] [t], ∀l,
191
(B.2.11)
which implies
γβl Tl [t] ≤ (λmin αmax + γβmax ) Tl∗ [t] [t], ∀l,
(B.2.12)
αl Ql [t] ≤ (λmin αmax + γβmax ) Tl∗ [t] [t], ∀l.
(B.2.13)
Thus, we have
γ
P
βl Tl [t]
L(λmin αmax + γβmax )
Tl∗ [t] [t]. (B.2.14)
≤
PL
b
2
l=1 αl Ql [t] + 4γCmax
l=1 βl Tl [t]
qP
L
l∈H∗
By substituting (B.2.14) into (B.2.5), we get
P
√
4γCmax Ll=1 βl
|kY[t + 1]k2 − kY[t]k2 | ≤ 2L max{Amax , Cmax } αmax +
b
4Cmax L(λmin αmax + γβmax )
+
Tl∗ [t] [t].
(B.2.15)
b
In addition, on event F3 , we have
kY[t]k22
=
L
X
αl Q2l [t]
+ 4γCmax
L
X
βl Tl [t]
l=1
l=1
2
≤ (λmin αmax + γβmax )
= F32 Tl2∗ [t] [t],
Tl2∗ [t] [t]
L
X
1
+ 4Cmax L(λmin αmax + γβmax )Tl∗ [t] [t]
α
l
l=1
(B.2.16)
where F32 , (λmin αmax + γβmax )2
L
X
1
+ 4Cmax L(λmin αmax + γβmax ). Hence, we have
α
l
l=1
kY[t + 1]k2 ≤ kY[t]k2 + |kY[t + 1]k2 − kY[t]k2 |
≤ F1 Tl∗ [t] [t] + F2 ,
(B.2.17)
√
4Cmax L(λmin αmax + γβmax )
where F1 ,
+ F3 and F2 , 2L max{Amax , Cmax } αmax +
b
P
4γCmax Ll=1 βl
. Thus, we have
b
E eηkY[t+1]k2 ; F3 Q[t], T[t] ≤ eηF2 eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} .
(B.2.18)
By substituting (B.2.4), (B.2.10) and (B.2.18) into (B.2.2), we have
E eηkY[t+1]k2 |Q[t], T[t] ≤ eηD1 + ρeηkY[t]k2 + eηF2 eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} .
192
By taking expectation on both sides, we have
h
i
.
E eηkY[t+1]k2 ≤ eηD1 + ρE eηkY[t]k2 + eηF2 E eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} (B.2.19)
It is shown in Appendix B.5 that there exists positive constants η2 and D3 such that
h
i
E eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} ≤ D3 ,
(B.2.20)
holds for any 0 < η < η2 . By substituting (B.2.20) into (B.2.19), then,
E eηkY[t+1]k2 ≤ ρE eηkY[t]k2 + D,
(B.2.21)
holds for any 0 < η < min{η1 , η2 }, where D , eηD1 + D3 eηF2 . By using inequality
(B.2.21) and iterating over t, we have
1 − ρt
D
E eηkY[t]k2 ≤ ρt eηkY[0]k +
D ≤ eηkY[0]k +
,
1−ρ
1−ρ
which implies the existence of all moments of steady-state queue length and TSLS
counter.
B.3
Proof of Inequality (B.2.1)
For notational convenience, we replace B(α, β, γ) with B in the rest of the proof.
∆V
, E [V (Y[t + 1]) − V (Y[t])|Q[t], T[t]]
= E [kY[t + 1]k2 − kY[t]k2 |Q[t], T[t]]
q
q
2
2
= E
kY[t + 1]k2 − kY[t]k2 Q[t], T[t]
1
≤
E kY[t + 1]k22 − kY[t]k22 Q[t], T[t] ,
2kY[t]k2
(B.3.1)
√
where the last step follows from the fact that f (x) = x is concave for x ≥ 0 so that
x1 − x2
f (x1 ) − f (x2 ) ≤ f 0 (x2 )(x1 − x2 ) = √
with x1 , kY[t + 1]k22 and x2 , kY[t]k22 .
2 x2
193
Note that the difference in (B.3.1) is exactly ∆W in (B.1.1). Next, we consider
the term ∆W . Since
L
X
l=1
#
E
αl Ql [t] ≥
αl Ql [t]Cl [t]Q[t], T[t]
Cmax
l=1
" L
#
X
1
∗ E
≥
αl Ql [t]Cl [t]Sl [t]Q[t], T[t] .
Cmax
1
"
L
X
l=1
By substituting above inequality into (B.1.13), we have
" L
#
X
2
∆W ≤ −
E
αl Ql [t]Cl [t]Sl∗ [t]Q[t], T[t]
Cmax
l=1
" L
#
X
−2γE
βl Tl [t]Cl [t]Sl∗ [t]Q[t], T[t] + B
l=1
#
" L
X
(αl Ql [t] + γβl Tl [t])Cl [t]Sl∗ [t]Q[t], T[t] + B
≤ −2h1 ()E
l=1
#
" L
X
(a)
1
≤ −2h1 () E
(αl Ql [t] + γβl Tl [t])Cl [t]Q[t], T[t] + B
L
l=1
L
X
2cmin
(αl Ql [t] + γβl Tl [t]) + B,
≤ −
h1 ()
L
l=1
where h1 () , min
γβk Tk [t])Ck [t]Sk∗ [t]
Cmax
L
X
, 1 , the step (a) follows from the fact that
(αk Qk [t] +
k=1
≥ (αl Ql [t] + γβl Tl [t])Cl [t], ∀l = 1, 2, ..., L, and cmin , min E[Cl [t]].
l
By substituting (B.3.2) into (B.3.1), we have
∆V
1
≤
2kY[t]k2
≤
(B.3.2)
1
2kY[t]k2
!
L
X
2cmin
−
h1 ()
(αl Ql [t] + γβl Tl [t]) + B
L
l=1
!
L
X
−h2 ()
(αl Ql [t] + 4γCmax βl Tl [t]) + B ,
l=1
194
where h2 () , h1 ()
2cmin
. Note that
L
4γCmax βl Tl [t] ≥ 4γCmax βl Tl [t]1{4γCmax βl Tl [t]≥1}
p
4γCmax βl Tl [t]1{4γCmax βl Tl [t]≥1}
≥
p
p
=
4γCmax βl Tl [t] − 4γCmax βl Tl [t]1{4γCmax βl Tl [t]<1}
p
≥
4γCmax βl Tl [t] − 1,
(B.3.3)
where 1{·} is the indicator function.
Also, for any l = 1, 2, ..., L, if αl ≥ 1, then αl ≥
L
X
√
1
≥ 1 ≥ αl . Hence,
αl
αk
k=1
√
αl ; if 0 < αl < 1, then
( L
)
X 1
√
max
, 1 αl ≥ αl , ∀l.
αk
k=1
(B.3.4)
Thus, by using (B.3.3) and (B.3.4), we have
∆V
1
≤
2kY[t]k2
!
!
PL √
L
X
p
α
Q
[t]
l=1
nP l l o +
4γCmax βl Tl [t] + B1
− h2 ()
L
1
max
,
1
l=1
l=1 αl
1
(−h3 ()kY[t]k1 + B1 )
2kY[t]k2
h3 ()
B1
≤ −
+
,
2
2kY[t]k2




1
nP
o , 1 , and B1 , B + Lh2 ().
where h3 () , h2 () min
L
1
 max

,1
≤
(B.3.5)
l=1 αl
Hence, for any 0 < δ <
h3 ()
, if
2
kY[t]k2 >
B1
, b,
h3 () − 2δ
(B.3.6)
then, we have
∆V = E [kY[t + 1]k2 − kY[t]k2 |Y[t]] < −δ.
195
(B.3.7)
B.4
Proof of Lemma B.4
If kY[t]k2 > b, then
|kY[t + 1]k2 − kY[t]k2 |
=
≤
kY[t + 1]k22 − kY[t]k22
kY[t + 1]k2 + kY[t]k2
P
PL
L
2 2
[t]
[t
+
1]
−
α
Q
α
Q
l=1 l l
l=1 l l
P
P
4γCmax Ll=1 βl Tl [t + 1] − Ll=1 βl Tl [t]
+
kY[t + 1]k2 + kY[t]k2
kY[t + 1]k2 + kY[t]k2
P
P
P
P
P
L
L
L
2
2
4γCmax l=1 βl − l∈H∗ βl − l∈H∗ βl Tl [t]
l=1 αl Ql [t + 1] − l=1 αl Ql [t]
(a)
qP
≤ qP
+
kY[t + 1]k2 + kY[t]k2
L
L
2
2
α
Q
[t
+
1]
+
α
Q
[t]
l=1 l l
l=1 l l
P
P
√
4γCmax Ll=1 βl + 4γCmax l∈H∗ βl Tl [t]
√
≤ k αQ[t + 1]k2 − k αQ[t]k2 +
kY[t + 1]k2 + kY[t]k2
PL
4γCmax l=1 βl
√
√
≤ k αQ[t + 1]k2 − k αQ[t]k2 +
b
P
4γCmax l∈H∗ βl Tl [t]
,
(B.4.1)
+ qP
PL
L
2
l=1 βl Tl [t]
l=1 αl Ql [t] + 4γCmax
where the step (a) follows the fact that
kY[t]k22
≥
L
X
αl Q2l [t], ∀t, and the equation
l=1
(B.1.2). Note that
√
√
√
√
k αQ[t + 1]k2 − k αQ[t]k2 ≤ k αQ[t + 1] − αQ[t]k2
√
√
≤ k αQ[t + 1] − αQ[t]k1
√
≤ L max αl |Ql [t + 1] − Ql [t]|
l
√
≤ 2L max{Amax , Cmax } αmax ,
(B.4.2)
where αmax , max αl . By substituting above inequality into (B.4.1), we have the
l
desired result.
196
B.5
Proof of Inequality (B.2.20)
By using the law of total probability, we have
h
i
ηF1 Tl∗ [t] [t]
E e
1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]}
=
∞
X
m=0
h
i
E eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} Tl∗ [t] [t] = m Pr{Tl∗ [t] [t] = m}.(B.5.1)
h
i
Consider E eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} Tl∗ [t] [t] = m .
h
i
E eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} Tl∗ [t] [t] = m
= eηF1 m Pr Ql∗ [t] [t] < λmin Tl∗ [t] [t]Tl∗ [t] [t] = m
( t
)
X
≤ eηF1 m Pr
Al∗ [t] [j] < λmin mTl∗ [t] [t] = m ,
(B.5.2)
j=t−m+1
where the last step follows from the fact that given Tl∗ [t] [t] = m (i.e., link l∗ [t] is not
t
X
scheduled in the past m slots), Ql∗ [t] [t] ≥
Al∗ [t] [j]. By substituting (B.5.2)
j=t−m+1
into (B.5.1), we have
h
i
E eηF1 Tl∗ [t] [t] 1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]}
( t
)
∞
X
X
eηF1 m Pr
≤
Al∗ [t] [j] < λmin m, Tl∗ [t] [t] = m
m=0
≤
∞
X
j=t−m+1
eηF1 m Pr
m=0
≤
∞
X
≤
m=0
t
X
Al∗ [t] [j] < λmin m
j=t−m+1
eηF1 m Pr
m=0
∞
X
(
(
L
[
l=1
eηF1 m
L
X
l=1
Pr
(
(
t
X
j=t−m+1
))
Al [j] < λmin m
j=t−m+1
t
X
)
)
Al [j] < λmin m .
For each link l, Al [j] are i.i.d. over time and its mean is greater than λmin =
by Cramer’s theorem, we have
( t
)
X
Pr
Al [j] < λmin m < e−mIl (λl −λmin ) ,
j=t−m+1
197
(B.5.3)
1
min λl ,
2 l
(B.5.4)
where Il (·) is the rate function corresponding to the arrivals for link l. Note that
Il (λl − λmin ) > 0, ∀l. Thus, we have
∞
h
i
X
ηF1 Tl∗ [t] [t]
E e
1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} ≤ L
eηF1 m e−m minl Il (λl −λmin ) .
(B.5.5)
m=0
Hence, for 0 < η <
minl Il (λl − λmin )
, η2 , we have
F1
∞
h
i
X
ηF1 Tl∗ [t] [t]
E e
1{Ql∗ [t] [t]<λmin Tl∗ [t] [t]} ≤ L
e−(minl Il (λl −λmin )−F1 η)m
m=0
L
=
B.6
1 − e−(minl Il (λl −λmin )−F1 η)
, D3 . (B.5.6)
Proof of Lemma 4.3.1
In the rest of proof, we will omit the superscript p for brevity.
Proof of identity (4.3.1):
L
X
βl λl Tl [t + 1] =
X
βl λl (Tl [t] + 1)
l∈H
/
l=1
=
L
X
βl λl Tl [t] −
X
βl λl Tl [t] +
βl λl −
X
βl λl , (B.6.1)
l∈H
l=1
l∈H
l=1
L
X
where H , {l : Sl [t]Cl [t] > 0}. Taking expectation on both sides with respect to the
steady state distribution of (Q, T) and rearranging terms, we have the desired result.
Proof of identity (4.3.2):
L
X
βl λl Tl2 [t + 1] =
X
βl λl (Tl [t] + 1)2
l∈H
/
l=1
=
X
βl λl Tl2 [t] + 2
l∈H
/
=
L
X
βl λl Tl [t] +
l∈H
/
βl λl Tl2 [t] −
l=1
−2
X
X
X
βl λl Tl2 [t] + 2
l∈H
X
βl λl Tl [t] +
l∈H
L
X
l=1
198
βl λl
l∈H
/
L
X
βl λl Tl [t]
l=1
βl λl −
X
l∈H
βl λl .
(B.6.2)
Taking expectation on both sides with respect to the steady state distribution of
(Q, T) and rearranging terms, we have




L
X
X
X
2
βl λl T l  + E 
βl λl T l 
2
βl λl E T l = 2E 
l=1
l∈H
−
L
X
βl λl − E
l=1
"
l∈H
X
l∈H
βl λl
#!
.
(B.6.3)
Using Identity (4.3.1), we have the desired result.
B.7
Proof of Proposition 4.3.3
1X
αl Q2l . We have
Consider the quadratic Lyapunov function WQ (Q, T) ,
2 l=1
L
∆WQ (Q, T) = E [WQ (Q[t + 1], T[t + 1]) − WQ (Q[t], T[t])|Q[t], T[t]]
" L
#
L
X
1X
1
= E
αl Q2l [t + 1] −
αl Q2l [t]Q[t], T[t]
2
2
l=1
≤
≤
L
1X
2
l=1
L
X
l=1
E αl (Ql [t] + Al [t] − Cl [t]Sl∗ [t])2 − αl Q2l [t]Q[t], T[t]
αl E [Ql [t] (Al [t] − Cl [t]Sl∗ [t])|Q[t], T[t]]
l=1
1X
+
αl E A2l [t] + Cl2 [t] .
2 l=1
L
(B.7.1)
Taking expectation on both sides with respect to the steady state distribution of
2
(Q, T), and using the fact that E[∆WQ (Q, T)] = 0 followed from E[Ql ] < ∞ for all
l ∈ L, we have
0≤
L
X
l=1
L
L
i 1X
h
X
∗
αl λl E Ql −
αl E Q l Cl S l +
αl E A2l + Cl2 ,
2 l=1
l=1
which implies
L
X
l=1
L
L
h
i X
1X
∗
αl E Ql Cl S l ≤
αl λl E Ql +
αl E A2l + Cl2 .
2 l=1
l=1
199
(B.7.2)
Hence, we have
L
X
l=1
L
X
≤
E
h
i
∗
αl Ql + γβl T l Cl S l
αl λl E Ql + γ
l=1
L
X
l=1
L
i 1X
h
∗
βl E T l S l C l +
αl E A2l + Cl2 .
2 l=1
(B.7.3)
Recall that given Q[t] = Q, T[t] = T and the channel state C[t], we have
L
X
(αl Ql [t] + γβl Tl [t]) Cl [t]Sl∗ [t]
l=1
= max
S∈S
L
X
(αl Ql [t] + γβl Tl [t]) Cl [t]Sl .
(B.7.4)
l=1
According to the definition of the capacity region R, we can show
L
X
E [(αl Ql [t] + γβl Tl [t]) Cl [t]Sl∗ [t]|Q[t] = Q, T[t] = T]
l=1
= max
r∈R
L
X
(αl Ql [t] + γβl Tl [t]) rl .
(B.7.5)
l=1
The proof is available in Appendix B.8.
Since λ ∈ Int(R), there exists an > 0 such that λ(1 + ) ∈ R. Hence, we have
L
X
≥
l=1
L
X
E [(αl Ql [t] + γβl Tl [t]) Cl [t]Sl∗ [t]|Q[t] = Q, T[t] = T]
λl (1 + )E [αl Ql [t] + γβl Tl [t]|Q[t] = Q, T[t] = T] .
l=1
Taking expectation on both sides with respect to the steady state distribution of
(Q, T), we have
L
X
l=1
E
h
αl Ql + γβl Tl
∗
Cl S l
i
≥
L
X
l=1
200
λl (1 + )E αl Ql + γβl T l .
By substituting above inequality into (B.7.3) and canceling the common term in both
sides, we have
L
X
l=1
L
L
i
h
X
1 X
1
∗
βl E T l S l Cl +
αl E A2l + Cl2
1 + l=1
2γ(1 + ) l=1


