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State-Space Collapse via Drift Conditions

State-Space Collapse via Drift Conditions. Atilla Eryilmaz (OSU) and R. Srikant (Illinois). Goal. Motivation. 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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State-Space Collapse via Drift Conditions

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  1. State-Space Collapse via Drift Conditions Atilla Eryilmaz (OSU) and R. Srikant (Illinois)

  2. Goal

  3. Motivation • 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

  4. 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

  5. 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?

  6. Heavy-Traffic Regime • Consider the traffic regime where the arrival rate approaches the system capacity

  7. Foschini and Gans (1978)

  8. Steady-State Result for JSQ

  9. Lower-Bounding Queue

  10. The Lower Bound

  11. State-Space Collapse (1,1) q q

  12. A Useful Property of JSQ

  13. Drift Conditions and Moments

  14. Moments & State-Space Collapse

  15. The Upper Bound

  16. Using State-Space Collapse

  17. Handling Cross Terms

  18. Theorem

  19. Three-Step Procedure

  20. Wireless Networks

  21. Example • Two links, four feasible rates: (0,2), (1,2), (3,1), (3,0) Capacity Region: Set of average service rates (1,2) (0,2) (3,1) (3,0)

  22. MaxWeight (MW) Algorithm Capacity Region: Set of average service rates (1,2) (0,2) (3,1) (3,0) Arrival rates can be anywhere in the capacity region; MW stabilizes queues

  23. Lower Bound Capacity Region: Set of average service rates (1,2) (0,2) (3,1) (3,0) Arrival rates can be anywhere in the capacity region; MW stabilizes queues

  24. Heavy-Traffic Regime Capacity Region: Set of average service rates (1,2) (0,2) . (3,1) (3,0) Arrival rates can be anywhere in the capacity region; MW stabilizes queues

  25. State-Space Collapse c q q

  26. Upper Bound

  27. Theorem

  28. Implications c q q

  29. 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 • Similar to Bertsimas, Paschalidis and Tsitsiklis (1995), Kumar and Kumar (1995), Shah and Wischik (2008)

  30. 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

  31. 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 • JSQ • MaxWeight • MaxWeight with fading is an easy extension

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