Super-Fast Delay Tradeoffs for Utility
Optimal Scheduling in Wireless Networks
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Michael J. Neely
University of Southern California
http://www-rcf.usc.edu/~mjneely/
*Sponsored by NSF OCE Grant 0520324
A multi-node network with N nodes and L links:
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Slotted time t = 0, 1, 2, …
0
1
2
3
…
t
Traffic (An(c)(t)) and channel states S(t) i.i.d. over timeslots.
Control for Optimal Utility-Delay Tradeoffs…
1) Flow Control:
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Ai(c)
Ri(c)(t)
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An(c)(t) = New Commodity c data during slot t (i.i.d)
E[An(c)(t)] = ln(c) ,
(ln(c) ) = Arrival Rate Matrix
Rn(c)(t) = Flow Control Decision at (i,c):
Rn(c)(t) < min[Ln(c)(t) + An(c)(t) , Rmax]
2) Resource Allocation:
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S(t) = (Sab(t))
Transmission Rate Matrix: m(t) = (mab(t))
Channel State Matrix:
WS(t) = Set of Feasible Rate Matrices for Channel State S.
Resource allocation: choose
m(t) WS(t)
3) Routing:
Lc = Set of all links
acceptable for
commodity c traffic
to traverse
Examples…
mab(c)(t) = Amount of commodity c data
transmitted over link (a,b)
c
mab(c)(t) < mab(t)
mab(c)(t) = 0 if (a,b) Lc
3) Routing:
Example 1:
Lc = All network links
(commodity c =
mab(c)(t) = Amount of commodity c data
transmitted over link (a,b)
c
mab(c)(t) < mab(t)
mab(c)(t) = 0 if (a,b) Lc
)
3) Routing:
Example 2:
Lc = a directed subset
(commodity c =
mab(c)(t) = Amount of commodity c data
transmitted over link (a,b)
c
mab(c)(t) < mab(t)
mab(c)(t) = 0 if (a,b) Lc
)
3) Routing:
downlink
uplink
Example 3:
Lc = Specifies a
one-hop network
(no routing decisions)
mab(c)(t) = Amount of commodity c data
transmitted over link (a,b)
c
mab(c)(t) < mab(t)
mab(c)(t) = 0 if (a,b) Lc
3) Routing:
one-hop ad-hoc network
Example 4:
Lc = Specifies a
one-hop network
(no routing decisions)
mab(c)(t) = Amount of commodity c data
transmitted over link (a,b)
c
mab(c)(t) < mab(t)
mab(c)(t) = 0 if (a,b) Lc
gn(c)(r)
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Utility functions
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L = Capacity region (considering all control algs.)
rn(c) = Time average of Rn(c)(t) admission decisions.
GOAL:
(Joint flow control,
resource allocation,
and routing)
Network Utility Optimization:
Static Optimization: (Lagrange Multipliers and convex duality)
Kelly, Maulloo, Tan [J. Op. Res. 1998]
Xiao, Johansson, Boyd [Allerton 2001]
Julian, Chiang, O’Neill, Boyd [Infocom 2002]
P. Marbach [Infocom 2002]
Steven Low [TON 2003]
B. Krishnamachari, Ordonez [VTC 2003]
M. Chiang [Infocom 2004]
Stochastic Optimization:
Lee, Mazumdar, Shroff [2005] (stochastic gradient)
Eryilmaz, Srikant [Infocom 2005] (fluid transformations)
Stolyar [Queueing Systems 2005] (fluid limits)
Neely , Modiano [2003, 2005] (Lyapunov optimization)
Network Utility Optimization:
Static Optimization: (Lagrange Multipliers and convex duality)
Kelly, Maulloo, Tan [J. Op. Res. 1998]
Xiao, Johansson, Boyd [Allerton 2001]
Julian, Chiang, O’Neill, Boyd [Infocom 2002]
P. Marbach [Infocom 2002]
Steven Low [TON 2003]
B. Krishnamachari, Ordonez [VTC 2003]
M. Chiang [Infocom 2004]
Stochastic Optimization:
Lee, Mazumdar, Shroff [2005] (stochastic gradient)
Eryilmaz, Srikant [Infocom 2005] (fluid transformations)
Stolyar [Queueing Systems 2005] (fluid limits)
Neely , Modiano [2003, 2005] (Lyapunov optimization)
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
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Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
any rate vector!