L
X
1
Cmax  X
βl T l  +
E
≤
αl E A2l + Cl2
1+
2γ(1 + ) l=1
∗
l∈H



L
L
X
X
X
1
Cmax 
βl − E 
βl  +
αl E A2l + Cl2 ,
=
1 + l=1
2γ(1 + ) l=1
∗
βl λl E T l ≤
l∈H
where the last step uses identity (4.3.1).
B.8
Proof of Equation (B.7.5)
We will use the following fact in linear programming.
max
x∈A
L
X
al xl = max
l=1
x∈CH{A}
L
X
al x l ,
(B.8.1)
l=1
where x = (xl )Ll=1 is a L−dimensional vector, A is a set of L− dimensional vectors,
CH{A} is a convex hull of the set A and al , ∀l = 1, 2, ..., L, are real numbers.
Given Q[t], T[t] and C[t], we have
L
X
(αl Ql [t] + γβl Tl [t])Cl [t]Sl∗ [t]
l=1
= max
S∈S
=
L
X
(αl Ql [t] + γβl Tl [t])Cl [t]Sl
l=1
max
(C[t])
v=(vl )L
l=1 ∈S
L
X
(αl Ql [t] + γβl Tl [t])vl ,
(B.8.2)
l=1
where we recall that S (c) , {Sc : S ∈ S}, and ab , (al bl )Ll=1 denotes the componentwise product of two vectors a and b.
201
Next, we will show that
L
X
E[(αl Ql [t] + γβl Tl [t])Cl [t]Sl∗ [t]|Q[t] = Q, T[t] = T]
l=1
=
max
r=(rl )L
l=1 ∈R
L
X
(αl Ql [t] + γβl Tl [t])rl .
(B.8.3)
l=1
On one hand,
L
X
E[(αl Ql [t] + γβl Tl [t])Cl [t]Sl∗ [t]|Q[t] = Q, T[t] = T]
l=1
(a)
= E
=
"
max
v∈S (C[t])
X
L
X
l=1
#
(αl Ql [t] + γβl Tl [t])vl Q[t] = Q, T[t] = T
Pr{C[t] = c} max
v∈S (c)
c
(b)
=
X
Pr{C[t] = c}
=
X
Pr{C[t] = c}
c
=
L
X
l=1
max
L
X
L
X
(αl Ql [t] + γβl Tl [t])vl
l=1
∗(c)
(αl Ql [t] + γβl Tl [t])vl
l=1
(αl Ql [t] + γβl Tl [t])
X
∗(c)
Pr{C[t] = c}vl
c
l=1
(d)
≤
(αl Ql [t] + γβl Tl [t])vl
v∈CH{S (c) }
c
(c)
L
X
max
r=(rl )L
l=1 ∈R
L
X
(αl Ql [t] + γβl Tl [t])rl ,
(B.8.4)
l=1
where the steps (a) and (b) follow from equation (B.8.2) and equation (B.8.1), respectively; step (c) is true for
v
∗(c)
=
∗(c)
(vl )Ll=1
∈ argmax
L
X
(αl Ql [t] + γβl Tl [t])vl ;
v∈CH{S (c) } l=1
and step (d) follows from the fact that v∗(c) ∈ CH{S (c) } and
X
c
R.
202
Pr{C[t] = c}v∗(c) ∈
On the other hand,
max
r=(rl )L
l=1 ∈R
(a)
=
(b)
=
L
X
l=1
L
X
L
X
(αl Ql [t] + γβl Tl [t])rl
l=1
(αl Ql [t] + γβl Tl [t])rl∗
(αl Ql [t] + γβl Tl [t])
X
Pr{C[t] = c}
c
≤
X
=
X
Pr{C[t] = c}
= E
(d)
=
max
Pr{C[t] = c} max
v∈S (c)
max
v∈S (C[t])
L
X
(c)
(αl Ql [t] + γβl Tl [t])vl
v∈CH{S (c) }
c
"
L
X
l=1
c
(c)
(c)
Pr{C[t] = c}vl
c
l=1
=
X
L
X
l=1
L
X
L
X
(αl Ql [t] + γβl Tl [t])vl
l=1
(αl Ql [t] + γβl Tl [t])vl
l=1
#
(αl Ql [t] + γβl Tl [t])vl Q[t] = Q, T[t] = T
E[(αl Ql [t] + γβl Tl [t])Cl [t]Sl∗ [t]|Q[t] = Q, T[t] = T],
(B.8.5)
l=1
L
X
(αl Ql [t] + γβl Tl [t])rl ; step (b)
r∈R
l=1
X
∗
∗
Pr{C[t] =
follows from the fact that r ∈ R and thus r can be written as r∗ =
where the step (a) is true for r∗ = (rl∗ )Ll=1 ∈ argmax
c
c}v(c) , where v(c) ∈ CH{S (c) } for each channel state c; and step (c) and (d) follow
from equation (B.8.1) and equation (B.8.2), respectively.
By combing (B.8.4) and (B.8.5), we have the desired result.
B.9
Detailed Heavy-Traffic Analysis
In this section, we prove Proposition 4.5.3 by using the analytical approach in [15],
which includes two parts: (i) showing state-space collapse; (ii) using the state-space
collapse result to obtain an upper bound on the mean queue lengths. Yet, it is
203
worth noting that the strong coupling between queue length processes and TSLS
counters in the RSG Algorithm poses significant challenges in heavy-traffic analysis.
In particular, it requires new Lyapunov functions and a novel technique to establish
heavy-traffic optimality of the RSG Algorithm.
B.9.1
State-Space Collapse
We have mentioned in Section 4.2 that the RSG Algorithm is throughput-optimal, i.e.,
it stabilizes all queues for any arrival rate vector that are strictly within the capacity
region. Let {Q() }t≥0 and {T() }t≥0 be queue-length processes and TSLS counters
under the RSG Algorithm, respectively. Also, we use Q
()
()
and T
to denote their
limiting queue-length random vector and limiting TSLS random vector, respectively.
Then, by the continuous mapping theorem, we have
()
()
()
()
()
()
Qk ⇒ Qk ,
Q⊥ ⇒ Q⊥ ;
()
Tk ⇒ Tk ,
(B.9.1)
()
T⊥ ⇒ T⊥ ,
(B.9.2)
where ⇒ denotes convergence in distribution, and we define the projection and the
perpendicular vector of any given L-dimensional vector I with respect to the normal
vector d as:
Ik , hd, Iid,
I⊥ , I − Ik .
()
Next, we will show that under the RSG Algorithm, the second moment of kQ⊥ k
()
is bounded, dependent on γ(), while the second moment of kT k is bounded by
some constant independent of .
Proposition B.9.1. If Pr{Cl [t] = 0} > 0, ∀l ∈ L, then, under the RSG Algorithm,
there exists a constant NT,2 , independent of , such that
()
E[kQ⊥ k2 ] = O (γ())4 (log γ())2 ,
()
E[kβ · T k2 ] ≤ NT,2 .
204
(B.9.3)
(B.9.4)
We prove Proposition B.9.1 by first studying the drift of the Lyapunov function
()
V⊥ (Q() , T() ) , k(Q⊥ ,
q
2γ()Cmax β · T() )k,
and show that when V⊥ (Q() , T() ) is sufficiently large, it has a strictly negative drift
independent of , which is characterized in the following key lemma.
Lemma B.9.2. Under the RSG Algorithm, there exist positive constants κ and ς,
independent of , such that whenever V⊥ (Q() [t], T() [t]) > κ, we have
E[∆V⊥ (Q() [t], T() [t])|Q() [t], T () [t]] < −ς,
(B.9.5)
where ∆V⊥ (Q() [t], T() [t]) , V⊥ (Q() [t + 1], T() [t + 1]) − V⊥ (Q() [t], T() [t]).
The proof of Lemma B.9.2 is available in Appendix B.10.
Note that the TSLS counters have bounded increment but unbounded decrement,
since they can at almost increase by 1 and drop to 0 once their corresponding links
are scheduled. Due to this characteristic of TLSL, the absolute value of the drift
∆V⊥ (Q() , T() ) has neither an upper bound nor an exponential tail given the current
system state (Q() , T() ). Thus, we cannot directly apply Theorem 2.3 in [26], which
requires either boundedness or the exponential tail of the Lyapunov drift to establish
the existence of the second moment of the stochastic process. Indeed, for a Markov
Chain with a strictly negative drift of Lyapunov function, if its Lyapunov drift has
bounded increment but unbounded decrement, its second moment may not exist.
Counterexample: Consider a Markov Chain {X[t]}t≥0 with the following transition
probability:
Pj,j+1 =



1








1
2
j−1
j+1
if j = 0;



