21 Mar 2002 First Israelli-Swedish Workshop on Next Generation Networking1 / 28 Stochastic Analysis...
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Transcript of 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
1 / 28
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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Analytical ModelDefinition of Scheduler
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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Analytical ModelDefinition of Scheduler
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
Random Error Model2SMC Error Model
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
Random Error Model2SMC Error Model
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
Random Error Model2SMC Error Model
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