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

2. Goal 4/10/20152

3. 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

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 4/10/20154

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? 4/10/20155

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

7. Foschini and Gans (1978) 4/10/20157

8. Steady-State Result for JSQ 4/10/20158

9. Lower-Bounding Queue 4/10/20159

10. The Lower Bound 4/10/201510

11. State-Space Collapse 4/10/201511 (1,1) q qq

12. A Useful Property of JSQ 4/10/201512

13. Drift Conditions and Moments 4/10/201513

14. Moments & State-Space Collapse 4/10/201514

15. The Upper Bound 4/10/201515

16. Using State-Space Collapse 4/10/201516

17. Handling Cross Terms

18. A Useful Identity 4/10/201518

19. Theorem 4/10/201519

20. Three-Step Procedure 4/10/201520

21. Wireless Networks 4/10/201521

22. Example Two links, four feasible rates: (0,2), (1,2), (3,1), (3,0) 4/10/201522 (0,2) (1,2) (3,1) (3,0) Capacity Region: Set of average service rates

23. MaxWeight (MW) Algorithm 4/10/201523 (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

24. Lower Bound 4/10/201524 (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

25. Heavy-Traffic Regime 4/10/201525 (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.

26. State-Space Collapse 4/10/201526 c q qq

27. Upper Bound 4/10/201527

28. Theorem 4/10/201528

29. Implications 4/10/201529 c q qq

30. 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

31. 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

32. 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