if j = 1; Pj,0 = 

if j ≥ 2.
1
2
if j = 1;
2
j+1
if j ≥ 1.
205
1
0
1
1/2
1/3
1/2
2
2/3
2/4
3
2/4
4
2/5
Figure B.1: Markov Chain {X[t]}t≥0
The state transition diagram of Markov Chain {X[t]}t≥0 is shown in Figure B.1.
Consider a linear Lyapunov function X. For any X ≥ 2, we have
E[X[t + 1] − X[t]|X[t] = X] =
X −1
2X
−
= −1.
X +1 X +1
Thus, the Lyapunov function X has a strictly negative drift when X ≥ 2 and hence
the steady-state distribution of the Markov Chain exists. Recall that its drift increases
at almost by 1, but has unbounded decrement, which has similar dynamics with the
system under the RSG Algorithm.
Next, we will show that even the first moment of this Markov Chain does not
exist, let alone its second moment. Let X be the limiting random variable of the
Markov Chain and πj , Pr{X = j}. According to the global balance equations, we
can easily calculate
1
π1 = π 0 = ,
3
πj =
1
.
3j(j − 1)
(B.9.6)
Thus, we have
∞
X
∞
1 X
1
E[X] =
jπj = +
= ∞.
3
3(j
−
1)
j=1
j=2
()
Fortunately, we can establish the boundedness of the second moment of kQ⊥ k
under the RSG Algorithm by exploiting its unique dynamics under a mild assumption
206
that Pr{Cl [t] = 0} > 0, ∀l ∈ L, which leads to the following lemma that all TSLS
counters have an exponential tail independent of .
Lemma B.9.3. If pl , Pr{Cl [t] = 0} > 0, ∀l ∈ L, then, under the RSG Algorithm,
there exists a ϑ ∈ (0, 1), independent of , such that
Pr{Tl [t] ≥ m} ≤ ϑm ,
∀t ≥ m ≥ 0, ∀l ∈ L.
(B.9.7)
The proof of Lemma B.9.3 is available in Appendix B.11.
Remark: We can also show that all TLSL counters still have an exponential tail
independent of in non-fading single-hop network topologies. The extension to the
more general setup is left for future search.
Lemma B.9.3 directly implies (B.9.4). The rest of proof mainly builds on the ana()
lytical technique in [26], while it requires carefully partitioning the space (Q⊥ , T() ).
The detailed proof can be found in Appendix B.12.
B.9.2
Proof of Heavy-Traffic Optimality
()
We first give an upper bound on E[hd, Q i] by using the methodology of “setting the
drift of a Lyapunov function equal to zero”. We will omit the superscript associated
with the queue lengths and TSLS counters for brevity in the rest of proof. To derive
an upper bound, we need the following fundamental identity (see Lemma 8 in [15]):
E hd, A − C · S∗ (Q, T, C)i2
E hd, U(Q, T, C)i2
+
2
2
+E hd, Q + A − C · S∗ (Q, T, C)ihd, U(Q, T, C)i
= E hd, Qihd, C · S∗ (Q, T, C) − Ai ,
(B.9.8)
which is derived through setting E[∆Wk (Q, T)] = 0.
Next, we give upper bounds for each individual term in the left hand side of
207
(B.9.8) and a lower bound for the right hand side of (B.9.8). By setting the mean
drift of hd, Qi equal to zero and using the fact that Ul ≤ Cmax for all l, we have
E hd, U(Q, T, C)i = hd, E[C · S∗ (Q, T, C)]i − hd, λi
(a)
(b)
= hd, E[C · S∗ (Q, T, C)]i − (b − ) ≤ ,
(B.9.9)
where the step (a) follows from the definition of λ(0) , λ + d and hd, λ(0) i = b; step
(b) follows from the facts that E[C · S∗ (Q, T, C)] must be within capacity region R
and that hd, ri ≤ b for any vector r ∈ R.
Using the fact that Ul ≤ Cmax for each link l, we have
1
1 E hd, U(Q, T, C)i2 ≤ hd, Cmax 1iE hd, U(Q, T, C)i ≤ hd, Cmax 1i.(B.9.10)
2
2
2
Inequality (B.9.10) means that there is almost no unused services under heavy-traffic
conditions.
By observing that the RSG Algorithm selects the schedule S which maximizes
hd, Si with high probability, we can show
X E hd, A − C · S∗ (Q, T, C)i2 ≤ ζ () +
2bbc + (bc )2 + hd, Cmax 1i2 (B.9.11)
,
γ
c
c∈C
2
where we recall that ζ () , hd2 , σ () i + Var(Ψ) + 2 is defined in Proposition 4.5.3,
and
πc , Pr hd, C · S∗ (Q, T, C)i = bc C = c ,
γc , min bc − hd, ri : for all r ∈ S (c) \ {w : bc = hd, wi} .
The detailed proof is provided in Appendix B.15. Inequality (B.9.11) indicates that
the second moment of d-weighted difference between arrivals and services is dominated by the d2 -weighted variance of the arrival process and the variance of the
channel fading process in the heavy-traffic limit.
208
In addition, by using similar arguments as in the proof for Proposition 4 in [15],
we have
E hd, Q + A − C · S∗ (Q, T, C)ihd, U(Q, T, C)i ≤
where dmin ,
min
m∈{l:dl >0}
r
E[kQ⊥ k2 ]
Cmax
, (B.9.12)
dmin
dm .
Finally, by using the definition of the RSG Algorithm and Proposition B.9.1, we
can show
E hd, Qihd, C · S∗ (Q, T, C) − Ai
q
≥ E kQk k − cot(θ) E[kQ⊥ k2 ] + (γ())2 kβ · Tk2 s
X 1
(bc )2 + hd, Cmax 1i2 ,
×
χc
c∈C
(B.9.13)
π
where θ ∈ (0, ] is an angle such that hd, R∗ (Q, T)i = b, for all Q and T satisfying
2
k(Q + γβ · T)k k
≥ cos(θ), and R∗ (Q, T) , E[C · S∗ (Q, T, C)|Q, T]. The detailed
kQ + γβ · Tk
proof is provided in Appendix B.16.
By substituting bounds (B.9.10), (B.9.11), (B.9.12) and (B.9.13) into identity
(B.9.8), we have
ζ ()
()
+B ,
E kQk k ≤
2
where
(B.9.14)
r
Cmax 1 X B , hd, Cmax 1i + E[kQ⊥ k2 ]
+
2bbc + (bc )2 + hd, Cmax 1i2
2
cmin
2 c∈C χc
s
q
X 1
+ cot(θ) 2 E[kQ⊥ k2 ] + (γ())2 NT,2 ×
(bc )2 + hd, Cmax 1i2 (B.9.15)
.
χ
c
c∈C
()
Thus, if lim B
↓0
()
= 0, then the RSG Algorithm is heavy-traffic optimal. Noting that
NT,2 is independent of , to satisfy lim B
↓0
()
= 0, it is enough to have
lim E[kQ⊥ k2 ] = 0 and lim (γ())2 = 0.
↓0
↓0
209
(B.9.16)
1
By using Proposition B.9.1, it is easy to see that γ() = O( √
) meets the above
5
requirements.
B.10
Proof of Lemma B.9.2
Since normal vector d 0, we have λ(0) 0. In addition, since λ(0) is a relative
interior point of dominant hyperplane H(d) , there exists a small enough δ > 0 such
that
Bδ , H
(d)
o
\n
(0)
r 0 : kr − λ k ≤ δ ,
(B.10.1)
representing the set of vectors on the hyperplane H(d) that are within δ distance from
\
λ(0) , lies strictly within the face F (d) , H(d) R.
In the rest of proof, we will omit associated with the queue length processes,
the TSLS counters and parameter γ() for brevity. Noting the difficulty to directly
study the drift of Lyapunov function V⊥ (Q, T), we relate it with the drift of other
proper Lyapunov functions, which is characterized in the following lemma.
Lemma B.10.1. Define the following Lyapunov functions:
W (Q, T) , k(Q,
p
2γCmax β · T)k2 ,
Wk (Q, T) , kQk k2 .
(B.10.2)
(B.10.3)
Then, given Q[t] = Q and T[t] = T, their one-step drifts denoted by:
∆W (Q, T) , [W (Q[t + 1], T[t + 1]) − W (Q[t], T[t])] ,
∆Wk (Q, T) , Wk (Q[t + 1], T[t + 1]) − Wk (Q[t], T[t]) ,
satisfy the following inequality:
∆V⊥ (Q, T) ≤
∆W (Q, T) − ∆Wk (Q, T)
√
.
2k(Q⊥ , 2γCmax β · T)k
210
(B.10.4)
The proof of Lemma B.10.1 is similar to that in [15] and is omitted here for brevity.
The rest of proof follows from Lemma B.10.1 by studying the conditional expectation of ∆W (Q, T) and ∆Wk (Q, T). We will omit the time reference [t] without
confusion.
We first consider E [∆W (Q, T)|Q[t] = Q, T[t] = T].
E [∆W (Q, T)|Q[t] = Q, T[t] = T]
= E[kQ[t + 1]k2 + 2γCmax kβ · T[t + 1]k1 − kQ[t]k2 − 2γCmax kβ · T[t]k1 |Q[t], T[t]]
= E[kQ + A − S∗ · C + Uk2 − kQk2 + 2γCmax (kβ · T[t + 1]k1 − kβ · T[t]k1 )|Q, T]
= E[kQ + A − S∗ · Ck2 + 2hQ + A − S∗ · C, Ui + kUk2
−kQk2 + 2γCmax (kβ · T[t + 1]k1 − kβ · T[t]k1 )|Q, T]
(a)
≤ E[kQ + A − S∗ · Ck2 − kQk2 + 2γCmax (kβ · T[t + 1]k1 − kβ · T[t]k1 )|Q, T]
"
(b)
= E 2hQ, A − S∗ · Ci + kA − S∗ · Ck2
+2γCmax
L
X
βl −
X
βl −
l∈H∗
l=1
∗
X
l∈H∗
#
!
βl Tl [t] Q, T
≤ 2E [hQ, A − S · Ci|Q, T] + K1 − 2γE [hβ · T, S∗ · Ci|Q, T] ,
where K1 ,
2
L max{A2max , Cmax
}
+ 2γCmax
L
X
(B.10.5)
βl and the step (a) uses the fact that
l=1
Ul [t](Ql [t] + Al [t] − Sl∗ [t]Cl [t]) = −Ul2 [t] ≤ 0, for each l; (b) follows from the fact
L
L
L
X
X
X
X
X
that
βl Tl [t + 1] −
βl Tl [t] =
βl −
βl −
βl Tl [t], where H∗ , {l ∈ L :
l=1
Sl∗ [t]Cl [t] > 0}.
l=1
l=1
l∈H∗
l∈H∗
Next, we consider E [hQ, A − S∗ · Ci|Q, T]. By using the definition of projection
211
λ(0) , we have
E [hQ, A − S∗ · Ci|Q, T] = hQ, λ(0) − di − E[hQ, S∗ · Ci|Q, T]
= −kQk k + hQ, λ(0) i − E[hQ + γβ · T, S∗ · Ci|Q, T]
+γE[hβ · T, S∗ · Ci|Q, T].
(B.10.6)
Given the queue-length vector Q[t], TSLS vector T[t] and the channel state C[t] at
the beginning of slot t, according to the definition of the RSG Algorithm, we have
hQ[t] + γβ · T[t], S∗ [t] · C[t]i = max hQ[t] + γβ · T[t], S · C[t]i,
S∈S (C[t])
which implies
hQ + γβ · T, E[S∗ · C|Q, T]i = maxhQ + γβ · T, ri
r∈R
≥ maxhQ + γβ · T, ri
r∈Bδ
≥ maxhQ, ri + hγβ · T, r∗ i,
r∈Bδ
where r∗ ∈ argmaxhQ, ri. Since λ(0) 0, we can find a δ > 0 sufficiently small such
r∈Bδ
that rl ≥ rmin for all r = (rl )l∈L ∈ Bδ and some rmin > 0. Hence, we have
hQ + γβ · T, E[S∗ · C|Q, T]i ≥ maxhQ, ri + γrmin kβ · Tk1 .
r∈Bδ
By substituting above inequality into (B.10.6), we have
E [hQ, A − S∗ · Ci|Q, T]
≤ −kQk k + minhQ, λ(0) − ri − γrmin kβ · Tk1 + γE[hβ · T, S∗ · Ci|Q, T].
r∈Bδ
Since λ(0) − r is perpendicular to the normal vector d for r ∈ Bδ , we have
minhQ, λ(0) − ri = minhQ⊥ , λ(0) − ri = −δkQ⊥ k.
r∈Bδ
r∈Bδ
Hence, we have
E [hQ, A − S∗ · Ci|Q, T]
≤ −kQk k − δkQ⊥ k − γrmin kβ · Tk1 + γE[hβ · T, S∗ · Ci|Q, T]. (B.10.7)
212
Thus, by substituting (B.10.7) into (B.10.5), we have
E[∆W (Q, T)|Q, T] ≤ −2kQk k − 2δkQ⊥ k − 2γrmin kβ · Tk1 + K1 . (B.10.8)
Next, we lower bound E[∆Wk (Q, T)|Q, T].
E[∆Wk (Q, T)|Q[t] = Q, T[t] = T]
= E[hd, Q[t + 1]i2 − hd, Q[t]i2 |Q[t] = Q, T[t] = T]
= E[hd, Q + A − S∗ · C + Ui2 − hd, Qi2 |Q, T]
= E[hd, Q + A − S∗ · Ci2 + hd, Ui2 + 2hd, Q + A − S∗ · Cihd, Ui − hd, Qi2 |Q, T]
= E[2hd, Qihd, A − S∗ · Ci + hd, A − S∗ · Ci2
+2hd, Q + A − S∗ · Cihd, Ui + hd, Ui2 |Q, T]
= 2hd, Qihd, λ − E[S∗ · C|Q, T]i − 2E[hd, S∗ · Cihd, Ui|Q, T] + E[hd, Ui2 |Q, T]
+E[hd, A − S∗ · Ci2 + 2hd, Q + Aihd, Ui|Q, T]
≥ 2hd, Qihd, λ − E[S∗ · C|Q, T]i − 2E[hd, S∗ · Cihd, Ui|Q, T]
(a)
≥ 2hd, Qihd, λ − E[S∗ · C|Q, T]i − K2
(b)
= −2kQk k − K2 + 2kQk k hd, λ(0) i − hd, E[S∗ · C|Q, T]i
(c)
≥ −2kQk k − K2 ,
(B.10.9)
2
where the step (a) is true for K2 , 2LCmax
; step (b) uses the definition of projection
λ(0) ; step (c) follows from the fact that R ⊂ {r 0 : hd, ri ≤ b} and b = hd, λ(0) i.
By using the bounds (B.10.8), (B.10.9) and Lemma B.10.1, we have
E ∆W (Q, T) − ∆Wk (Q, T)|Q, T
√
E[∆V⊥ (Q, T)|Q, T] ≤
2k(Q⊥ , 2γCmax β · T)k
P
−2δkQ⊥ k − 2γrmin Ll=1 βl Tl + K1 + K2
√
≤
.
2k(Q⊥ , 2γCmax β · T)k
p
p
p
Note that γβl Tl ≥ γβl Tl 1{αTl ≥1} ≥ γβl Tl 1{γβl Tl ≥1} = γβl Tl − γβl Tl 1{γβl Tl <1} ≥
213
p
1
γβl Tl − 1, and kQ⊥ k ≥ √ kQ⊥ k1 , where 1{·} is an indicator function. Thus, we
L
have
P √
− √2δL kQ⊥ k1 − 2rmin Ll=1 γβl Tl + K1 + K2 + 2Lrmin
√
E[∆V⊥ (Q, T)|Q, T] ≤
2k(Q⊥ , 2γCmax β · T)k
δ
rmin
K1 + K2 + 2Lrmin
√
≤ − min √ , √
+
.
2k(Q⊥ , 2γCmax β · T)k
L 2Cmax
rmin
δ
, by taking
Hence, for any 0 < ς < min √ , √
L 2Cmax
κ,
we have the desired result.
B.11
K1 + K2 + 2Lrmin
n
o
,
rmin
2 min √δL , √2C
−
ς
max
(B.10.10)
Proof of Lemma B.9.3
If the event
Ej , {Cl [j] > 0, Ci [j] = 0, ∀i 6= l; Al [j − 1] > 0}
(B.11.1)
happens for some j ∈ [t − m + 1, t), then under the RSG Algorithm, link l should be
()
scheduled at least once during the past m slots and thus Tl [t] < m. This implies
()
Pr{Tl [t] ≥ m} ≤ Pr{Ej does not happen for all j ∈ [t − m + 1, t)} = ϑm
l .
where ϑl , 1 − (1 − ql )(1 − pl )Πi6=l pi , and ql , Pr{Al [t] = 0}, ∀l ∈ L. Since λl > 0,
we have ql < 1 and thus ϑl ∈ (0, 1). Thus, by taking ϑ , max ϑl , we have the desired
l∈L
result.
B.12
Proof of Proposition B.9.1
In the rest of proof, we will omit associated with the queue length processes, the
TSLS counters and parameter γ() for brevity. It is quite challenging to directly
214
give an upper bound on E[kQ⊥ k2 ]. Instead, we upper-bound the moment generation
function of kQ⊥ k, and use the relationship between the moments of a random variable and its moment generation function to upper-bound E[kQ⊥ k2 ] as shown in the
following lemma.
Lemma B.12.1. For a random variable X with E[eηX ] < ∞ for some η > 0, we
have
E[X n ] ≤
for n = 1, 2, 3 · · · .
n
1
log en−1 E[eηX ]
,
n
η
(B.12.1)
Please see the Appendix B.14 for the proof of Lemma B.12.1.
p
Let Z[t] , Q⊥ [t], 2γCmax β · T[t] . We first give an upper bound on
E eηkZ[t+1]k Q[t], T[t] .
To that end, let l∗ [t] ∈ argmax βl Tl [t]. We partition (Q⊥ [t], T[t]) into sets F1 , F2 and
l
F3 , where
F1 , {kZ[t]k ≤ κ} ;
F2 , kZ[t]k > κ, kQ⊥ [t]k > Tl∗ [t] [t] ;
F3 , kZ[t]k > κ, kQ⊥ [t]k ≤ Tl∗ [t] [t] .
Then, we have
3
X
E eηkZ[t+1]k Q[t], T[t] =
E eηkZ[t+1]k ; Fi Q[t], T[t] .
i=1
Next, we consider each term in (B.12.2) individually.
(i) On event F1 , we have
v
u
L
u
X
kZ[t]k = tkQ⊥ [t]k2 + 2γCmax
βl Tl [t] ≤ κ,
l=1
215
(B.12.2)
which implies kQ⊥ [t]k ≤ κ.
For Q⊥ [t + 1], we have
Q⊥ [t + 1] = Q[t + 1] − hd, Q[t + 1]id
= (Q[t] + A[t] − S[t] + U[t]) − hd, Q[t] + A[t] − S[t] + U[t]id
= Q[t] − hd, Q[t]id + (A[t] + U[t] + hd, S[t]id) − (S[t] + hd, A[t] + U[t]id)
(a)
√
Q⊥ [t] + Amax 1 + Cmax 1 + LCmax d
√
Q⊥ [t] + (Amax + 2 LCmax )1,
(B.12.3)
where step (a) uses the following inequality
hd, S[t]i ≤ kdkkS[t]k ≤
√
LCmax .
Hence, we have
kZ[t + 1]k
2
2
= kQ⊥ [t + 1]k + 2γCmax
L
X
βl Tl [t + 1]
l=1
L
X
√
≤ kQ⊥ [t] + (Amax + 2 LCmax )1k2 + 2γCmax
βl (Tl [t] + 1)
(a)
2
√
l=1
= kZ[t]k + 2 Amax + 2 LCmax kQ⊥ [t]k1
√
+L Amax + 2 LCmax
2
+ 2γCmax
√
√
≤ κ2 + 2 Amax + 2 LCmax κ L
(b)
√
+L Amax + 2 LCmax
2
+ 2γCmax
L
X
βl
l=1
L
X
βl , G21 ,
(B.12.4)
l=1
where step (a) uses the inequality (B.12.3); step (b) utilizes the inequality kxk1 ≤
√
Lkxk for any L-dimensional vector x. Hence, we have
E eηkZ[t+1]k ; F1 Q[t], T[t] ≤ eηG1
To consider other two terms in (B.12.2), we need the following lemma.
216
(B.12.5)
Lemma B.12.2. Under the RSG Algorithm, if kZ[t]k > κ, then
P
2γCmax Ll=1 βl
|kZ[t + 1]k − kZ[t]k| ≤ 2L max{Amax , Cmax } +
κ
P
β
T
[t]
l∈H∗ l l
,(B.12.6)
+2γCmax q
P
kQ⊥ [t]k2 + 2γCmax Ll=1 βl Tl [t]
where H∗ , {l : Sl∗ [t]Cl [t] > 0}.
The proof is available in Appendix B.13.
(ii) On event F2 , we have
P
L
X
Lβl∗ [t] Tl∗ [t] [t]
l∈H∗ βl Tl [t]
q
≤
≤L
βl .
P
kQ⊥ [t]k
l=1
kQ⊥ [t]k2 + 2γCmax Ll=1 βl Tl [t]
By substituting above inequality into (B.12.6), we get
|kZ[t + 1]k − kZ[t]k| ≤ G2 ,
(B.12.7)
P
L
X
2γCmax Ll=1 βl
+ 2γCmax L
βl . Noting that
where G2 , 2L max{Amax , Cmax } +
κ
l=1
(B.9.5) and (B.12.7) satisfy conditions of Lemma 2.2 in [26], there exists η1 > 0, and
ρ = eηG2 − η(G2 + ς) ∈ (0, 1), independent of , such that
Thus, we have
E eη(kZ[t+1]k−kZ[t]k) ; F2 Q[t], T[t] ≤ ρ, ∀0 < η < η1 .
E eηkZ[t+1]k ; F2 Q[t], T[t] ≤ ρeηkZ[t]k .
(iii) On event F3 , we have
P
Lβl∗ [t] Tl∗ [t] [t]
l∈H∗ βl Tl [t]
q
p
≤
P
2γCmax βl∗ [t] Tl∗ [t] [t]
kQ⊥ [t]k2 + 2γCmax Ll=1 βl Tl [t]
q
L
= √
βl∗ [t] Tl∗ [t] [t]
2γCmax
qP
L
q
L
l=1 βl
≤ √
Tl∗ [t] [t]
2γCmax
qP
L
L
l=1 βl
≤ √
Tl∗ [t] [t].
2γCmax
217
(B.12.8)
(B.12.9)
By substituting (B.12.9) into (B.12.6), we get
|kZ[t + 1]k − kZ[t]k| ≤ 2L max{Amax , Cmax } +
v
u
L
u
X
l=1 βl
t
+ L 2γCmax
βl Tl∗ [t] [t].
PL
2γCmax
κ
l=1
In addition, on event F3 , we have
v
u
L
u
X
t
2
kZ[t]k =
kQ⊥ [t]k + 2γCmax
βl Tl [t]
l=1
q
Tl2∗ [t] [t] + 2γCmax Lβl∗ [t] Tl∗ [t] [t]
≤
v
u
L
u
X
t
2
≤
Tl∗ [t] [t] + 2γCmax LTl∗ [t] [t]
βl
l=1
v
u
L
u
X
t
βl Tl∗ [t] [t]
1 + 2γCmax L
≤
(B.12.10)
l=1
Hence, we have
kZ[t + 1]k ≤ kZ[t]k + |kZ[t + 1]k − kZ[t]k|
≤ F1 Tl∗ [t] [t] + F2 ,
(B.12.11)
v
v
u
u
P
L
L
u
u
X
X
2γCmax Ll=1 βl
t
t
where F1 , L 2γCmax
βl + 1 + 2γCmax L
βl and F2 ,
+
κ
l=1
l=1
2L max{Amax , Cmax }. Thus, we have
E eηkZ[t+1]k ; F3 Q[t], T[t] ≤ eηF2 eηF1 Tl∗ [t] [t] .
(B.12.12)
By substituting (B.12.5), (B.12.8) and (B.12.12) into (B.12.2), we have
E eηkZ[t+1]k |Q[t], T[t] ≤ eηG1 + ρeηkZ[t]k + eηF2 eηF1 Tl∗ [t] [t] .
By taking expectation on both sides, we have
E eηkZ[t+1]k ≤ eηG1 + ρE eηkZ[t]k + eηF2 E eηF1 Tl∗ [t] [t]
≤ e
ηG1
L
X
ηkZ[t]k ηF2
+ ρE e
+e
E eηF1 Tl [t] .
l=1
218
(B.12.13)
Next, we upper bound the term E eηF1 Tl [t] .
t
ηF1 T [t] (a) X
l
=
eηF1 m Pr{Tl [t] = m}
E e
≤
(b)
≤
≤
m=0
t
X
m=0
t
X
m=0
∞
X
eηF1 m Pr{Tl [t] ≥ m}
eηF1 m ϑm
eηF1 m ϑm
m=0
(c)
=
1
,
1 − eηF1 ϑ
(B.12.14)
where step (a) uses the fact that Tl [t] ≤ t for any t ≥ 0; step (b) follows from Lemma
B.9.3; step (c) is true for 0 < η < η2 and ϑeη2 F1 < 1.
By substituting (B.12.14) into (B.12.13), we have
E eηkZ[t+1]k ≤ ρE eηkZ[t]k + G,
holding for 0 < η < η0 , min{η1 , η2 }, where G , eηG1 +
(B.12.15)
LeηF2
. By using
1 − eηF1 ϑ
inequality (B.12.15) and iterating over t, we have
G
1 − ρt
G ≤ eηkZ[0]k +
,
E eηkZ[t]k ≤ ρt eηkZ[0]k +
1−ρ
1−ρ
G
which implies E eηkQ⊥ [t]k ≤ eηkZ[0]k +
.
1−ρ
Thus, we have
h
i
G
ηkQ⊥ [t]k
E e
≤ eηkZ[0]k +
1−ρ
(B.12.16)
LeηF2
, ρ , eηG2 − η(G2 + ς) ∈ (0, 1), κ = O(γ), G1 = O(γ),
ηF
1
1−e ϑ
√
G2 = O(γ), F1 = O( γ) and F2 = O(1). Note that we need to choose a η > 0 such
where G , eηG1 +
that
ϑeηF1 < 1
(B.12.17)
eηG2 − η(G2 + ς) < 1.
(B.12.18)
219
It is not hard to verify that
(B.12.19)
satisfies above requirements. If γ is large enough such that
ς
< 1 and G2 F1 ,
G2
1
0 < η ≤ min
2
1
1 1
G2 + ς
ln ,
ln
F1 ϑ G2
G2
then we have
1
G2 + ς
1
1
ln
≤
ln .
G2
G2
F1 ϑ
Thus, we can take η ∗ ,
η ∗ = O(
1
).
γ2
1
G2 + ς
ln
to meet the above requirements, and hence
2G2
G2
Taking η = η ∗ and noting that η ∗ <
G
1−ρ
=
1−
1
1
ln , we have
2F1 ϑ
∗
Leη F2
∗
1−ϑeη F1
∗
(eη G2 − η ∗ (G2 +
eη
∗G
1
η∗ G
≤
(B.12.20)
e
+
1
+
1 − (eη∗ G2 −
ς))
∗
Leη F2
√
1− ϑ
η ∗ (G2
∗G
+ ς))
η∗ F
2
√
+ Le
1− ϑ
=
12
1 − 1 + Gς2
+ 12 1 + Gς2 ln 1 + Gς2


