1. On Control of Queueing Networks and The Asymptotic Variance Rate of Outputs Ph.d Summary Talk Yoni Nazarathy Supervised by Prof. Gideon Weiss Haifa Statistics Seminar, November 19, 2008

2. 2 PLANT OUTPUT The Problem Domain Finite Horizon [0,T] Desired: 1.Low Holding Costs 2.Low Resource Idleness 3.Low Output Variability

3. 3 Queues and Networks A Brief Survey

4. 4 Phenomena of Queues

5. 5 Key Phenomena Stability / instability Congestion increases with utilization Variability of primitives causes larger queues Steady state Little’s law Flashlight principle State space collapse …

6. 6 Queueing Networks

7. 7 Multi-Class = 2

8. 8 Infinite Inputs

9. 9 Miracles

10. 10 PLANT OUTPUT The Problem Domain Finite Horizon [0,T] Desired: 1.Low Holding Costs 2.Low Resource Idleness 3.Low Output Variability

11. 11 Server 1Server 2 1 2 3 Attempt to minimize: Near Optimal Finite Horizon Control

12. 12 s.t. Separated Continuous Linear Program (SCLP) Fluid Relaxation Server 1Server 2 1 2 3

13. 13 SCLP – Bellman, Anderson, Pullan, Weiss Piecewise linear solution Simplex based algorithm, finite time (Weiss) Optimal Solution: Fluid Solution

14. 14 Fluid Tracking

15. 15 seed 1 seed 2 seed 3 seed 4 Asymptotic Optimality

16. 16 PLANT OUTPUT The Problem Domain Finite Horizon [0,T] Desired: 1.Low Holding Costs 2.Low Resource Idleness 3.Low Output Variability

17. 17 2 job streams, 4 steps Queues at 2 and 4 Infinite job supply at 1 and 3 2 servers The Push-Pull Network 12 3 4 Control choice based on No idling, FULL UTILIZATION Preemptive resume Push Pull Push Pull

18. 18 Configurations Inherently stable network Inherently unstable network Assumptions (A1) SLLN (A2) I.I.D. + Technical assumptions (A3) Second moment Processing Times Previous Work (Kopzon et. al.): 12 3 4

19. 19 Policies Inherently stable Inherently unstable Policy: Pull priority (LBFS) Policy: Linear thresholds 12 3 4 Typical Behavior: 2,4 3 4 2 1 1,3 Typical Behavior: Server: “don’t let opposite queue go below threshold” Push Pull Push 1,3

20. 20 KSRS 12 3 4

21. 21 Push pull vs. KSRS Push Pull KSRS with “Good” policy

22. 22 Stability Result 12 3 4 QueueResidual is strong Markov with state space Theorem: Under assumptions (A1) and (A2), X(t) is positive Harris recurrent. Proof follows framework of Jim Dai (1995) 2 Things to Prove: 1.Stability of fluid limit model 2.Compact sets are petite Positive Harris Recurrence:

23. 23 PLANT OUTPUT The Problem Domain Finite Horizon [0,T] Desired: 1.Low Holding Costs 2.Low Resource Idleness 3.Low Output Variability

24. 24 Example 1: Stationary stable M/M/1, D(t) is PoissonProcess( ): Example 2: Stationary M/M/1/1 with. D(t) is RenewalProcess(Erlang(2, )): Variability of Outputs Asymptotic Variance Rate of Outputs For Renewal Processes:

25. 25 Taken from Baris Tan, ANOR, 2000. Previous Work: Numerical

26. 26 BRAVO Effect

27. 27 BRAVO Effect: A Phenomena Using a “renewal-reward” method for regenerative simulation for. Queues with Restricted Accessibility (Perry et. al.)

28. 28 Summary of Results Queueing System Without LossesFinite Capacity Birth Death Queue Push Pull Queueing NetworkInfinite Supply Re-Entrant Line

29. 29 Infinite Supply Re-entrant Line 4 2 1 3 5 6 7 8 10 9

30. 30 “Renewal Like” 4 2 1 3 5 6 7 8 10 9 1 6 8

31. 31 A Future Direction

32. 32 Finite Q Rate 1 Infinite Q Rate 2 α α 1 Steady State Total Mean Queue Sizes An Implication of BRAVO? ? IT DOESN’T “WORK”

33. Finite Q Rate 1/4 Rate 1/4 Finite Q Infinite Q Rate 2 Rate 1/2 Infinite Q Poisson(α) Overflow Overflows Priority Infinite Q Rate 1 α Steady State Mean Queue Sizes 11/4 When rate exceeds ¼ overflows of first queue cause the second server to mostly give priority to the fast stream. Non Monotonic Networks ?

34. 34 Now Lets Do לחיים !