Delay Analysis and
Optimality of Scheduling Policies
for Multihop Wireless Networks

Gagan Raj Gupta
Post-Doctoral Research Associate with the Parallel Programming
Laboratory, University of Illinois at Urbana–Champaign.

Ness B. Shroff
Ohio Eminent Scholar in Networking and Communications
Chaired Professor of ECE and CSE, Ohio State University
Published in IEEE/ACM Transactions on Networking, Feb. 2011
Outline
Introduction
 System model
 Deriving lower bounds on average delay
 Design of delay-efficient policies
 Illustrative examples
 Conclusion

2
Introduction

A large number of studies on multihop wireless
networks have been devoted to system stability
while maximizing metrics like throughput or utility.

The delay performance of wireless networks,
however, has largely been an open problem.
◦ the mutual interference inherent in wireless networks.

This paper presented a new, systematic
methodology to obtain a fundamental lower
bound on the average packet delay under any
scheduling policy.
3
Introduction (cont’d)
The delay performance of any scheduling policy is
primarily limited by the interference.
 Many bottlenecks to be formed in the network

◦ The transmission medium is shared
◦ A bottleneck contains multiple links
4
Introduction (cont’d)

In this paper, the authors development of a new queue grouping
technique to handle the complex correlations of the service
process resulting from the multihop nature of the flows
◦ (K,X)-bottlenecks
 Queueing model
5
System model

The service structure is slotted. Each packet has a
deterministic service time equal to one unit.

A(t)=(A1(t),…,AN(t)) : the vector of exogenous arrivals
◦ Ai(t) : the number of packets injected into the system by the
source si during time slot t.

=(1,…, N) : the corresponding arrival rate vector.

Pi=(vi0, vi1,…, vi|Pi|) : the path on which flow i is routed
◦ vij is a node at a j-hop distance from the source node
6
System model (cont’d)


The queue length vector is denoted by
Q(t) = (Qij(t): i=1,2,…N)
At each time slot, an activation vector I(t) is scheduled
depending on the scheduling policy and the underlying
interference model.
◦ Iij(t) indicates whether or not flow i received service at the j-th
hop from source si at time slot t.
7
(K,X)-bottleneck

We partition the flows into several groups.
◦ Each group passes through a (K,X)-bottleneck, and
the queueing for each group is analyzed individually.

(K,X)-bottleneck : a set of links X such that no more
than K of its links can be scheduled simultaneously

(K,X)-bottleneck  G/D/K queue
8
Characterizing Bottlenecks in the System



1{iX} : indicate whether the flow passes through the
(K,X)-bottleneck.
The total flow rate X crossing the bottleneck X is
given by
Let the flow I enter the (K,X)-bottleneck at the node
viki and leave it at the node vili .
number of hops in bottleneck
9
Deriving lower bounds on average delay


The sum of queues upstream of each link in X at time t
is given by SX(t)
packet
Si1=1
packet
Si2=1
Si3=1
Si4=2
Si4=1
bottleneck
Si5=2
Si5=2
Si6=2
Si6=2
SX=6
SX=5
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Reduced System


Let
be the queue length of this system at time t.
The queue evolution of the reduced system is given by
the following equation:
11
Bound on Expected Delay

delay from
vi1 to vili
delay from vili to vi|Pi|
where
12
Flow Partition

How to compute the lower bound on the average
delay for a system containing multiple bottlenecks ?
13
Flow Partition (cont’d)
Assume that we have precomputed a list of
bottlenecks in the system
 Let Z be the set of flows in the system.
 Let π be a partition on Z such that each
element p π is a set of flows passing through
a common (Kp, Xp)-bottleneck.


Our objective is to compute a partition π such
that the lower bound on can be maximized.
14
Flow Partition (cont’d)

Greedily search for a set of flows pP and the
corresponding (Kp,Xp)-bottleneck that yields
the maximum lower bound
15
Design of delay-efficient policies

Such a scheduler must satisfy the following
properties:
◦ Ensure high throughput
K
◦ Allocate resources equitably
 Starvation leads to an increase in the average delay
in the system.
16
The clique network

A clique network is one in which
the interference constraints
allow only one link to be scheduled
at any given time.
◦ (1,X)-bottleneck
◦ Any work-conserving policy will achieve the lower
bound on SX.
◦ Note that a policy that minimizes SX may not
minimize the sum of queue lengths in the system at all
times, nor is it guaranteed to be delay-optimal.
17
The clique network (cont’d)

The optimal policy
◦ Last Buffer First Serve (LBFS)
 Scheduling the packet that is closest to its
destination is optimal.
1-hop to dest.
2-hop to dest.
Delay time: 1, 3, 6
3-hop to dest.
Delay time: 3, 5, 6
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Back-Pressure Policy
A throughput-optimal scheduling policy.
 Define the differential backlog
of flow i passing
through a link
as



For each link, the flow with the maximum differential
backlog is chosen.
The link-scheduling component schedules the
activation vector with the maximum weight at every
time slot.
19
Back-Pressure Policy (cont’d)
20
Illustrative examples

Tandem Queue
The differential backlog at the last hop becomes comparatively
large for small values of , thereby increasing the relative priority
of the last link.
21
Illustrative examples (cont’d)

Simulation results for Tandem Queue
22
Illustrative examples (cont’d)

Clique
23
Illustrative examples (cont’d)

Dumbbell Topology
24
Illustrative examples (cont’d)

Tree topology
25
Illustrative examples (cont’d)

Cycle topology
K=2: X={1,2,3,4,5,6,7,8}
K=1: X1={1,2,3}
X2={6,7,8}
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Conclusion
This paper develop a new approach to reduce the
bottlenecks in a multihop wireless to singlequeue systems to carry out lower bound analysis.
 The analysis is very general and admits a large
class of arrival processes.
 The analysis can be readily extended to handle
channel variations.

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Comments

How to identify the bottlenecks in a wireless
mesh network ?

The analysis model can only obtain the lower
bound of “expected delay time”

How good is the lower bound ?
◦ especially when K is large.
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