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
any rate vector!
l
r
Utility functions
Cross-Layer Control Algorithm (with control parameter V>0):
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
gn(c)(r)
any rate vector!
l
r
Utility functions
Achieves: [O(1/V), O(V)] utility-delay tradeoff!
Uses theory of Lyapunov Optimization [Neely, Modiano 2003, 2005]
Generalizes classical Lyapunov Stability results of:
-Tassiulas, Ephremides [Trans. Aut. Control 1992]
-Kumar, Meyn [Trans. Aut. Control 1995]
-McKeown, Anantharam, Walrand [Infocom 1996]
-Leonardi et. al., [Infocom 2001]
Question:
Is [O(1/V), O(V)] the optimal utility-delay tradeoff?
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Avg. Delay
Results: For a large class of overloaded networks, we
can do much better by achieving O(log(V)) average delay.
O(log(V))
V
Question:
Is [O(1/V), O(V)] the optimal utility-delay tradeoff?
l
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Avg. Delay
Results: For a large class of overloaded networks, we
can do much better by achieving O(log(V)) average delay.
O(log(V))
V
Question:
Is [O(1/V), O(V)] the optimal utility-delay tradeoff?
l
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Avg. Delay
Results: For a large class of overloaded networks, we
can do much better by achieving O(log(V)) average delay.
O(log(V))
V
Question:
Is [O(1/V), O(V)] the optimal utility-delay tradeoff?
l
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Avg. Delay
Results: For a large class of overloaded networks, we
can do much better by achieving O(log(V)) average delay.
O(log(V))
V
Question:
Is [O(1/V), O(V)] the optimal utility-delay tradeoff?
l
e
e
Avg. Delay
Results: For a large class of overloaded networks, we
can do much better by achieving O(log(V)) average delay.
O(log(V))
V
Question:
Is [O(1/V), O(V)] the optimal utility-delay tradeoff?
l
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Avg. Delay
Results: For a large class of overloaded networks, we
can do much better by achieving O(log(V)) average delay.
O(log(V))
V
Overloaded and Fully Active Assumptions:
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Assumption 1 (Overloaded): Optimal operating point r* has
all positive entries, and the input rate matrix l is outside of
the capacity region and strictly dominates r*. That is, there
exists an e>0 such that:
e < rn(c) < ln(c) - e
Overloaded and Fully Active Assumptions:
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*Assumption 2 (Fully Active): All queues Un(c)(t) that
can be positive are also active sources of commodity c
data.
*Used implicitly in proofs of conference version (Infocom 2006) but
not stated explicitly. Described in more detial in JSAC 2006 (on web).
Overloaded and Fully Active Assumptions:
downlink
uplink
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*Assumption 2 (Fully Active): All queues Un(c)(t) that
can be positive are also active sources of commodity c
data.
*Natural assumption for overloaded one-hop networks.
(Network is defined by all active links)
Overloaded and Fully Active Assumptions:
one-hop ad-hoc network
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*Assumption 2 (Fully Active): All queues Un(c)(t) that
can be positive are also active sources of commodity c
data.
*Natural assumption for overloaded one-hop networks.
(Network is defined by all active links)
Overloaded and Fully Active Assumptions:
one-hop ad-hoc network
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*Assumption 2 (Fully Active): All queues Un(c)(t) that
can be positive are also active sources of commodity c
data. Holds for a large class of multi-hop networks.
Example: 1 or more commodities, all nodes are independent
sources of each of these commodities (as in “all-to-all” traffic)
Overloaded and Fully Active Assumptions:
Fully Active assumption can be restrictive in general
multi-hop networks with stochastic channels:
Logarithmic Utility-Delay Tradeoffs Unknown:
l1
1
m1(t)
2
m2(t)
Logarithmic Utility-Delay Tradeoffs Achievable:
l2
l1
1
m1(t)
2
m2(t)
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
Automatically satisfied if we
stabilize the network.
