On Control of Queueing Networks and The Asymptotic Variance Rate of Outputs

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

3. Queues and Networks A Brief Survey

4. Phenomena of Queues

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. Queueing Networks

7. Multi-Class = 2

8. Infinite Inputs

9. Miracles

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

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

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

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

14. Fluid Tracking

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

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

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

18. 1 2 3 4 Processing Times Assumptions (A1) SLLN (A2) I.I.D. + Technical assumptions (A3) Second moment • Configurations • Inherently stable network • Inherently unstable network Previous Work (Kopzon et. al.):

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

20. 1 2 3 4 KSRS

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

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

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

24. Variability of Outputs Asymptotic Variance Rate of Outputs For Renewal Processes: 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, )):

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

26. BRAVO Effect

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

28. Summary of Results Queueing System Without Losses Finite Capacity Birth Death Queue Push Pull Queueing Network Infinite Supply Re-Entrant Line

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

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

31. A Future Direction

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

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

34. NowLets Do לחיים!