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. ShroffOhio Eminent Scholar in Networking and Communications Chaired Professor of ECE and CSE, Ohio State University

Published in IEEE/ACM Transactions on Networking, Feb. 2011

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OutlineIntroductionSystem modelDeriving lower bounds on

average delayDesign of delay-efficient policiesIllustrative examplesConclusion

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IntroductionA 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.

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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

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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

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System modelThe 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, vi

1,…, vi|Pi|) : the path on which flow i is

routed◦ vi

j is a node at a j-hop distance from the source node

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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.

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(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

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Characterizing Bottlenecks in the System

1{iX} : 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 vi

ki and leave it at the node vili .

number of hops in bottleneck

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Deriving lower bounds on average delay

The sum of queues upstream of each link in X at time t is given by SX(t)

bottleneckpacket packet

Si1=1 Si

2=1 Si3=1 Si

4=2 Si5=2 Si

6=2 SX=6Si

4=1 Si5=2 Si

6=2 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:

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Bound on Expected Delay

delay from vili to vi

|Pi|delay from vi

1 to vi

li

where

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Flow PartitionHow to compute the lower bound on the

average delay for a system containing multiple bottlenecks ?

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Flow Partition (cont’d)Assume that we have precomputed a

list of bottlenecks in the systemLet 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.

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Flow Partition (cont’d)Greedily search for a set of flows pP

and the corresponding (Kp,Xp)-bottleneck that yields the maximum lower bound

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Design of delay-efficient policiesSuch a scheduler must satisfy the

following properties:◦Ensure high throughput

◦Allocate resources equitably Starvation leads to an increase in the

average delay in the system.

K

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The clique networkA clique network is one in which

the interference constraints allow only one link to be scheduledat 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.

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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.

3-hop to dest.

Delay time: 1, 3, 6

Delay time: 3, 5, 6

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Back-Pressure PolicyA 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.

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Back-Pressure Policy (cont’d)

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Illustrative examplesTandem Queue

The differential backlog at the last hop becomes comparativelylarge for small values of , thereby increasing the relative priorityof the last link.

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Illustrative examples (cont’d)Simulation results for Tandem

Queue

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Illustrative examples (cont’d)Clique

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Illustrative examples (cont’d)Dumbbell Topology

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Illustrative examples (cont’d)Tree topology

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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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ConclusionThis paper develop a new approach to

reduce the bottlenecks in a multihop wireless to single-queue 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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CommentsHow 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.