21 mar 2002 first israelli-swedish workshop on next generation networking1 / 28 stochastic analysis...

21 Mar 2002 First Israelli-Swedish Workshop on Next Generation Networking

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Stochastic Analysis of Wireless-fair Scheduling

Hwee Pink Tan and Raphael Rom

Dept of Elect. Eng, Technion

Israel Institute of Technology


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Outline

Introduction

Wireless-Fair Scheduling

Analytical Model

Performance Evaluation

Summary and Future Directions


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Introduction

Several Wireless Scheduling algorithms proposed recently based on fair queuing paradigm

Analytical bounds inadequate to characterize scheduling performance Error-free guarantees Channel-conditioned deterministic bound

Stochastic nature of wireless channel enables stochastic analysis of wireless scheduling algorithms


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Wireless-fair Scheduling

Wireless Scheduling Scenario

Need for Wireless Adaptation

Definition of Wireless-Fair Scheduling

Lead / Lag Accounting Service

Compensation Service


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Scheduling Scenario

Single Channel shared amongst N users rj = demand of user (flow) j

Fixed-size time-slotted transmission

Perfect Knowledge at Scheduler Queue status of user j at slot i, Qi

j {backlogged, idle}

Channel state of user j at slot i, CSij {Good, Bad}

r1

r2

rN

T T

CentralizedWireless Scheduler

Wireless Link


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Wireless-fair SchedulingNeed for Wireless Adaptation

Fair Scheduling in wired link (WiredFS) guarantees throughput, delay and fairness

Direct application of WiredFS in wireless link results in degradation of

Throughput and Delay guarantees‘wasted’ slots due to channel errors

Fairness guaranteetime and spatial dependence of channel errors


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Wireless-fair SchedulingDefinition

Wireless-Fair Scheduling WiredFSWireless-Adaptation Service

Wireless-Adaptation ServiceReassigns transmission slots based on channel stateCompensates for reassignment

Lead / Lag Accounting ServiceCompensation Service


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Wireless-fair Scheduling Lead / Lag Accounting Service

Notion of Lag (Lead)

A flow’s lag (lead) = amount of service it is entitled to (needs to relinquish) in the future to compensate for service lost (gained) in the past

Lagj = Lag of flow j = - Lead of flow j


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Wireless-fair Scheduling Lead / Lag Accounting Service

Update of Lagj

Lagj = Lagj + 1 whenFlow j gives up an allocated slot (due to channel error) to another flow

Lagj = Lagj -1 whenFlow j transmits in a slot given up by another flow

Effects of Bounded Lagj

Tradeoff between fairness and channel efficiency


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Wireless-fair Scheduling Compensation Service

Allows lagging flows to reclaim ‘lost’ service from leading flows

Defines the ‘how, when and which’ for which a flow transmits in a slot given up by another flow

a flow gives up its allocated slot to allow another flow to catch up


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Analytical Model

Assumptions

Definition of Analysis Intervals

Definition of Symmetric Two-Flow Wireless-Fair Scheduler

Scheduling Mechanism

Characterization of Scheduler


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Flow characteristicsr1=r2=0.5 Independent ArrivalsEqual-sized packets (tx time = T)Infinite Buffer Length

Channel characteristicsSimilar channel conditions for f1 and f2

Analytical ModelAssumptions

r1

r2

T T

Rate- and Channel-Symmetric(Symmetric) Two-Flow Scheduling


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Analytical ModelDefinition of Analysis Interval

Regenerative Interval

Performance Interval

Legend:

t 1 t 2 t 3 t 4

Duration where flow 1is backlogged

Duration where flow 2is backlogged

time


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Analytical ModelDefinition of Scheduler

Notationsj = flow index; i = slot index

xij = lead of flow j at the end of slot i

Ai = allocation in slot i {S1,S2}Flow status

Flow j is


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Wired-Fair Service

Determines primary allocation policy, Ai Within any performance interval, alternate slot allocation suffices

Lead/Lag Accounting Servicexi

1 + xi2 = 0

Hence, sufficient to define xi = xi1

Unbounded xi


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Compensation ServiceDetermines secondary slot allocation policy, based on input from Lead/Lag Accounting Service

Absolute transmission priority to lagging flowsAllocated leading flow always relinquish transmission priority

Slot ‘wasted’ only when no error-free flow exists


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Analytical ModelImplementation of Scheduler

Slot allocation policySlots are always allocated to the lagging flow if it exists;