1
(a)
 = O γ2 ,
= O
1 − 1 + 2Gς 2 + 12 1 + Gς2 Gς2
eη
1
1
1
∗
and η F2 = O
. Thus, we have
where the step (a) uses η G1 = O
γ
γ2
h ∗
i
E eη kQ⊥ k = O(γ 2 ). By using Lemma B.12.1, we have
∗
E[kQ⊥ k2 ] ≤
1 h η∗ kQ⊥ [t]k i2
2
4
log
eE
e
=
O
γ
(log
γ)
.
(η ∗ )2
220
B.13
Proof of Lemma B.12.2
If kZ[t]k > κ, then
|kZ[t + 1]k − kZ[t]k|
=
|kZ[t + 1]k2 − kZ[t]k2 |
kZ[t + 1]k + kZ[t]k
2
2
≤
|kQ⊥ [t + 1]k − kQ⊥ [t]k |
+
kZ[t + 1]k + kZ[t]k
≤
|kQ⊥ [t + 1]k2 − kQ⊥ [t]k2 |
+
kQ⊥ [t + 1]k + kQ⊥ [t]k
P
P
2γCmax Ll=1 βl Tl [t + 1] − Ll=1 βl Tl [t]
kZ[t + 1]k + kZ[t]k
P
P
P
L
2γCmax l=1 βl − l∈H∗ βl − l∈H∗ βl Tl [t]
2γCmax
≤ |kQ⊥ [t + 1]k − kQ⊥ [t]k| +
P
PL
kZ[t + 1]k + kZ[t]k
P
P
L
β
β
−
∗
l + 2γCmax
l∈H∗ βl Tl [t]
l∈H
l=1 l
kZ[t + 1]k + kZ[t]k
2γCmax l=1 βl
≤ |kQ⊥ [t + 1]k − kQ⊥ [t]k| +
κ
P
β
T
[t]
∗
l l
l∈H
+2γCmax q
.
PL
2
kQ⊥ [t]k + 2γCmax l=1 βl Tl [t]
(B.13.1)
Note that
(a)
|kQ⊥ [t + 1]k − kQ⊥ [t]k| ≤ kQ⊥ [t + 1] − Q⊥ [t]k
(b)
= kQ[t + 1] − Q[t] − Qk [t + 1] + Qk [t]k
≤ kQ[t + 1] − Q[t]k + kQk [t + 1] − Qk [t]k
(c)
≤ 2kQ[t + 1] − Q[t]k
(d)
≤ 2kQ[t + 1] − Q[t]k1
≤ 2L max |Ql [t + 1] − Ql [t]|
l
≤ 2L max{Amax , Cmax },
(B.13.2)
where the step (a) uses the inequality |kxk2 − kyk2 | ≤ kx − yk2 for any vector x and
y; step (b) follows from the definition of Q = Q⊥ + Qk ; step (c) follows from the
non-expansive property of the projection onto the a convex set; step (d) is true since
221
kxk ≤ kxk1 for any vector x. By substituting (B.13.2) into (B.13.1), we have the
desired result.
B.14
Proof of Lemma B.12.1
E[X n ] =
(a)
≤
(b)
≤
1 ηX n
E
log
e
ηn
n 1 n−1 ηX
E
log
e
e
ηn
n
1
n−1
ηX
log
e
E[e
]
,
ηn
(B.14.1)
where the step (a) follows from the fact that f (y) = (log y)n is increasing in y ∈ [1, ∞)
n
for n = 1, 2, · · · ; (b) uses the fact that g(y) = log en−1 y
is concave in [1, ∞) for
n = 1, 2, · · · , and Jensen’s Inequality.
B.15
Proof of Inequality (B.9.11)
To show inequality (B.9.11), we need the following lemma.
Lemma B.15.1. Let
πc , Pr hd, c · S∗ (Q, T, c)i = bc C = c and
χc , min bc − hd, ri : for all r ∈ S (c) \ {w : bc = hd, ri} .
Then, for each channel state c ∈ C, and any ∈ (0, χc ψc ), we have
1 − πc ≤
,
χc ψc
where we recall that ψc , Pr{C = c}.
222
(B.15.1)
The proof is similar to that of Claim 1 in [15] and is omitted here for conciseness.
Lemma B.15.1 implies that
i
h
2 ∗
E bc − hd, C · S (Q, T, C)i C = c
h
i
2 ∗
∗
= E bc − hd, C · S (Q, T, C)i hd, C · S (Q, T, C)i =
6 bc × (1 − πc )
(bc )2 + hd, Cmax 1i2 .
(B.15.2)
≤
χc ψc
Similarly, we have
bc
E bc − hd, C · S∗ (Q, T, C)iC = c ≤
.
χc ψc
For E hd, A − C · S∗ (Q, T, C)i2 C = c , we have
(B.15.3)
E hd, A − C · S∗ (Q, T, C)i2 C = c
h
2 i
= E hd, Ai − b + b − bc + bc − hd, c · S∗ (Q, T, c)i
h
2 i
= E (hd, Ai − b)2 + (b − bc )2 + E bc − hd, c · S∗ (Q, T, c)i
+2 (hd, λi − b) (b − bc ) + 2 (hd, λi − b) E bc − hd, c · S∗ (Q, T, c)i
+2 (b − bc ) E bc − hd, c · S∗ (Q, T, c)i .
(B.15.4)
Next, we give upper bounds for each individual term in the right hand side of (B.15.4).
We will repeatedly use the identity
hd, λi − b = −,
(B.15.5)
where it follows from the definition of λ(0) . By noting that bc , max hd, c · si, we
s∈S (c)
have
(hd, λi − b) E bc − hd, c · S∗ (Q, T, c)i
= −E bc − hd, c · S∗ (Q, T, c)i ≤ 0.
(B.15.6)
In addition, by using inequality (B.15.3), we have
(b − bc ) E bc − hd, c · S∗ (Q, T, c)i
bc
≤ bE bc − hd, c · S∗ (Q, T, c)i ≤ b
.
χ c ψc
223
(B.15.7)
For E (hd, Ai − b)2 , we have
E (hd, Ai − b)2 = E (hd, Ai − hd, λi + hd, λi − b)2
= E (hd, Ai − hd, λi − )2
D
E
2
() 2
+ 2 .
= d, σ
(B.15.8)
Thus, by substituting (B.15.2), (B.15.5), (B.15.6), (B.15.7) and (B.15.8) into (B.15.4),
we have
E hd, A − C · S∗ (Q, T, C)i2 C = c
D
E
2
() 2
+ 2 + (b − bc )2 +
≤ d, σ
2bbc + (bc )2 + hd, Cmax 1i2 − 2 (b − bc ) .
χc ψc
By taking the expectation on both sides of the above inequality, we have the desired
result.
B.16
Proof of Inequality (B.9.13)
Let R∗ (Q[t], T[t]) , E[C · S∗ (Q[t], T[t], C[t])|Q[t], T[t]]. For the face F , H(d) ∩ R
π
of the region R, there exists an angle θ ∈ (0, ] such that hd, R∗ (Q, T)i = b, for all
2
k(Q + γβ · T)k k
Q and T satisfying
≥ cos(θ). Note that
kQ + γβ · Tk
E hd, Qihd, C · S∗ (Q, T, C) − Ai
= E hd, Q + γβ · Tihd, C · S∗ (Q, T, C) − Ai
−E hd, γβ · Tihd, C · S∗ (Q, T, C) − Ai .
(B.16.1)
For E hd, Q + γβ · Tihd, C · S∗ (Q, T, C) − Ai , we have
E hd, Q + γβ · Tihd, C · S∗ (Q, T, C) − Ai
= E hd, Q + γβ · Ti (b − hd, Ai)
−E hd, Q + γβ · Ti b − hd, C · S∗ (Q, T, C)i .
224
(B.16.2)
By using the fact that the arrivals are independent of system state, we have
E hd, Q + γβ · Ti (b − hd, Ai) = E hd, Q + γβ · Ti
h
i
= E kQk k + γE k β · T k k .(B.16.3)
Consider E hd, Q + γβ · Ti b − hd, C · S∗ (Q, T, C)i .
=
=
(a)
=
(b)
=
(c)
=
(d)
≤
(e)
≤
≤
E hd, Q + γβ · Ti b − hd, C · S∗ (Q, T, C)i
E E k(Q + γβ · T)k k b − hd, C · S∗ (Q, T, C)i Q, T
E k(Q + γβ · T)k k b − hd, R∗ (Q, T)i
E kQ + γβ · Tk cos(φ) b − hd, R∗ (Q, T)i
E kQ + γβ · Tk cos(φ)1{φ>θ} b − hd, R∗ (Q, T)i
E k(Q + γβ · T)⊥ k cot(φ)1{φ>θ} b − hd, R∗ (Q, T)i
cot(θ)E k(Q + γβ · T)⊥ k b − hd, R∗ (Q, T)i
r
h
2 i
cot(θ) E k(Q + γβ · T)⊥ k2 E b − hd, R∗ (Q, T)i
r h
q 2 i
2
2
2
∗
,
cot(θ) E k(Q⊥ k + γ kβ · Tk × E b − hd, R (Q, T)i (B.16.4)
where the step (a) is true for that φ is the angle between vector Q + γβ · T and the
normal vector d; (b) follows from the definition of θ; (c) uses k(Q + γβ · T)⊥ k =
kQ + γβ · Tk sin(φ); (d) follows from the fact that cotangent function is decreasing
πi
in 0, ; (e) uses Cauchy-Schwartz Inequality.
2
225
Next, let’s consider E
=
(b)
≤
=
=
2 i
b − hd, R∗ (Q, T)i .
2 i
b − hd, R∗ (Q, T)i
h 2 i
∗
E E Ψ − hd, C · S (Q, T, C)i Q, T
ii
h h
2 E E Ψ − hd, C · S∗ (Q, T, C)i Q, T
h
2 i
E Ψ − hd, C · S∗ (Q, T, C)i
h
i
X
2 ∗
ψc E bc − hd, C · S (Q, T, C)i C = c
E
(a)
h
h
c∈C
(c)
≤ X 1
(bc )2 + hd, Cmax 1i2 ,
χc
c∈C
(B.16.5)
where the step (a) follows from the definition of Ψ with distribution Pr{Ψ = bc } = ψc
for c ∈ C, and the definition of R∗ (Q, T); (b) uses Jensen’s Inequality; (c) uses
inequality (B.15.2).
Thus, by substituting (B.16.3), (B.16.4) and (B.16.5) into (B.16.2), we have
E hd, Q + γβ · Tihd, C · S∗ (Q, T, C) − Ai
p
≥ E kQk k + γE k(β · T)k k − cot(θ) (kQ⊥ k2 + γ 2 kβ · Tk2 ) s
X 1
×
(bc )2 + hd, Cmax 1i2 .
(B.16.6)
χ
c
c∈C
For E hd, γβ · Tihd, C · S∗ (Q, T, C) − Ai , we have
=
=
=
(a)
≤
(b)
=
E hd, γβ · Tihd, C · S∗ (Q, T, C) − Ai
E E hd, γβ · Tihd, C · S∗ (Q, T, C) − AiQ, T
E E hd, γβ · Ti hd, R∗ (Q, T)i − hd, Ai Q, T
E hd, γβ · Ti hd, R∗ (Q, T)i − b + b − hd, λi
E hd, γβ · Ti (b − hd, λi)
γE k(β · T)k k ,
226
(B.16.7)
where the step (a) uses the fact that hd, R∗ (Q, T)i ≤ b; (b) follows from the definition
of λ(0) and b = hd, λ(0) i. By substituting (B.16.6) and (B.16.7) into (B.16.1), we have
the desired result.
227
APPENDIX C: PROOFS FOR CHAPTER 5
C.1
Properties of Functional Classes
The following remarks explore more properties of classes A, B and C.
f (x + a)
exists for any a ∈ R, then this limit should be equal to 1.
(1) In B, if lim
x→∞
f (x)
f (x + a)
f (x + 2)
Indeed, let lim
= b for any a ∈ R, where b > 0. Then b = lim
=
x→∞
x→∞
f (x)
f (x)
f (x + 2) f (x + 1)
lim
·
= b2 . Thus, b = 1.
x→∞ f (x + 1)
f (x)
(2) If the definition of C is not constrained by the set B, then C is not necessarily a
f (x + a)
does
subset of B. In fact, we can construct a function f ∈ C for which lim
x→∞
f (x)
not exist and hence f 6∈ B.
(3) In C, if f ∈ F, then the lower bound of f (x1 + x2 ) always exists. Also if there
exists w > 0 such that f (2x) ≤ wf (x) for any x ≥ 0, then the upper bound of
f (x1 + x2 ) always exists. Indeed, since f (·) is nondecreasing, f (x1 + x2 ) ≥ f (xi ),
1
1
for i = 1 or 2. Hence f (x1 + x2 ) ≥ (f (x1 ) + f (x2 )). Thus, let K1 = , then
2
2
we always have K1 (f (x1 ) + f (x2 )) ≤ f (x1 + x2 ). On the other hand, f (x1 + x2 ) ≤
max{f (2x1 ), f (2x2 )} ≤ f (2x1 ) + f (2x2 ) ≤ w(f (x1 ) + f (x2 )). Thus, let K2 = w, we
have f (x1 + x2 ) ≤ K2 (f (x1 ) + f (x2 )).
(4) If f ∈ C, then given n ∈ N, there exist K10 and K20 satisfying 0 < K10 ≤ K20 < ∞
m
m
m
X
X
X
such that K10
f (xi ) ≤ f (
xi ) ≤ K20
f (xi ), for m = 1, ..., n, where xi ≥ 0, i =
i=1
i=1
i=1
1, ..., m. This directly follows from the induction.
\
f (2x)
(5) A C = ∅. Indeed, if f ∈ A, then lim
= ∞. Thus, for any c > 0, ∃M > 0
x→∞ f (x)
228
such that f (2x) > cf (x) for any x > M . Hence, f 6∈ C. On the other hand, if f ∈ C,
f (2x)
then ∃d > 0 such that f (2x) ≤ df (x). Hence, lim sup
≤ d and thus f 6∈ A.
f (x)
x→∞
C.2
Proof of Lemma 5.3.1
Proof. If n = 1, because λ1 ∈ (0, 1), by assumption, there exists a 0 < δ1 <
a21
≥ a21 (1 + δ1 ).
such that
λ1
Assume that n = k, it is true. That is, if
k
X
1
− 1,
λ1
λi < 1 and λi > 0 (i = 1, ..., k), then
i=1
there exists a δk = δ(λ1 , ..., λk ) > 0 such that
1 2
1
a1 + ... + a2k ≥ (a1 + ... + ak )2 (1 + δk ).
λ1
λk
(C.2.1)
Then for n = k + 1 and λ1 + ... + λk + λk+1 < 1, we have
1 2
1
1 2
a1 + ... + a2k +
a
λ1
λk
λk+1 k+1
1 2
1
1 2
λk + λk+1 2 λk + λk+1 2
a + ... +
a
+
ak +
ak+1
=
λ1 1
λk−1 k−1 λk + λk+1
λk
λk+1
s
!2
λk + λk+1 2 λk + λk+1 2
ak +
ak+1 · (1 + δk+1 ), (C.2.2)
≥
a1 + ... + ak−1 +
λk
λk+1
where the last step follows from the assumption. Since
λk + λk+1 2 λk + λk+1 2
ak +
ak+1 − (ak + ak+1 )2
λk
λk+1
λk 2
λk+1 2
=
− 2ak ak+1
ak +
a
λk
λk+1 k+1
s
λk+1 2 λk 2
≥ 2
a ·
a
− 2ak ak+1 = 0,
λk k λk+1 k+1
(C.2.3)
hence
s
λk + λk+1 2 λk + λk+1 2
ak +
ak+1 ≥ (ak + ak+1 ).
λk
λk+1
229
(C.2.4)
Thus, equation (C.2.2) becomes
k+1
X
1 2
a ≥
λ i
i=1 i
C.3
k+1
X
i=1
ai
!2
(1 + δk+1 ).
Proof of Inequality (5.3.4)
∆V
, E [V (Q[t + 1]) − V (Q[t])|Q[t] = Q]
|Si |
N X
X
1
i
i
=
E i (h(Ql [t + 1]) − h(Ql [t]))|Q[t] = Q .
λl
i=1 l=1
By the mean-value theorem, we have h(Qil [t + 1]) − h(Qil [t]) = f (Rli [t])(Qil [t + 1] −
Qil [t]) = f (Rli [t])(Ail [t] − Sli [t] + Uli [t]), where Rli [t] lies between Qil [t] and Qil [t + 1].
Hence, we get
∆V
|Si |
N X
X
1
i
l
i
i
=
E i f (Rl [t])(Al [t] − Sl [t] + Ul [t])|Q[t] = Q
λl
i=1 l=1
|Si |
N X
X
1
i
i
E i f (Rl [t])Ul [t]|Q[t] = Q
=
λl
|i=1 l=1
{z
}
,∆V1
|Si |
+
N X
X
|i=1
l=1
1
i
i
i
E i f (Rl [t])(Al [t] − Sl [t])|Q[t] = Q .
λl
{z
}
,∆V2
For ∆V1 , if Qil [t] = Qil > 0, then Uli [t] = 0. If Qil [t] = Qil = 0, then Uli [t] may
be equal to 1. But in this case, Qil [t + 1] ≤ Amax (since Ail [t] ≤ Amax ). Hence,
f (Rli [t]) ≤ f (Amax ) < ∞. Thus,
i
∆V1 =
|S |
N X
X
i=1 l=1
1
i
i
E i f (Rl [t])Ul [t]|Q[t] = Q 1{Qil =0}
λl
i
i
|S |
|S |
N X
N X
X
X
1
≤
f (Amax ) ≤ D
f (Amax ),
λi
i=1 l=1 l
i=1 l=1
230
(C.3.1)
1
< ∞ and 1{·} is the indicator function.
min{λil }
Next, let’s focus on ∆V2 . We know that f (Rli [t]) = f (Qil [t] + ail ) (|ail | ≤ Amax ).
where D ,
According to the definition of function f ∈ B, given > 0, there exists M > 0, such
f (Rli [t])
i
i
that for any Ql [t] = Ql > M , we have − 1 < , that is, (1 − )f (Qil ) <
i
f (Ql )
i
i
f (Rl [t]) < (1 + )f (Ql ). Thus, we have
f (Rli [t])(Ail [t] − Sli [t])
= f (Rli [t]) (Ail [t] − Sli [t])+ − (Ail [t] − Sli [t])−
≤ (1 + )f (Qil )(Ail [t] − Sli [t])+ − (1 − )f (Qil )(Ail [t] − Sli [t])−
= f (Qil )(Ail [t] − Sli [t]) + f (Qil ) Ail [t] − Sli [t]
≤ f (Qil )(Ail [t] − Sli [t]) + Amax f (Qil ),
(C.3.2)
where (x)+ = max{x, 0}, (x)− = − min{x, 0}, and |Ail [t] − Sli [t]| ≤ |Ail [t]| ≤ Amax .
Thus, we divide ∆V2 into two parts:
i
|S |
N X
X
∆V2 =
|i=1
+
l=1
1
i
i
i
E i f (Rl [t])(Al [t] − Sl [t])|Q[t] = Q 1{Qil >M }
λl
{z
}
,∆V3
|Si |
N X
X
|i=1
l=1
1
i
i
i
E i f (Rl [t])(Al [t] − Sl [t])|Q[t] = Q 1{Qil ≤M } .
λl
{z
}
,∆V4
For ∆V3 , by using (C.3.2), we have
i
i
|S |
|S |
N X
N X
X
X
1
i
i
i
f (Ql )(λl − Pl )1{Qil >M } + DAmax f (Qil )1{Qil >M } , (C.3.3)
∆V3 ≤
i
λ
i=1 l=1 l
i=1 l=1
where
Pli
P|Si |
i
f (Qil )
= E Sl [t]|Q[t] = Q = PN l=1
.
P|Sk |
k
f
(Q
)
l=1
l
k=1
231
i
|S |
N X
X
1
Next, let’s consider the term
f (Qil )(λil − Pli ), which can be expressed as:
i
λ
i=1 l=1 l
P|Si |
|Si |
|Si |
|Si |
N X
N X
N X
i
X
X
X
1
f (Qil )
i
i
i
i
l=1 f (Ql )
f
(Q
)(λ
−
P
)
=
f
(Q
)
−
l
l
l
l
λi
λil PN P|Sk | f (Qk )
i=1 l=1 l
i=1 l=1
i=1 l=1
l
l=1
k=1
PN P|Si |
PN P|Si | f (Qil ) P|Si |
i 2
( i=1 l=1 f (Ql )) − i=1 ( l=1 λi )( l=1 f (Qil ))
l
=
.
PN P|Si |
i
f
(Q
)
l=1
l
i=1
Since
i
i
i
|S |
|S |
|S |
N X
N
X
X
f (Qil ) X
1 X
i
)(
(
f (Ql )) ≥
(
f (Qil ))2 ,
i
i
λl
λ l=1
i=1 l=1
i=1
l=1
where λi =
max
{l=1,...,|Si |}
λil , and by Lemma 1, there exists a δ > 0 such that
i
i
|S |
|S |
N X
N
X
X
1 X
i 2
(
f (Ql )) ≥ (
f (Qil ))2 (1 + δ),
i
λ l=1
i=1 l=1
i=1
(C.3.4)
we have
i
i
i
|S |
|S |
|S |
N X
N X
X
X
f (Qil ) X
i
(
)(
f (Ql )) ≥ (
f (Qil ))2 (1 + δ).
i
λ
l
i=1 l=1
i=1 l=1
l=1
Thus, we get
i
i
|S |
|S |
N X
N X
X
X
1
i
i
i
f (Qil ).
f (Ql )(λl − Pl ) ≤ −δ
i
λ
i=1 l=1
i=1 l=1 l
(C.3.5)
Hence, we have
i
|S |
N X
X
1
f (Qil )(λil − Pli )1{Qil >M }
i
λ
i=1 l=1 l
i
≤ −δ
|S |
N X
X
f (Qil )1{Qil >M }
−δ
i=1 l=1
≤ −δ
|Si |
N X
X
i=1 l=1
|Si |
≤ −δ
N X
X
i=1 l=1
i
i
|S |
N X
X
f (Qil )1{Qil ≤M }
i=1 l=1
f (Qil )1{Qil >M } +
|Si |
N X
X
i=1 l=1
f (Qil )1{Qil >M } + D
|S |
N X
X
1
−
f (Qil )(λil − Pli )1{Qil ≤M }
λ
i=1 l=1 l
1
f (Qil )Pli 1{Qil ≤M }
λil
i
|S |
N X
X
f (M ).
i=1 l=1
232
(C.3.6)
Thus, we can choose small enough such that γ = δ − DAmax > 0, and thus we have
i
∆V3 ≤ −γ
|S |
N X
X
i
f (Qil )1{Qil >M }
+D
i=1 l=1
≤ −γ
|Si |
N X
X
|S |
N X
X
f (M )
i=1 l=1
f (Qil ) + (D + γ)
i=1 l=1
|Si |
N X
X
f (M )
i=1 l=1
For ∆V4 , we have
∆V4
|Si |
N X
X
1
i
i
i
≤
E i f (Rl [t])|Al [t] − Sl [t]||Q[t] = Q 1{Qil ≤M }
λl
i=1 l=1
i
i
|S |
|S |
N X
N X
X
X
1
≤
Amax f (M + Amax ) ≤ DAmax
f (M + Amax ).
λi
i=1 l=1 l
i=1 l=1
Thus, we get
i
∆V ≤ −γ
|S |
N X
X
f (Qil ) + G,
(C.3.7)
i=1 l=1
where G , D
∞.
i=1 l=1
i
i
i
|S |
N X
X
f (Amax )+DAmax
|S |
N X
X
i=1 l=1
233
f (M +Amax )+(D +γ)
|S |
N X
X
i=1 l=1
f (M ) <
APPENDIX D: PROOFS FOR CHAPTER 6
D.1
Proof of Lemma 6.4.4
If T = 1, we have LHS = min{c[0]s[0], a} and RHS = min{c[0], a}s[0]. Since s[0] =
0 or 1, LHS = RHS.
Assume that T = k, (6.4.5) is true, that is,

