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State-Space Collapse via Drift Conditions Atilla Eryilmaz (OSU) and R. Srikant (Illinois) 4/10/20151

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Goal 4/10/20152

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Motivation 3 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

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Multiple 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 4/10/20154

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Multi-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? 4/10/20155

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Heavy-Traffic Regime Consider the traffic regime where the arrival rate approaches the system capacity 4/10/20156

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Foschini and Gans (1978) 4/10/20157

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Steady-State Result for JSQ 4/10/20158

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Lower-Bounding Queue 4/10/20159

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The Lower Bound 4/10/201510

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State-Space Collapse 4/10/ (1,1) q qq

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A Useful Property of JSQ 4/10/201512

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Drift Conditions and Moments 4/10/201513

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Moments & State-Space Collapse 4/10/201514

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The Upper Bound 4/10/201515

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Using State-Space Collapse 4/10/201516

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Handling Cross Terms

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A Useful Identity 4/10/201518

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Theorem 4/10/201519

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Three-Step Procedure 4/10/201520

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Wireless Networks 4/10/201521

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Example Two links, four feasible rates: (0,2), (1,2), (3,1), (3,0) 4/10/ (0,2) (1,2) (3,1) (3,0) Capacity Region: Set of average service rates

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MaxWeight (MW) Algorithm 4/10/ (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

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Lower Bound 4/10/ (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

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Heavy-Traffic Regime 4/10/ (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.

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State-Space Collapse 4/10/ c q qq

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Upper Bound 4/10/201527

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Theorem 4/10/201528

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Implications 4/10/ c q qq

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Use Beyond Heavy-Traffic Regime 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 obtained o Similar to Bertsimas, Paschalidis and Tsitsiklis (1995), Kumar and Kumar (1995), Shah and Wischik (2008) 4/10/201530

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Stability 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 4/10/201531

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Conclusions 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 for o JSQ o MaxWeight o MaxWeight with fading is an easy extension 4/10/201532

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