Otherwise, alternate slot allocation is employed

Update of xi

Flow transmits in non-allocated slot

Lagging flow transmits in allocated slot


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Analytical ModelScheduling Mechanism

i Packet i of Flow 1 (F1) becomes HOL i Packet i of Flow 1(F1) departs

1 2 3 4 xi 0 0 -1 0 1 2 1 0 0 Slot Number 1 2 3 4 5 6 7 8 9 Allocation S1 S2 S1 S1 S2 S2 S2 S2 S1 Actual Transmission

f1 f2 f1 f1 f1 f2 f2 f1

Error Status Flow 1 Flow 2 CSj = Bad CSj = Good


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Analytical ModelCharacterization of Scheduler

Within any performance interval, the Wireless Scheduler can be characterized as 2-D Markov Chain, {(xi,Ai), i=1,2,3,…}

State variables: xi, Ai

Markov Points: slot intervals

Simplification to 1-D Markov Chain, {yq, q=1,2,3}State variable : yq=xiq

Markov Points : departure instants of packet q of flow 1, iq, q=1,2,3,…


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Performance Evaluation

Packet delay distribution, G(n)

Fairness distribution, F(y)

Channel Error Model

Results


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Performance EvaluationDelay distribution, G(n)

Allocation S1 S2 Slot Number k k+1 k+n-1 k+n

Packet q-1 departs Packet q departs

yq-1=xk-1=xinit xk+n-1 yq=xk+n=xfin

Markov points

Consider packet q of flow 1 that becomes HOL in slot k and departs at the end of slot n+kMarkov Interval = Delay of packet q = n slotsG(n) = cdf of n


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Performance EvaluationFairness distribution, F(y)

Allocation S1 S2 Slot Number k k+1 k+n-1 k+n

Packet q-1 departs Packet q departs

yq-1=xk-1=xinit xk+n-1 yq=xk+n=xfin

Markov variables

Markov variable, yq ≈ disparity (or ‘unfairness’) in cumulative service received by both flows

F(y) = cdf of y


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G(n) and F(y) can be computed given the channel error model

Performance EvaluationChannel Error Model

Good Bad

pge

peg

Two-state Markov Chain Error Modelpcorr = pge + peg = 0.1

PB = {0.2,0.8}

Random Error ModelUncorrelated average error rate = PE= {0.2,0.8}


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Performance EvaluationDelay Performance

pE=0.2 pE=0.8

mean std mean std

Random Error Model 2.08 0.80 5.50 5.17

2SMC Error Model 1.93 3.80 4.83 19.81

0 2 4 6 8 10 12 14 16 18 2010

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Delay Bound

Loss

Rat

e

Delay Distribution for Symmetric Two-Flow Wireless-Fair Scheduling

Random Error Model2SMC Error Model

0 5 10 15 20 25 3010-3

10-2

10-1

100

Delay Bound

Loss

Rat

e

Delay Distribution for Symmetric Two-Flow Wireless-Fair Scheduling


pE = 0.2 pE = 0.8


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Performance EvaluationFairness Performance

pE=0.2 pE=0.8

mean std mean std

Random Error Model 0.25 0.56 2.71 2.79

2SMC Error Model 7.65 8.22 13.63 15.98

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x=lead of flow 1

F(x

)

Fairness Distribution for Symmetric Two-Flow Wireless-Fair Scheduling


0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x=lead of flow 1F

(x)

Fairness Distribution for Symmetric Two-Flow Wireless-Fair Scheduling


pE = 0.2 pE = 0.8


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Summary

Analytical Performance Model for Symmetric Two-Flow Wireless-Fair SchedulingBy proper choice of analysis interval and time

instants of observation, scheduler can be modeled as a 1-D Markov Chain

Performance Evaluation Packet DelayFairness


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Future Directions

Asymmetric Two-Flow SchedulingMore general scenarioPossible to approximate performance of

symmetric N-flow scheduling

Channel-independent FairnessFlow-dependent channel errors induce

‘unfairness’


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References

Hwee Pink Tan and Raphael Rom. Stochastic Analysis of Wireless-Fair Scheduling. Submitted to Mobicom 2002

Hwee Pink Tan and Raphael Rom. Performance Evaluation of Wireless-Fair Scheduling. Submitted to Globecom 2002

Hwee Pink Tan and Raphael Rom. Stochastic Analysis of Symmetric Two-Flow Wireless-Fair Scheduling . Technical Report, Technion EE publication CITT #371, March 2002

http://www.ee.technion.ac.il/people/hweepink

21 mar 2002 first israelli-swedish workshop on next generation networking1 / 28 stochastic analysis...

Documents

lag of flow j

flow lag j

wirelessfair scheduling

past lag j

lead of flow j slide

israelliswedish workshop

wireless link results

channel state of user