!+ 
( k−1
) k−1
t−1


X
X
X
min c[t], a −
c[j]s[j]
s[t].
min
c[t]s[t], a =


t=0
t=0
(D.1.1)
j=0
Then, for T = k + 1, we have

!+ 
k
t−1


X
X
s[t]
min c[t], a −
c[j]s[j]


t=0
j=0


!+ 
!+ 
t−1
k−1
k−1




X
X
X
min c[t], a −
c[j]s[j]
s[t] + min c[k], a −
c[j]s[j]
s[k]
=




t=0
j=0
j=0

( k−1
)
!+ 
k−1


X
X
(a)
= min
c[t]s[t], a + min c[k]s[k], a −
c[j]s[j]


t=0
j=0

( k−1
)
!+ 
k
k−1
X

X
X
(b)
c[t]s[t] , a + a −
c[j]s[j]
,
= min
c[t]s[t], a + c[k]s[k], max a,


t=0
t=0
j=0
(D.1.2)
where step (a) follows from the induction assumption and uses the fact that s[t] only
takes 0 or 1; step (b) uses the following identity: min{a, b} + min{c, d} = min{a +
c, a + d, b + c, b + d}.
234
(1) If a ≥
k−1
X
c[t]s[t], then
t=0
(D.1.2) = min
= min

k
X

t=0
( k
X
c[t]s[t], a + c[k]s[k], a, a +
a−
k−1
X
c[j]s[j]
j=0
)
c[t]s[t], a ,
t=0
a−
k−1
X
c[j]s[j]
j=0
0.
(2) If a <

(D.1.3)
where the last step follows the fact that c[k]s[k] ≥ 0 and
k−1
X
!+ 

!+
≥
c[t]s[t], then
t=0
(D.1.2) = min
( k
X
c[t]s[t], a + c[k]s[k],
t=0
= min
( k
X
)
k−1
X
c[t]s[t], a
t=0
c[t]s[t], a .
t=0
)
(D.1.4)
Thus, by induction, we have the desired result.
D.2
Proof of Proposition 6.4.8
Consider the Lyapunov function V (Z) ,
L
X
h(Zl ), where h0 (x) = f (x). Then, by
l=1
using a similar argument to the proof of Lemma 1 in [39] (also see [43]), it is not hard
to show that if for any process A ∈ Λ1 (ρ, C), there exists γ > 0 and H ≥ 0 such that
∆V (Z) ,
L
X
E [f (Zl )(Dl [kT ] − Il [kT ])|Z[kT ] = Z]
l=1
≤ −γ
L
X
f (Zl ) + H.
l=1
By the telescoping technique, we have
K
L
1 XX
H
lim sup
E[f (Zl [kT ])] ≤
< ∞,
γ
K→∞ K
k=1 l=1
235
(D.2.1)
which implies the stability-in-the-mean and thus the Markov Chain is positive recurrent [61]. Next, we will show inequality (D.2.1) to complete the proof. By substituting
the expression of Dl [kT ] (see the discussion before (6.4.1)) into ∆V (Z), we have
∆V (Z) =
L
X
E [f (Zl )(Al [kT ] − Il [kT ])|Z[kT ] = Z]
|l=1
{z
}
,∆V1 (Z)




(k+1)T −1
L


X
X
−E
f (Zl ) min
Cl [kT ]SlF [t], Al [kT ] Z[kT ],


l=1
t=kT
{z
}
|
,∆V2 (Z)
where SF [t] = (SlF [t])Ll=1 denotes the schedule chosen by FCSMA algorithm at time
t. Let
Wl [t] = f (Zl [kT ]) min



Cl [kT ], Al [kT ] − Cl [kT ]
t−1
X
Sl [j]
j=kT
!+ 


,
for any t = kT, kT + 1, ..., (k + 1)T − 1, where S[j] = (Sl [j])Ll=1 is a feasible schedule
at time slot j. Let W G [t] be the weight of link picked by the Greedy Algorithm with
CSI at time slot t. Recall that W G [t] = max Wl [t]. Next, we will derive an upper
l
bound for ∆V1 (Z) by using Lemma 6.4.1 and give a lower bound for ∆V2 (Z).
First, let’s focus on ∆V1 . By Lemma 6.4.1, there exist non-negative numbers
α(a, c; s0 , s1 , ..., sT −1 ) satisfying (6.4.2) and for a δ > 0 small enough, we have
λl (1 − ρl ) ≤
X
a
PA (a)
X
PC (c)
c
X
s0 ,s1 ,...,sT −1 ∈S
α(a, c; s0 , s1 , ..., sT −1 ) min
(T −1
X
j=0
236
cl sjl , al
)
− δ.
(D.2.2)
By using (D.2.2), we have
∆V1 =
L
X
f (Zl )λl (1 − ρl )
l=1
≤
X
PA (a)
a
L
X
X
c
f (Xl ) min
≤
(T −1
X
PA (a)
a
X
α(a, c; s0 , s1 , ..., sT −1 )
s0 ,s1 ,...,sT −1 ∈S
cl sjl , al
j=0
l=1
X
X
PC (c)
)
−δ
PC (c)
c
t=kT
f (Zl )
l=1
(k+1)T −1
X
L
X
G
W [t] − δ
L
X
f (Zl ),
(D.2.3)
l=1
where the last step follows
Theorem
6.4.6 that the Greedy Algorithm with CSI
(Tfrom
)
L
−1
X
X
maximizes
f (Zl ) min
cl sjl , al for any feasible schedules s0 , s1 ,...,sT −1 , given
j=0
l=1
virtual queue lengths, channel state information and arrivals, and uses (6.4.2).
Thus, we have


L
X
G 

f (Zl )
≤ E
W [t]Z[kT ] = Z − δ
l=1
t=kT
 


(k+1)T −1
L
X
X
G 



f (Zl ).
W [t]Z[kT ], A[kT ], C[kT ] Z[kT ] − δ
= E E
l=1
t=kT
(k+1)T −1
∆V1
X
Note that W G [t] is non-increasing in t within each frame, since the number of remaining packets cannot increase as t increases. Pick any W > 0 and let
F0 , {W G [kT ] ≤ W , W G [kT + 1] ≤ W , ..., W G [(k + 1)T − 1] ≤ W };
Fj , {W G [kT + j − 1] > W , W G [kT + j] ≤ W }, ∀j = 1, ..., T − 1;
FT , {W G [(k + 1)T − 1] > W },
where Fj corresponds to the event where the weight chosen by Greedy Algorithm is
237
greater than W in the first j slots in frame k. Thus, (Fj )Tj=0 forms a partition of a
set {W G [kT ], W G [kT + 1], ..., W G [(k + 1)T − 1]}. Then, we have


(k+1)T −1
X
E
W G [t]Z[kT ], A[kT ], C[kT ]
t=kT


(k+1)T −1
T
X X
= E
W G [t]1Fj Z[kT ], A[kT ], C[kT ]
j=0
t=kT
!
#
" T
+j−1
X kTX
≤ E
W G [t]1Fj + (T − j)W Z[kT ], A[kT ], C[kT ]
j=0
t=kT
" T kT +j−1
#
X X
T (T + 1)W
= E
.
W G [t]1Fj Z[kT ], A[kT ], C[kT ] +
2
j=1 t=kT
Thus, ∆V1 becomes
#
" T kT +j−1
L
X
X X
T (T + 1)W
G
−δ
f (Zl ).(D.2.4)
∆V1 ≤ E
W [t]1Fj Z[kT ] = Z +
2
j=1 t=kT
l=1
Second, let’s consider ∆V2 . Let

!+ 
t−1


X
SlF [j]
.
WlF [t] = f (Zl [kT ]) min Cl [kT ], Al [kT ] − Cl [kT ]


j=kT
Then, by using Lemma 6.4.4 and switching the summations over l and t, we have


