State-Space Collapse via State-Space Collapse via Drift ConditionsDrift Conditions

Atilla Eryilmaz (OSU) and R. Srikant (Illinois)

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GoalGoal

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MotivationMotivation

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• Parallel servers• Jobs are buffered at

a single queue• When a server

becomes idle, it grabs the first job from the queue to serve

• All servers are fully utilized whenever possible

Multiple queuesMultiple queues• Jobs arrive and

choose to join the shortest queue upon arrival

• Total queue length is the same as in the case of a single queue if jobs “defect” to a different queue whenever one becomes empty

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Multi-Path RoutingMulti-Path Routing• Choice of paths from source to destination: route

each packet on currently least-congested path• JSQ is an abstraction of such routing scheme. It is

not possible for packets to defect from one path to another.

• Is JSQ still optimal in the sense of minimizing queue lengths?

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Heavy-Traffic RegimeHeavy-Traffic Regime• Consider the traffic regime where the arrival rate

approaches the system capacity

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Foschini and Gans (1978)Foschini and Gans (1978)

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Steady-State Result for JSQSteady-State Result for JSQ

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Lower-Bounding QueueLower-Bounding Queue

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The Lower BoundThe Lower Bound

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State-Space CollapseState-Space Collapse

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(1,1)

q

q

A Useful Property of JSQA Useful Property of JSQ

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Drift Conditions and MomentsDrift Conditions and Moments

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Moments & State-Space Moments & State-Space

CollapseCollapse

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The Upper BoundThe Upper Bound

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Using State-Space CollapseUsing State-Space Collapse

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Handling Cross TermsHandling Cross Terms

TheoremTheorem

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Three-Step ProcedureThree-Step Procedure

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Wireless NetworksWireless Networks

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ExampleExample• Two links, four feasible rates: (0,2), (1,2), (3,1),

(3,0)

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(0,2)(1,2)

(3,1)

(3,0)

Capacity Region:Set of average service rates

MaxWeight (MW) AlgorithmMaxWeight (MW) Algorithm

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(0,2)(1,2)

(3,1)

(3,0)

Capacity Region:Set of average service rates

Arrival rates can be anywhere in the capacity region; MW stabilizes queues

Lower BoundLower Bound

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(0,2)(1,2)

(3,1)

(3,0)

Capacity Region:Set of average service rates

Arrival rates can be anywhere in the capacity region; MW stabilizes queues

Heavy-Traffic RegimeHeavy-Traffic Regime

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(0,2)(1,2)

(3,1)

(3,0)

Capacity Region:Set of average service rates

Arrival rates can be anywhere in the capacity region; MW stabilizes queues

.

State-Space CollapseState-Space Collapse

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c

q

q

Upper BoundUpper Bound

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TheoremTheorem

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ImplicationsImplications

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c

q

q

Use Beyond Heavy-Traffic Use Beyond Heavy-Traffic

RegimeRegime

• Each face of the capacity region provides an upper and lower bound

• Treat these as constraints

• From this the best upper and lower bounds can be obtainedo Similar to Bertsimas, Paschalidis and Tsitsiklis (1995),

Kumar and Kumar (1995), Shah and Wischik (2008)

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Stability and PerformanceStability and Performance• Stability of control policies can be shown by

considering the drift of a Lyapunov function

• Setting this drift equal to zero gives bounds on queue lengths in steady-state

• But these are not tight in heavy-traffic

• The moment-based interpretation of state-space collapse and the upper bounding ideas to use this information provide tight upper bounds

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ConclusionsConclusions• An approach to state-space collapse using

exponential bounds based on drift conditions

• A technique to use to these bounds in obtaining tight upper bounds

• Demonstrated foro JSQo MaxWeighto MaxWeight with fading is an easy extension

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