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
Difficult to achieve “super-fast”
logarithmic delay tradeoffs working
Directly with this constraint.
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
However: For any queueing system (stable or not):
Un
(c)(t)
(actual bits
transmitted)
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
However: For any queueing system (stable or not):
Un
(c)(t)
(actual bits
transmitted)
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
Want to Solve:
We Know:
Also:
IF EDGE EFFECTS SMALL:
Un(c)(t)
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
Want to Solve:
We Know:
Also:
IF EDGE EFFECTS SMALL:
Introduce a virtual queue [Neely Infocom 2005]:
Zn(c)(t)
The Tradeoff Optimal Control Algorithm:
Define the aggregate “bi-modal” Lyapunov Function:
Designing “gravity”
into the system:
Q
Minimize:
Un(c)
[Buffer partitioning
Concept similar to
Berry-Gallager 2002]
Utility-Delay Optimal Algorithm (UDOA):
(stated here in special case of zero transport layer storage)
(1) Flow Control (a): At node n, observe queue backlog Un(c)(t).
ln(c)
Rn(c)(t)
Un(c)(t)
Rest of Network
If Un(c)(t) > Q then Rn(c)(t) = 0
(reject all new data)
If Un(c)(t) < Q then Rn(c)(t) = An(c)(t) (admit all new data)
(where V is a parameter that affects network delay)
Utility-Delay Optimal Algorithm (UDOA):
(stated here in special case of zero transport layer storage)
(1) Flow Control (b): At node n, observe virtual queue Zn(c)(t).
ln(c)
Rn(c)(t)
Un(c)(t)
Rest of Network
Then Update the Virtual Queues Zn(c)(t).
(2) Routing: Observe neighbor’s queue length Un(c)(t), compute:
Node n
link (n,b)
cnb*(t) =
Define Wnb*(t) = maxmizing weight over all c
(where (n,c) Lc)
Define cnb*(t) as the arg maximizer.
(This is the best commodity to send over link (n,b)
if Wnb*(t) >0. Else send nothing over link (n,b)).
Note: Routing Algorithm is related to the Tassiulas-Ephremides
Differential backlog policy [1992], but uses weights that switch
Aggressively and discontinuously ON and OFF to yield optimal
delay tradeoffs.
(3) Resource Allocation: Observe Channel State S(t).
Choose mab(c)(t) such that
Theorem (UDOA Performance): If the overloaded
And fully active assumptions are satisfied, then with
Suitable choices of parameters Q, w (as functions of V),
we have for any V>0:
Theorem (Optimality of logarithmic delay): For one-hop
networks with zero transport layer storage space (all
admission/rejection decisions made upon packet arrival),
then any average congestion tradeoff is necessarily
logarithmic in V. (details in paper)
Two Queue Downlink Simulation:
l1
Pr[ON] = p1
l2
Pr[ON] = p2
Observation: The coefficient Q can be reduced by a factor of 30
without Effecting edge probability, leading to further (constant factor)
reductions in average delay with no affect on utility. Shown below is
Reduction by 30 (original Q would have delay multiplied by 30)).
Utility Optimal
Throughput point
V parameter
“Super-Fast” Flow Control.
(Input Traffic exceeds
network capacity).
Thruput 1
Delay (slots)
Thruput 2
input rate
Simulation
Bound
V parameter
V (Log scale x-axis)
Conclusions:
Utility Optimal
Throughput point
V parameter
“Super-Fast” Flow Control.
(Input Traffic exceeds
network capacity).
Thruput 1
Delay (slots)
input rate
Simulation
Bound
V parameter
V (Log scale x-axis)
1) “Super-Fast” Logarithmic Delay Tradeoff Achievable via
Dynamic Scheduling and Flow Control.
2) Logarithmic Delay is Optimal for one-hop Networks.
Fundamental Utility-Delay Tradeoff: [O(1/V), O(log(V))]
3) Novel Lyapunov Optimization Technique for Achieving
Optimal Delay Tradeoffs.