(k+1)T −1 L
X X
∆V2 = E 
WlF [t]SlF [t]Z[kT ] = Z .
(D.2.5)
t=kT
l=1
Let > 0 and ζ > 0. For each event Fj , ∀j = 1, 2, ..., T , we have W G [kT ] >
W , ..., W G [kT + j − 1] > W . By using Lemma 6.1.4, we obtain that for any ζ 0 > 0,
choose W such that
)
( L
X
F
G F
Pr
Wl [t]Sl [t] ≥ (1 − )W [t]Fj ≥ 1 − ζ 0 ,
l=1
238
(D.2.6)
holds for any t = kT, kT + 1, ..., kT + j − 1. Hence, we have
)
(kT +j−1 L
kT +j−1
X X
X
F
F
G Pr
Wl [t]Sl [t] ≥ (1 − )
W [t]Fj
t=kT l=1
t=kT
)
( L
X
≥ Pr
WlF [t]SlF [t] ≥ (1 − )W G [t], ∀t = kT, ..., kT + j − 1Fj
l=1
≥ 1 − jζ 0 ≥ 1 − T ζ 0 ,
(D.2.7)
where we use the fact that given any two events E1 and E2 such that Pr{E1 } ≥ 1−1
\
and Pr{E2 } ≥ 1 − 2 , we have Pr{E1 E2 } ≥ 1 − 1 − 2 . By picking ζ 0 small enough
such that 1 − T ζ 0 ≥ 1 − ζ, we have
)
(kT +j−1 L
kT +j−1
X
X X
F
F
G Wl [t]Sl [t] ≥ (1 − )
W [t]Fj ≥ 1 − ζ,
Pr
t=kT
l=1
t=kT
for j = 1, ..., T, which implies that
"kT +j−1 L
#
X X
WlF [t]SlF [t]1Fj Z[kT ] = Z
E
t=kT l=1
"kT +j−1 L
#
X X
= Pr{Fj }E
WlF [t]SlF [t]Z[kT ] = Z, Fj
t=kT l=1
#
"kT +j−1
X
G ≥ Pr{Fj }(1 − )(1 − ζ)E
W [t]Z[kT ] = Z, Fj
t=kT
"kT +j−1
#
X
W G [t]1Fj Z[kT ] = Z ,
= (1 − )(1 − ζ)E
(D.2.8)
t=kT
for j = 1, ..., T. Thus, we have


T (k+1)T
L
X
X−1 X
∆V2 = E 
WlF [t]SlF [t]1Fj Z[kT ] = Z
j=0
t=kT
l=1
#
" T kT +j−1 L
X X X
≥ E
WlF [t]SlF [t]1Fj Z[kT ] = Z
j=1 t=kT l=1
" T kT +j−1
#
X X
≥ (1 − )(1 − ζ)E
W G [t]1Fj Z[kT ] .
j=1
239
t=kT
(D.2.9)
Thus, by using (D.2.4) and (D.2.9), ∆V becomes
" T kT +j−1
#
X X
∆V ≤ ( + ζ − ζ)E
W G [t]1Fj Z[kT ]
j=1
−δ
L
X
t=kT
T (T + 1)W
.
2
f (Zl ) +
l=1
(D.2.10)
Since
#
E
W G [t]1Fj Z[kT ]
j=1 t=kT


(k+1)T −1
L
X
X
G 

W [t]Z[kT ] ≤ Amax T
f (Zl ),
≤ E
t=kT
l=1
"
we have
∆V
+j−1
T kTX
X
≤ ( + ζ − ζ)Amax T
L
X
f (Zl ) − δ
l=1
= −γ
L
X
f (Zl ) + H,
L
X
l=1
f (Zl ) +
(D.2.11)
T (T + 1)W
2
(D.2.12)
l=1
T (T + 1)W
and γ = δ − Amax ( + ζ − ζ)T . We can choose β, , ζ small
2
enough such that γ > 0.
where H =
240
APPENDIX E: PROOFS FOR CHAPTER 7
E.1
Proof of Proposition 7.2.1
Proof. Let Rl and θj be the rate that lth user can achieve and the probability that j
users probe the channel, respectively, where l = 1, 2, ..., L and j = 0, 1, ..., L. Then,
we can get the average probing rate as follows:
" L
#
L
X
1X
1
E
Xl =
lθl ,
L
L l=1
l=1
where we use the fact that
L
X
(E.1.1)
Xl = j with probability of θj .
l=1
When j users probe the channel, by recalling our assumption that only probing
L
X
Rl = 1 − (1 − p)j . Thus, the average
users are allowed to transmit, we have
l=1
achievable rate can be expressed as follows:
" L
#
L
X
1
1X
E
Rl =
θj 1 − (1 − p)j .
L
L j=1
l=1
(E.1.2)
We want to select a probability distribution {θl }Ll=0 such that the average achiev-
able rate is maximized.
1X
θl 1 − (1 − p)l
L l=1
L
max
θ=(θl )L
l=1
Subject to
L
X
l=1
L
X
(E.1.3)
θl ≤ 1
(E.1.4)
lθl ≤ Lm
(E.1.5)
l=1
θl ≥ 0, ∀l = 1, ..., L,
241
(E.1.6)
where (E.1.4) is true since
L
X
θl = 1 and θ0 ≥ 0, and (E.1.5) holds since the average
l=0
probing rate is not greater than m.
By associating Lagrangian Multipliers µ1 ≥ 0 and µ2 ≥ 0 with constraints (E.1.4)
and (E.1.5) respectively, we get the following partial Lagrangian function L(θ, µ1 , µ2 ):
!
!
L
L
L
X
X
1X
F (θ, µ1 , µ2 ) =
θl 1 − (1 − p)l − µ1
θl − 1 − µ2
lθl − N m
L l=1
l=1
l=1
L
X 1
1 − (1 − p)l − µ1 − µ2 l θl + µ1 + µ2 Lm.
=
L
l=1
Then, the dual function q(µ1 , µ2 ) can be expressed as follows:

1

 µ1 + µ2 Lm , if
1 − (1 − p)l ≤ µ1 + µ2 l


L

q(µ1 , µ2 ) = sup F (θ, µ1 , µ2 ) =
∀l = 1, ..., L;

θ≥0




+∞ , otherwise.
Since the original optimization problem is just a linear programming, there is no
duality gap and thus it is equivalent to solve the following dual problem:
min
µ1 + µ2 Lm
Subject to
µ1 + µ2 l ≥
µ1 ≥0,µ2 ≥0
(E.1.7)
1
1 − (1 − p)l , ∀l = 1, ..., L.
L
Since the objective function and constraint function are linear functions representing lines in R2 , we call the objective function and constraint function as the objective
line and constraint line respectively. Note that the normal vector of the objective line
is [1, Lm]T and the normal vector of the constraint line l is [1, l]T . If 0 ≤ Lm ≤ 1, by
p
the optimality condition [5], the optimal objective line should pass the point (0, ),
L
p
and thus the maximum achievable rate is 0 + Lm = mp; if l ≤ Lm ≤ l + 1
L
(l = 1, ..., L − 1), the optimal objective line should pass the intersection point of two
1
1
constraint lines µ1 + µ2 l =
1 − (1 − p)l and µ1 + µ2 (l + 1) =
1 − (1 − p)l+1 ,
L
L
242
1 − (1 + lp)(1 − p)l p(1 − p)l
which is
,
, and hence the maximum achievable rate
N
N
1
l
1
1 − (1 + lp)(1 − p)l p(1 − p)l
+
Lm = + m −
p(1 − p)l − (1 − p)l .
is
L
L
L
L
L
E.2
Proof of Lemma 7.3.1
Proof. (1) (Necessity) Suppose all data queues are strongly stable and each user
satisfies its allowable probing rate constraint under some policy Φ which determines
the probing schedule X[t] and the transmission schedule S[t] in every slot t. For
M
1 X
Xl (τ )Cl (τ )Sl (τ ) and
some positive integer number M , we define µl (M ) ,
M τ =1
1
pl (M ) ,
Xl (τ ) as the empirical average service rate and probing rate for user l,
M
respectively.
Let TxM be the set of slots in [1, M ] in which the probing schedule is x, and TxM (c)
be the set of slots in TxM in which the channel state vector is c. First, we consider
the empirical average service rate µl (M ).
M
1 X
Xl (τ )Cl (τ )Sl (τ )
µl (M ) =
M τ =1
1 XX X
=
xl cl Sl (τ )
M x∈X c
x
τ ∈TM (c)
X
X |Tx | X |Tx (c)|
1
M
M
·
xl cl Sl (τ )
M c |TxM |
|TxM (c)|
x∈X
τ ∈Tx
(c)
M
X
X
=
αM (x)
σM (c)yM (x, c),
=
x∈X
where αM (x) ,
(E.2.1)
c
|TxM |
|Tx (c)|
1
, σM (c) , Mx
and yM (x, c) , x
M
|TM |
|TM (c)|
X
xl cl Sl (τ ).
τ ∈Tx
M (c)
Observe that yM (x, c) is a convex combination of the set {0, xl cl }. By Caratheodory’s
X
theorem, there exists a non-negative real sequence {βM (x, c; s)}s∈S with
βM (x, c; s) =
s∈S
243
1, such that yM (x, c) can be rewritten as
yM (x, c) =
X
βM (x, c; s)xl cl sl .
(E.2.2)
s∈S
Hence, we have
µl (M ) =
X
αM (x)
X
σM (c)
c
x∈X
X
βM (x, c; s)xl cl sl .
(E.2.3)
s∈S
Next, we consider the empirical average probing rate pl (M ).
M
X
X |Tx |
1 X
1 X X
M
pl (M ) =
Xl (τ ) =
xl =
αM (x)xl .
xl =
M τ =1
M x∈X τ ∈Tx
M
x∈X
x∈X
(E.2.4)
M
For each positive integer number M , the number of αM (x) and βM (x, c; s) is
bounded. By compactness, we can find a subsequence of integers {Mk } such that
Mk → ∞, and such that there exist limiting probabilities α(x) and β(x, c; s) satisfying:
αMk (x) → α(x), ∀x ∈ X ,
βMk (x, c; s) → β(x, c, s), ∀x ∈ X , c.
In addition, channel states are i.i.d. over time, we have
σMk (c) → Pr{C[t] = c}.
(E.2.5)
Hence, the sequences {µl (Mk )} and {pl (Mk )} converge to
X
α(x)
x∈X
and
X
X
Pr{C[t] = c}
c
X
x l cl s l
s∈S
α(x)xl , respectively.
x∈X
Since the policy Φ makes all data queues strongly stable, by Lemma 1 in [65], the
arrival rate to each queue should be no greater than its service rate, i.e.,
λl ≤
X
x∈X
α(x)
X
Pr{C[t] = c}
c
X
s∈S
244
x l cl s l .
(E.2.6)
Also, each user satisfies the allowable probing rate constraint under policy Φ, which
implies that
X
α(x)xl ≤ ml .
(E.2.7)
x∈X
(2) (Sufficiency) We will show that any arrival rate vector λ strictly inside Λ(m, C)
can be supported by a simple randomized probing and transmission policy that selects
probing schedule X with probability α(X) and chooses transmission schedule S with
probability β(X, C; S) at each slot. First, we should note that the average probing
rate of each user under this policy is not greater than its allowable probing rate, since
X
α(x)xl ≤ ml , ∀l. Next, we will show that all data queues are strongly stable under
x
this policy.
1X 2
Q [t]. Then, we have
2 l=1 l
L
Consider the Lyapunov function V [t] , V (Q[t]) =
∆V
, E[V [t + 1] − V [t]|Q[t] = Q]
L
1X
E[(Ql [t] + Al [t] − Xl [t]Sl [t]Cl [t])2 − Q2l [t]|Q[t]]
≤
2 l=1
L
X
=
E [Ql [t](Al [t] − Xl [t]Sl [t]Cl [t])|Q[t]] + B1 ,
(E.2.8)
l=1
where
1X
,
E[(Al [t] − Xl [t]Sl [t]Cl [t])2 |Q[t]]
2 l=1
L
B1
1X
E[A2l [t] + Cl2 [t]] , B1,max < ∞.
2 l=1
L
≤
Thus, we have
∆V ≤ B1,max +
L
X
λl −
Ql
X
α(x)
x
l=1
X
c
P (C[t] = c)
X
β(x, c; s)xl cl sl
s∈S
Since λ is strictly within Λ(m, C), there exists a > 0 such that
λl ≤
X
x
α(x)
X
c
P (C[t] = c)
X
s∈S
245
β(x, c; s)xl cl sl − , ∀l.
!
.
Then, by using the above inequality, ∆V becomes
∆V ≤ −
L
X
Ql + B1,max .
(E.2.9)
l=1
By using Theorem 4.1 in [66], all data queues are strongly stable.
E.3
Proof of Lemma 7.3.2
(1) (Necessity) For any λ ∈ Λ(m, C), there exist non-negative numbers α(x) and
β(x, c; s) satisfying (7.3.1), (7.3.2), (7.3.3) and (7.3.4). Thus, for any set of users
A ⊆ N, we have
X
l∈A
λl ≤
X
x
α(x)
X
P (C[t] = c)
c
X
β(x, c; s)
s∈S
X
x l cl s l .
(E.3.1)
l∈A
For any given x and c, since at most one user can be scheduled at each slot, we have


 = 0, xl cl = 0, ∀l ∈ A;
X
X
β(x, c; s)
x l cl s l
(E.3.2)

 ≤ 1, otherwise.
s∈S
l∈A
By substituting (E.3.2) into (E.3.1), we get (7.3.5).
(2) (Sufficiency) Since λ ∈ Γ(m, C), there exists non-negative numbers α(x) satisfying (7.3.5), (7.3.6) and (7.3.7). Consider the following policy: in each slot t, during
the probing stage, it selects the probing schedule XR [t] with probability α(XR [t]);
during the transmission stage, it selects user l∗ satisfying l∗ ∈ arg max Ql [t]XlR [t]Cl [t].
l
We first note that each user satisfies the average probing constraint under this policy.
Next, we will show that this policy can makes all data queues strongly stable for any
arrival rate vector λ strictly inside Γ(m, C).
The following proof is similar to that in [92]. By choosing the same Lyapunov
function and following the same argument as in Lemma 7.3.1, we have
" L
#
L
X
X
∆V ≤
λl Ql − E
Ql [t]XlR [t]Sl∗ [t]Cl [t]|Q[t] + B1,max .
l=1
l=1
246
Next, let’s focus on the term E
"
L
X
#
Ql [t]XlR [t]Sl∗ [t]Cl [t]|Q[t] . Consider a permuta-
l=1
tion el , l = 1, ..., L of the integers 1 to L which is such that Qel ≥ Qel−1 , for l = 2, ..., L,
and if Qel = Qel−1 then el > el−1 . For any given x and c, we define the following sets:
R0 , {xl cl = 0, ∀l = 1, 2, ..., L};
(E.3.3)
Rl , {xel cel = 1, xej cej = 0 for L ≥ j > l} for l = 1, 2, ..., L − 1; (E.3.4)
RL , {xeL ceL = 1};
(E.3.5)
Tl , {xej cej = 0, for L ≥ j ≥ l}, ∀l = 1, 2, ..., L.
(E.3.6)
Thus, we have
E
=
"
L
X
X
Ql [t]XlR [t]Sl∗ [t]Cl [t]|Q[t] = Q
l=1
α(x)
x
=
X
x
X
Pr{C[t] = c}
c
α(x)
X
Pr{C[t] = c}
c
L
X
#
xl cl Ql Sl∗ [t]
l=1
L X
L
X
xl cl Ql Sl∗ [t]1Rj .
(E.3.7)
j=1 l=1
Since
N
X
xl cl Ql Sl∗ [t]1Rj = Qej ,
(E.3.8)
l=1
we have
" N
#
N
X
X
X
X
R
∗
E
α(x)
Pr{C[t] = c}
Qei 1Ri . (E.3.9)
Qi [t]Xi [t]Si [t]Ci [t]|Q[t] =
x
i=1
c
i=1
Observe that
L
X
l=1
Qel 1Rl = Qe1 (1 − 1T1 ) +
L
X
j=2
247
(Qej − Qej−1 )(1 − 1Tj ),
where we use facts that 1Tl+1 = 1Tl + 1Rl , ∀l = 1, 2, ..., L − 1, and 1 = 1TL + 1RL .
Thus, we have
E
"
L
X
#
Ql [t]XlR [t]Sl∗ [t]Cl [t]|Q[t]
l=1
1−
= Qe1
X
α(x)
x
X
Pr{C[t] = c}1T1
c
!
L
X
X
X
+
(Qej − Qej−1 ) 1 −
α(x)
Pr{C[t] = c}1Tj
x
j=2
c
!
.
Since
L
X
λl Ql =
l=1
L
X
λel Qel = Qe1
l=1
L
X
l=1
L
L
X
X
λel +
(Qej − Qej−1 )
λel ,
j=2
l=j
∆V becomes
∆V
≤ B1,max + Qe1
L
X
λel − 1 +
X
α(x)
x
l=1
X
Pr{C[t] = c}1T1
c
!
L
L
X
X
X
X
+
(Qej − Qej−1 )
λel − 1 +
α(x)
Pr{C[t] = c}1Tj
j=2
x
l=j
c
!
We define
ζ , min
A⊆N
(
1−
X
x
α(x)
X
Pr{C[t] = c}1{xl cl =0,∀l∈A} −
c
X
l∈A
λl
)
.
Since the arrival rate vector λ is strictly inside the region Γ(m, C), we have ζ > 0.
Thus, we have
∆V
L
X
≤ B1,max − Qe1 ζ − ζ
(Qej − Qej−1 )
j=2
= B1,max − ζQeL
L
ζ X
Ql .
≤ B1,max −
L l=1
By using Theorem 4.1 in [66], all data queues are strongly stable.
248
E.4
Proof of Proposition 7.3.4
Proof. Consider the Lyapunov function
1X 2
(Q [t] + Ul2 [t]).
2 l=1 l
L
W [t] , W (Q[t], U[t]) =
(E.4.1)
Then, we have
∆W , E[W [t + 1] − W [t]|Q[t] = Q, U[t] = U]
L
1X
≤
E[(Ql [t] + Al [t] − Xl∗ [t]Sl∗ [t]Cl [t])2 − Q2l [t]|Q[t], U[t]]
2 l=1
1X
+
E[(Ul [t] + Xl∗ [t] − Il [t])2 − Ul2 [t]|Q[t], U[t]]
2 l=1
L
=
L
X
E [Ql [t](Al [t] − Xl∗ [t]Sl∗ [t]Cl [t])|Q[t], U[t]]
l=1
+
L
X
E [Ul [t](Xl∗ [t] − Il [t])|Q[t], U[t]] + B2 ,
(E.4.2)
l=1
where
1X
1X
,
E[(Al [t] − Xl∗ [t]Sl∗ [t]Cl [t])2 |Q[t], U[t]] +
E[(Xl∗ [t] − Il [t])2 |Q[t], U[t]]
2 l=1
2 l=1
L
B2
L
1X
E[A2l [t] + Cl2 [t] + Il2 [t] + 1] , B2,max < ∞.
2 l=1
L
≤
Hence, we have
∆W ≤
L
X
λl Ql −
l=1
−
=
ml Ul + B2,max
l=1
L
X
E [Ql [t]Xl∗ [t]Sl∗ [t]Cl [t] − Ul [t]Xl∗ [t]|Q[t], U[t]]
l=1
L
X
λl Ql −
l=1
L
X
"
L
X
ml Ul + B2,max
l=1
−E max Ql [t]Xl∗ [t]Cl [t] −
l
L
X
l=1
249
#
Ul [t]Xl∗ [t]|Q[t], U[t] .
Since λ ∈ Int(Λ(m, C)), there exists a > 0 such that
λl ≤
X
α(x)
x
X
P (C[t] = c)
c
X
β(x, c; s)xl cl sl − , ∀l.
s∈S
Then, by using the above inequality, ∆W becomes
!
L
L
X
X
X
∆W ≤ −
Ql +
Ul
α(x)xl − ml + B2,max
l=1
+
X
X
α(x)
x
"
x
l=1
P (C[t] = c)
c
X
β(x, c; s)
s∈S
L
X
−E max Ql [t]Xl∗ [t]Cl [t] −
l
L
X
xl cl sl Ql −
l=1
L
X
U l xl
l=1
#
Ul [t]Xl∗ [t]|Q[t], U[t] .
l=1
!
(E.4.3)
By using (7.3.4), we have
∆W ≤ −
L
X
Ql + B2,max +
α(x)
x
l=1
"
X
= −
L
X
Ql + B2,max +
l=1
−
X
α(x)E
x
≤ −
L
X
L
X
l=1
"
X
x
P (C[t] = c) max xl cl Ql −
l
c
−E max Ql [t]Xl∗ [t]Cl [t] −
l
X
L
X
l=1
#
U l xl
!
Ul [t]Xl∗ [t]|Q[t], U[t]
"
α(x)E max xl Cl [t]Ql [t] −
max Ql [t]Xl∗ [t]Cl [t]
l
l
−
L
X
Ul [t]xl |Q[t], U[t]
l=1
L
X
Ul [t]Xl∗ [t]|Q[t], U[t]
l=1
Ql + B2,max .
#
#
(E.4.4)
l=1
By using Theorem 4.1 in [66], all data queues are strongly stable and all virtual
queues are mean rate stable.
E.5
Some Properties of Function f (E, e)
Lemma E.5.1.
max {min{xi , y}} = min{ max xi , y}.
1≤i≤n
1≤i≤n
250
(E.5.1)
Proof. (i) If n = 1, LHS = min{x1 , y} = RHS. (ii) Assume (E.5.1) is true for all
n ≤ k. Then for n = k + 1, we have
max {min{xi , y}}
1≤i≤k+1
= max{ max {min{xi , y}}, min{xk+1 , y}}
1≤i≤k
= max{min{ max xi , y}, min{xk+1 , y}}(by assumption(n = k))
1≤i≤k
= min{ max xi , y}(by assumption(n = 2)).
(E.5.2)
1≤i≤k+1
Remarks:
We can use similar argument to show that min {max{xi , y}} =
1≤i≤n
max{ min xi , y}.
1≤i≤n
Consider a set E of users and e ∈
/ E over a symmetric ON-OFF fading channel
with Pr{Cl = 1} = p, ∀l. We assume that there are K users in E whose queue
lengths are less than or equal to Qe . Without loss of generality, we assume that
Q1 ≤ Q2 ≤ ... ≤ QK ≤ Qe ≤ QK+1 ≤ ... ≤ Q|E| . We denote E1 , {1, 2, ..., K}
and E2 , {K + 1, K + 2, ..., |E|}. Let H be the event that at least one users in E2
have the available channel. Let Ii be the event that Ci = 1, Cj = 0 for K ≥ j > i,
i = 1, 2, ..., K − 1, and IK be the event that CK = 1.
Lemma E.5.2.


 Qe , if H happens;
max min{Ql Cl , Qe } =
T
l∈E


Qi , if Hc Ii happens for i = 1, 2, ..., K.
(E.5.3)
Proof.
max min{Ql Cl , Qe }
l∈E
= max max min{Ql Cl , Qe }, max min{Ql Cl , Qe }
l∈E1
l∈E2
= max min{Ql Cl , Qe } + max min{Ql Cl , Qe }
l∈E1
l∈E2
− min max min{Ql Cl , Qe }, max min{Ql Cl , Qe }
l∈E1
l∈E2
251
(E.5.4)
Note that
max min{Ql Cl , Qe } =
l∈E2
Thus,


 Qe , if event H happens;


=
=
(E.5.5)
0 , otherwise.
min max min{Ql Cl , Qe }, max min{Ql Cl , Qe }
l∈E1
l∈E2


 min max min{Ql Cl , Qe }, Qe
, if event H happens;
l∈E1


0 , otherwise


 max min{Ql Cl , Qe } , if event H happens;
l∈E1


(E.5.6)
0 , otherwise.
where we use Lemma E.5.1. In addition, if event Ii (i = 1, 2, ..., K) happens, we have
max min{Ql Cl , Qe } = Qi
l∈E1
(E.5.7)
By substituting (E.5.5), (E.5.6) and (E.5.7) into (E.5.4), we have (E.5.3).
Corollary E.5.3.
f (E, e) =
K
X
p2 (1 − p)|E|−k Qk + p(1 − (1 − p)|E|−K )Qe
(E.5.8)
k=1
Proof. Pr{H} = 1 − (1 − p)|E|−K and Pr{Hc
Thus,
\
Ii } = p(1 − p)|E|−i for i = 1, 2, ..., K.
f (E, e) , E[max min{Ql Cl , Qe Ce }]
l∈E
= pE[max min{Ql Cl , Qe }]
l∈E
=
K
X
p2 (1 − p)|E|−k Qk + p(1 − (1 − p)|E|−K )Qe ,
k=1
where we use Lemma E.5.2.
252
E.6
Proof of Basic Iterative Equation
According to the maximum-minimums identity, we have
max
Ql Cl = max{max Ql Cl , Qr Cr }
S
{r}
l∈F
l∈F
= max Ql Cl + Qr Cr − min{max Ql Cl , Qr Cr }
l∈F
l∈F
= max Ql Cl + Qr Cr − max min{Ql Cl , Qr Cr },
l∈F
(E.6.1)
l∈F
where we use Lemma E.5.1. By taking expectation and subtracting the term
X
l∈F
on both sides of (E.6.1), we get (7.4.2).
S
Ul
{r}
Proof of f (D∗ , d) ≤ f (B, d) in Lemma 7.4.2
E.7
Proof. (1) If A∗
\
D∗ = ∅ or Qd ≤ min Ql , then, by Corollary E.5.3, we have
l∈B
f (B, d) = p 1 − (1 − p)|B| Qd .
(E.7.1)
Without loss of generality, we assume there are K1 users in D∗ whose queue lengths
are less than or equal to Qd , that is, Qj1 ≤ Qj2 ≤ ... ≤ QjK1 ≤ Qd ≤ QjK1 +1 ≤ Qj|D∗ | .
Then, by Corollary E.5.3, we have
∗
f (D , d) =
K1
X
p2 (1 − p)|D
∗ |−k
Qjk + p 1 − (1 − p)|D
∗ |−K
1
k=1
Hence, by noting that |D∗ | = |B|, we have
∗
f (D , d) − f (B, d) =
K1
X
p2 (1 − p)|D
∗ |−k
Qd .
∗
Qjk + p (1 − p)|D | − (1 − p)|D
k=1
Since
−
K1
X
p2 (1 − p)|D
∗ |−k
∗
= p (1 − p)|D | − (1 − p)|D
k=1
we have
∗
f (D , d) − f (B, d) =
K1
X
p2 (1 − p)|D
k=1
253
∗ |−k
∗ |−K
1
∗ |−K
1
Qd .
,
(Qjk − Qd ) ≤ 0.
(E.7.2)
\
6 ∅ and there are some users in
Thus, we have f (D∗ , d) ≤ f (B, d). (2) If A∗ D∗ =
\
\
A∗ D∗ whose queue lengths are less than Qd , let T , A∗ D∗ , B0 , B \ T and
D0 , D∗ \ T. Figure E.1 characterizes the relationship among all these sets.
A*
B
d
B’
T
T
D’
D*
Figure E.1: The relations among all sets
We define
g(E, F, e) , −E min max min{Ql Cl , Qe Ce }, max min{Ql Cl , Qe Ce } ,
where E
\
l∈E
l∈F
F = ∅ and e ∈
/ E, e ∈
/ F. Then, we have
f (B, d) = E max min{Ql Cl , Qd Cd }
l∈B
= E max max0 min{Ql Cl , Qd Cd }, max min{Ql Cl , Qd Cd }
l∈T
l∈B
= E max0 min{Ql Cl , Qd Cd } + E max min{Ql Cl , Qd Cd }
l∈T
l∈B
−E min max0 min{Ql Cl , Qd Cd }, max min{Ql Cl , Qd Cd }
l∈T
l∈B
0
0
= f (B , d) + f (T, d) + g(B , T, d),
254
(E.7.3)
where we use the maximum-minimums identity. Similarly, we have
f (D∗ , d) = f (D0 , d) + f (T, d) + g(D0 , T, d).
(E.7.4)
Thus, to show f (D∗ , d) ≤ f (B, d), we only need to show
f (D0 , d) + g(D0 , T, d) ≤ f (B0 , d) + g(B0 , T, d).
(E.7.5)
Note that Qd ≤ min0 Ql . Without loss of generality, we assume that K2 users in
l∈B
0
D whose queue lengths are less than or equal to Qd , that is Qj1 ≤ Qj2 ≤ ... ≤
QjK2 ≤ Qd ≤ QjK2 +1 ≤ ... ≤ Qj|D0 | . We denote D0 1 , {j1 , j2 , ..., jK2 } and D0 2 ,
{jK2 +1 , jK2 +2 , ..., j|D0 | }. By using similar technique in deriving equation (E.7.2), we
have
0
0
f (D , d) − f (B , d) =
K2
X
0
p2 (1 − p)|D |−k (Qjk − Qd ).
(E.7.6)
k=1
0
Next, let’s focus on the term g(B , T, d).
0
g(B , T, d) = −E min max0 min{Ql Cl , Qd Cd }, max min{Ql Cl , Qd Cd }
l∈T
l∈B
= −pE min max0 min{Ql Cl , Qd }, max min{Ql Cl , Qd } .
l∈T
l∈B
Let J be the event that at least one user in B0 has the available channel. Then, we
have
max0 min{Ql Cl , Qd } =
l∈B
Thus, we get
=
=


 Qd , if event J happens;


(E.7.7)
0 , otherwise.
min max0 min{Ql Cl , Qd }, max min{Ql Cl , Qd }
l∈T
l∈B


 min Qd , max min{Ql Cl , Qd }
, if event J happens;
l∈T


0 , otherwise


 max min{Ql Cl , Qd } , if event J happens;
l∈T


0 , otherwise.
255
(E.7.8)
0
where we use Lemma E.5.1. Since Pr{J } = (1 − (1 − p)|B | ), we have
0
g(B0 , T, d) = −p 1 − (1 − p)|B | E[max min{Ql Cl , Qd }]
l∈T
0
= (1 − p)|B | − 1 f (T, d).
(E.7.9)
Let’s consider the term g(D0 , T, d).
0
g(D , T, d) = −E min max0 min{Ql Cl , Qd Cd }, max min{Ql Cl , Qd Cd }
l∈T
l∈D
= −pE min max0 min{Ql Cl , Qd }, max min{Ql Cl , Qd } .
l∈T
l∈D
Let K be the event that at least one user in D0 2 has the available channel. Let Lk be
the event that Cjk = 1, Cji = 0 for k < i ≤ K2 , k = 1, 2, ..., K2 and LK2 be the event
that CjK2 = 1. Then, by using Lemma E.5.2, we have



Qd , if event K happens;



T
max0 min{Ql Cl , Qd } =
Qjk , if event Kc Lk happens,
l∈D





for k = 1, 2., , , ., K2 .
Thus, we get
min max0 min{Ql Cl , Qd }, max min{Ql Cl , Qd }
l∈T
l∈D



min Qd , max min{Ql Cl , Qd }
, if K happens;


l∈T


\
c
=
min
Q
,
max
min{Q
C
,
Q
}
,
if
K
Lk happens,
j
l
l
d
k

l∈T





for k = 1, 2., , , ., K2



max min{Ql Cl , Qd } , if K happens;


 l∈T
\
c
=
max
min{Q
Lk happens,
l Cl , Qjk } , if K

l∈T




for k = 1, 2., , , ., K ,
2
256
where we use Lemma E.5.1. Hence, we have
g(D , T, d) = −pE max min{Ql Cl , Qd } Pr{K}
0
l∈T
−
K2
X
pE max min{Ql Cl , Qjk } Pr K
k=1
l∈T
= − Pr{K}f (T, d) −
K2
X
k=1
0
have
c
\
Lk
o
n \ o
Pr Kc Lk f (T, jk ).
Note that Pr{K} = 1 − (1 − p)|D |−K2 and Pr{Kc
g(D0 , T, d) =
n
\
(E.7.10)
0
Lk } = p(1 − p)|D |−k . Thus, we
K2
X
0
0
(−p)(1 − p)|D |−k f (T, jk ) + (1 − p)|D |−K2 − 1 f (T, d).
k=1
Note that |B0 | = |D0 |. Thus, we have
K2
X
0
g(D , T, d) − g(B , T, d) =
(−p)(1 − p)|D |−k f (T, jk )
0
0
k=1
0
0
+ (1 − p)|D |−K2 − (1 − p)|D | f (T, d).(E.7.11)
|D0 |−K2
Note that (1 − p)
|D0 |
− (1 − p)
=p
K2
X
0
(1 − p)|D |−k . Thus, (E.7.11) becomes
k=1
g(D0 , T, d) − g(B0 , T, d) =
K2
X
0
(−p)(1 − p)|D |−k (f (T, jk ) − f (T, d)) . (E.7.12)
k=1
Consider the term f (T, jk ) − f (T, d). Without loss of generality, we assume nk users
in T whose queue lengths are less than or equal to Qjk and nd users whose queue
lengths are less than or equal to Qd , that is, Qi1 ≤ Qi2 ≤ ... ≤ Qink ≤ Qjk ≤ Qink +1 ≤
... ≤ Qind ≤ Qd ≤ Qind +1 ≤ Qi|T| . Note that nk ≤ nd . Thus, by using Corollary
257
E.5.3, we have
f (T, jk ) − f (T, d)
nk
X
=
p2 (1 − p)|T|−l Qil + p 1 − (1 − p)|T|−nk Qjk
l=1
nd
X
−
l=1
p2 (1 − p)|T|−l Qil − p 1 − (1 − p)|T|−nd Qd
= p 1 − (1 − p)
|T|−nk
|T|−nd
Qjk − p 1 − (1 − p)
Qd −
nd
X
l=nk +1
nd
X
≥ p 1 − (1 − p)|T|−nk Qjk − p 1 − (1 − p)|T|−nd Qd − Qd
p2 (1 − p)|T|−l Qil
p2 (1 − p)|T|−l
l=nk +1
= p 1 − (1 − p)|T|−nk Qjk − p 1 − (1 − p)|T|−nd Qd − p (1 − p)|T|−nd − (1 − p)|T|−nk Qd
= p 1 − (1 − p)|T|−nk (Qjk − Qd ) .
(E.7.13)
Thus, we have
g(D0 , T, d) − g(B0 , T, d)
K2
X
0
=
(−p)(1 − p)|D |−k (f (T, jk ) − f (T, d))
≤
=
k=1
K2
X
k=1
K2
X
k=1
0
p2 (1 − p)|D |−k (Qjk − Qd ) (1 − p)|T|−nk − 1
0
0
(Qjk − Qd )p2 (1 − p)|D |+|T|−nk −k − (1 − p)|D |−k .
Hence, we have
f (D∗ , d) − f (B, d)
K2
K2
X
X
0
0
0
≤
(Qjk − Qd )p2 (1 − p)|D |−k +
(Qjk − Qd )p2 (1 − p)|D |+|T|−nk −k − (1 − p)|D |−k
=
k=1
K2
X
k=1
0
(Qjk − Qd )p2 (1 − p)|D |+|T|−nk −k ≤ 0.
k=1
Thus, we have the desired result.
258
(E.7.14)
E.8
Proof of Lemma 7.5.3
Proof. We only prove the first part in (7.5.2). The second part follows the similar
argument. It is enough to show
max al Clmin [t] ≤st max al Cl [t],
l
l
∀t,
(E.8.1)
holds for any constants al ≥ 0, ∀l, where Y1 ≤st Y2 means that the random variable
Y1 is stochastically smaller than the random variable Y2 [82]. In each time slot t,
according to the definition of stochastically smaller, we need to show
Pr{max al Cl [t] ≤ b} ≤ Pr{max al Clmin [t] ≤ b},
l
l
∀b,
(E.8.2)
which is equivalent to showing
Pr{al Cl [t] ≤ b, ∀l ∈ L} ≤ Pr{al Clmin [t] ≤ b, ∀l ∈ L},
∀b.
If b < 0, we have
Pr{al Cl [t] ≤ b, ∀l ∈ L} = Pr{al Clmin [t] ≤ b, ∀l ∈ L} = 0,
since al ≥ 0 and Cl [t] ≥ 0, ∀l. Thus, we assume b ≥ 0 in the rest of the proof. Let
G , {l ∈ L : al > 0}, we have
Pr{al Cl [t] ≤ b, ∀l ∈ L} = Pr{al Cl [t] ≤ b, ∀l ∈ G},
Pr{al Clmin [t] ≤ b, ∀l ∈ L} = Pr{al Clmin [t] ≤ b, ∀l ∈ G}.
Thus, we only need to show
Pr{al Cl [t] ≤ b, ∀l ∈ G} ≤ Pr{al Clmin [t] ≤ b, ∀l ∈ G},
which is equivalent to proving
b
b
min
Pr Cl [t] ≤ , ∀l ∈ G ≤ Pr Cl [t] ≤ , ∀l ∈ G .
al
al
259
(E.8.3)
Next, we will show that (E.8.3) is true. Let H , {l ∈ G :
b
≥ c1 }. We have
al
b
b
Pr Cl [t] ≤ , ∀l ∈ G ≤ Pr{Cl [t] ≤ , ∀l ∈ G \ H}
al
al
= Pr{Cl [t] = 0, ∀l ∈ G \ H}.
(E.8.4)
From the construction of independent ON-OFF fading channel Cmin , we have
Pr{Cl [t] = 0} ≤ Pr{Clmin [t] = 0}, ∀l.
(E.8.5)
Since condition (7.5.1) holds, we have
Pr{Cl [t] = 0, ∀l ∈ G \ H} ≤ Pr{Clmin [t] = 0, ∀l ∈ G \ H}.
Thus, we get
b
Pr Cl [t] ≤ , ∀l ∈ G
al
min
≤ Pr Cl [t] = 0, ∀l ∈ G \ H
b
min
= Pr Cl [t] ≤ , ∀l ∈ G \ H (since Clmin [t] = 0 or c1 )
al
b
min
= Pr Cl [t] ≤ , ∀l ∈ G .
al
E.9
Proof of Lemma 7.5.4
Consider a system over fading channel CII . We show that the JPT algorithm where
we use channel statistics and rates of channel CI in the probing component can
support any arrival rate vector λ ∈ Λ(m, CI ). By choosing the same Lyapunov
function and following the same steps as in the proof for Proposition 7.3.4, we have
∆W ≤
L
X
l=1
λl Ql −
"
L
X
ml Ul + B2,max
l=1
−E max Ql [t]Xl∗ [t]ClII [t] −
l
L
X
l=1
260
#
Ul [t]Xl∗ [t]|Q[t], U[t] .
Given any value of Q[t] and U[t] in slot t, Ql [t]Xl∗ [t], ∀l, are just non-negative constant
numbers. Thus, by the condition (7.5.3), we have
E[max Ql [t]Xl∗ [t]ClI [t]|Q[t], U[t]] ≤ E[max Ql [t]Xl∗ [t]ClII [t]|Q[t], U[t]]. (E.9.1)
l
l
Hence, we get
∆W ≤
L
X
l=1
λl Ql −
L
X
ml Ul + B2,max
l=1
"
−E max Ql [t]Xl∗ [t]ClI [t] −
l
L
X
Ul [t]Xl∗ [t]|Q[t], U[t]
l=1
I
#
For any λ ∈ Λ(m, C ), by Proposition 7.3.4, the considered JPT algorithm can
support this arrival rate vector and thus we have the desired result.
E.10
Proof of Lemma 7.5.5
Proof. By maximum-minimums identity, we have
X
1 X
h(p) =
E[Xl Wl ] −
E[min{Xl1 Wl1 , Xl2 Wl2 }]
p l∈A
l ,l ∈A
1 2
l1 <l2
+... + (−1)
E[min{X1 W1 , ..., X|A| W|A| }]
X
X
=
E[Xl ] − p
E[min{Xl1 , Xl2 }]
|A|−1
l1 ,l2 ∈A
l1 <l2
l∈A
+... + (−1)|A|−1 p|A|−1 E[min{X1 , ..., X|A| }].
Y
Xl , ∀L ⊆ A. Thus, we have
Note that min Xl =
l∈L
h(p) =
X
l∈L
E[Xl ] − p
X
E[Xl1 Xl2 ] + ... + (−1)|A|−1 p|A|−1 E[
l1 ,l2 ∈A
l1 <l2
l∈A
(E.10.1)
Y
Xl ]. (E.10.2)
l∈A
Hence, we have
h0 (p) = −
X
E[Xl1 Xl2 ] + 2p
l1 ,l2 ∈A
l1 <l2
X
E[Xl1 Xl2 Xl3 ]
l1 ,l2 ,l3 ∈A
l1 <l2 <l3
+... + (−1)|A|−1 (|A| − 1)p|A|−2 E[
Y
l∈A
261
Xl ].
(E.10.3)
Let γi (i = 0, 1, 2, ..., |A|) be the probability that i users in set A probe the channel.
k
X
Y
Let’s consider the term
E[ Xli ] (k = 2, 3, ..., |A|). Let U be the event
l1 ,l2 ,...,lk ∈A
l1 <l2 <...<lk
i=1
that users l1 , l2 , ..., lk probe the channel and Vj (j = 0, 1, 2, ..., |A|) be the event that
j users probe the channel. By the law of total probability, we have
E[
k
Y
Xli ] = Pr{U} =
i=1
|A|
X
γj Pr{U|Vj }.
(E.10.4)
j=k
Thus, we have
X
l1 ,l2 ,...,lk ∈A
l1 <l2 <...<lk
k
Y
E[ Xli ] =
i=1
=
X
|A|
X
γj Pr{U|Vj }
l1 ,l2 ,...,lk ∈A j=k
l1 <l2 <...<lk
|A|
X
j=k
γj
X
Pr{U|Vj }.
(E.10.5)
l1 ,l2 ,...,lk ∈A
l1 <l2 <...<lk
Note that if {l1 , l2 , ..., lk } is a subset of the set of selected j users, then Pr{U|Vj } = 1;
otherwise, Pr{U|Vj } = 0. Thus, we have
X
j
γj
.
E[ Xli ] =
k
i=1
j=k
∈A
l1 ,l2 ,...,lk
l1 <l2 <...<lk
|A|
X
k
Y
By substituting (E.10.6) into (E.10.3), we have
2
3
4
|A|
0
h (p) = −
γ2 +
γ3 +
γ4 + ... +
γ|A|
2
2
2
2
3
4
|A|
+2p
γ3 +
γ4 + ... +
γ|A| + ...
3
3
3
|A|−1
|A|−2 |A|
+(−1)
(|A| − 1)p
γ|A|
|A|
|A|
k
X
X
i k
= −
γk
(−1)
(i − 1)pi−2 .
i
i=2
k=2
Using mathematical induction, it is not hard to show
n
X
1 − np(1 − p)n−1 − (1 − p)n
i n
,
(−1)
(i − 1)pi−2 =
p2
i
i=2
262
(E.10.6)
(E.10.7)
n−1
for any n ≥ 2. Noting that 1 − np(1 − p)
n X
n k
− (1 − p) =
p (1 − p)n−k ≥ 0
k
k=2
n
n
for any n ≥ 2, we have
(i − 1)pi−2 ≥ 0 for any n ≥ 2. Hence, we have
(−1)
i
i=2
h0 (p) ≤ 0, and thus h(p) is a decreasing function.
n
X
E.11
i
Proof of Proposition 7.5.6
Proof. Under Assumption 7.5.1, by Lemma 7.5.3 and Lemma 7.5.4, we have
Λ(m, Cmin ) ⊆ Λ(m, C) ⊆ Λ(m, Cmax ).
By Lemma 7.3.2, we have
Λ(m, Cmax ) = {λ : ∃ a probability distribution of probing schedule Xmax
X
λl ≤ E[max Xlmax Clmax [t]], ∀A ⊆ L and E[Xlmax ] ≤ ml , ∀l},
such that
l∈A
l∈A
and
Λ(m, Cmin ) = {λ : ∃ a probability distribution of probing schedule Xmin
X
such that
λl ≤ E[max Xlmin Clmin [t]], ∀A ⊆ L and E[Xlmin ] ≤ ml , ∀l}.
l∈A
l∈A
Let
ρ|A|
1
E[maxl∈A Xlmax Clmin [t]]
E[maxl∈A Xlmax Clmin [t]]
pmin c1
,
=ρ 1
.
E[maxl∈A Xlmax Clmax [t]]
E[maxl∈A Xlmax Clmax [t]]
pmax cM
(E.11.1)
By Lemma 7.5.5, we have ρ|A| ≥ ρ. Hence, for any λ ∈ ρΛ(m, Cmax ), we have
λ
λ ∈ Λ(m, Cmin ). Indeed, for any λ ∈ ρΛ(m, Cmax ), we have
∈ Λ(m, Cmax ), that
ρ
X λl
max max
max
is,
≤ E[max Xl Cl [t]], ∀A ⊆ L, and E[Xl ] ≤ ml , ∀l. Hence, for any
l∈A
ρ
l∈A
A ⊆ L, we have
X
l∈A
λl ≤ ρE[max Xlmax Clmax [t]] ≤ ρ|A| E[max Xlmax Clmax [t]] = E[max Xlmax Clmin [t]].
l∈A
l∈A
263
l∈A
By taking the probability distribution of Xmin the same as Xmax , we have λ ∈
Λ(m, Cmin ). Thus, for any λ ∈ ρΛ(m, C), we have λ ∈ Λ(m, Cmin ). Next, we will
show that the MSGP algorithm, combined with the MWS algorithm in the transmission stage, can support any arrival rate vector λ ∈ Λ(m, Cmin ), implying that it can
pmin c1
of the capacity region.
at least achieve a fraction ρ =
pmax cM
By choosing the same Lyapunov function and following the same argument as in
the proof for Proposition 7.3.4, we have
∆W , E[W [t + 1] − W [t]|Q[t] = Q, U[t] = U]
L
L
X
X
λl Ql −
ml Ul + B2,max
≤
l=1
l=1
"
−E max Ql [t]XlM [t]Cl [t] −
M
where X [t] =
l
(XlM [t])Ll=1
L
X
#
Ul [t]XlM [t]|Q[t], U[t] ,
l=1
is a probing schedule chosen by MSGP algorithm. Given
any value of Q[t] and U[t] at slot t, Ql [t]XlM [t], ∀l are just non-negative constant
numbers. Thus, by Lemma 7.5.3, we have
h
i
h
i
E max Ql [t]XlM [t]Cl [t]|Q[t], U[t] ≥ E max Ql [t]XlM [t]Clmin [t]|Q[t], U[t] .
l
l
Thus, ∆W becomes
∆W ≤
L
X
λl Ql −
L
X
ml Ul + B2,max
l=1
l=1
L
X
M
min
M
−E max Ql [t]Xl [t]Cl [t] −
Ul [t]Xl [t]|Q[t], U[t] .
l
l=1
Then, by using the fact that the MSGP algorithm can find the optimal probing schedule in the symmetric and independent ON-OFF fading channel Cmin and following
the same argument as in Proposition 7.3.4, we have the desired result.
E.12
Proof of Proposition 7.6.3
To prove this proposition, we need the following claim:
264
Claim 3. All edge weights with strictly positive value are lower bounded by a strictly
positive constant value.
Proof. Recall that c0 = 0. Since pij , ∀i = 1, ..., L, j = 0, 1, ..., M , and cj , j = 1, ..., M ,
aj
qij
are rational numbers, let cj =
and pij =
, where aj and bj are co-prime for any
bj
dij
j = 1, ..., M , and qij and dij are co-prime for any i = 1, ..., L, j = 0, 1, ..., M . First,
we will show that if φi > 0, we have
φi , Qi E[Ci ] − Ui ≥ QM
1
j=1 bj dij
Indeed, we have
Qi E[Ci ] − Ui
M
X
aj qij
= Qi
− Ui
b d
j=1 j ij
= QM
1
j=1 bj dij
Note that Qi
M
X
j=1
QM
1
j=1 bj dij
aj qij
Y
bl dil − Ui
M
Y
Qi
M
X
aj qij
j=1
, ∀i.
Y
(E.12.1)
bl dil − Ui
l6=j
M
Y
bj dij
j=1
!
bj dij is an integer. If φi > 0, then φi ≥
j=1
l6=j
. Next, we will show that if the weight φi − f ({i1 , ..., ik−1 }, i) > 0, then
φi − f ({i1 , ..., ik−1 }, i) ≥ QM
j=1 bj
QM
j1 =0
di1 j1 ...
265
1
QM
jk−1 =0
dik−1 jk−1
QM
j=0
dij
. (E.12.2)
Indeed, we have
φi − f ({i1 , ..., ik−1 }, i)
= Qi E[Ci ] − Ui − E
max
l∈{i1 ,...,ik−1 }
min{Ql Cl , Qi Ci }
M
M
X
X
aj 1
aj qij
− Ui −
= Qi
Pr Ci1 =
b
d
bj1
j
ij
j1 =0
j=1
M
M
X
ajk−1 X
aj
...
Pr Ci =
Pr Cik−1 =
bjk−1 j=0
bj
jk−1 =0
Qik1 ajk−1 Qi aj
Qi1 aj1 Qi aj
,
, ..., min
,
max min
bj 1
bj
bjk−1
bj
M
X
aj qij
= Qi
− Ui
b
d
j
ij
j=1
(I)
M
M
M
X
X
qik−1 jk−1 X qij
qi1 j1
−
...
d
d
d
j1 =0 i1 j1
jk−1 =0 ik−1 jk−1 j=0 ij
Qik1 ajk−1 Qi aj
Qi1 aj1 Qi aj
max min
,
, ..., min
,
bj 1
bj
bjk−1
bj
≥ QM
j=1 bj
QM
j1 =0
di1 j1 ...
1
QM
jk−1 =0
dik−1 jk−1
where (I) follows the same argument as in (E.12.1).
QM
j1 =0
dij
,
(E.12.3)
Thus, by combining (E.12.1) and (E.12.2), it is easy to see that all edge weights
M
L M
Y
1 YY 1
with strictly positive value should be lower bounded by
> 0.
b
d
j=1 j i=1 j=0 ij
[Proof of Proposition 7.6.3:]
Proof. Assume that the node with the optimal value is in level K. Given any τ > 0
and δ > 0. Let WkDM SGP and WkM SGP be the weight of an edge selected by DMSGP
algorithm and MSGP algorithm from level k − 1 to level k respectively. Note that
WkDM SGP and WkM SGP are strictly positive. By claim 3, WkDM SGP and WkM SGP are
lower bounded by a strictly positive constant value. Thus, by using similar argument
266
in [40], we can show that given any τ 0 > 0, ∃Gk > 0 such that for any G > Gk , we
have
Pr{WkDM SGP > WkM SGP (1 − δ)} > 1 − τ 0 .
Let W
DM SGP
=
K
X
WkDM SGP
and W
M SGP
=
k=1
K
X
(E.12.4)
WkM SGP . Thus, for any G ≥
k=1
max{G1 , G2 , ..., GK }, we have
Pr{W DM SGP > W M SGP (1 − δ)} ≥ Pr{WkDM SGP > WkM SGP (1 − δ), ∀k = 1, ..., K}
> 1 − Kτ 0 ,
(E.12.5)
where we use the fact [11] that given any two events E and F such that Pr{E} > 1−1
\
and Pr{F} > 1 − 2 , we have Pr{E
F} > 1 − 1 − 2 . We can pick τ 0 small enough
such that 1 − Kτ 0 > 1 − τ . Hence, we have
Pr{W DM SGP > W M SGP (1 − δ)} > 1 − τ.
(E.12.6)
Then, we have
E[W DM SGP |Q[t], U[t]] ≥ (1 − δ)(1 − τ )E[W M SGP |Q[t], U[t]].
By choosing the same Lyapunov function as in the proof for Proposition 7.3.4, the
remaining argument follows the similar reasoning as in the proof for Theorem 1 in
[40].
267
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