1. Positive Harris Recurrence and Diffusion Scale Analysis of a Push-Pull Queueing Network Yoni Nazarathy and Gideon Weiss University of Haifa ValueTools Conference Athens, 21 – 23 October, 2008

2. 2 Full Utilization Without Congestion

3. 3 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

4. 4 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

5. 5 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

6. 6 Similar to KSRS But different

7. 7 KSRS 12 3 4

8. 8 Push pull vs. KSRS Push Pull KSRS with “Good” policy

9. 9 Results

10. 10 Contribution Inherently stable Inherently unstable Pull priority policy Linear threshold policies 12 3 4 Results: Assumptions: (A3) Second moments Thm 1: Fluid limit model stability Thm 2: Positive Harris recurrence Thm 3: Diffusion limit (A1) SLLN (A2) I.I.D. + technical

11. 11 Fluid Stability

12. 12 Stochastic Model and Fluid Limit Model or Assume (A1), SLLN Fluid limits exists and w.p. 1, satisfy the fluid limit model

13. 13 Fluid Stability Thm 1: Under assumption (A1), the fluid limit model is stable. Definition: A fluid limit model is stable if there exists such that for every fluid solution, whenever then for any.

14. 14 Lyapounov Proof Inherently stable Pull priority policy Inherently unstable Linear threshold policies When, it stays at 0. When, at regular points of t,. For every solution of fluid model:

15. 15 Positive Harris Recurrence

16. 16 is strong Markov with state space. A Markov Process 12 3 4 Assume (A2), I.I.D. Queue Residual

17. 17 Positive Harris Recurrence Thm 2: Under assumptions (A1) and (A2), the state process is positive Harris recurrent. Proof follows framework of Jim Dai (1995). 2 Things to Prove: 1.Stability of fluid limit model (Thm 1). 2.Compact sets are petite (minorization).

18. 18 Diffusion Limit

19. 19 Diffusion Scaling

20. 20 Diffusion Limit Thm 3: Under assumptions (A1), (A2), (A3), With. 10 dimensional Brownian motion Expressions of are simple, yield asymptotic variance rate of outputs. Proof Outline: Use positive Harris recurrence to show,, simple calculations along with functional CLT for renewal processes yields the result.

21. 21 Consequences of Diffusion Limit 1) Negative correlation of outputs 2) Diffusion limit does not depend on policy!!!

22. 22 Open Questions Instability when push rate = pull rate State space collapse General MCQNs with infinite inputs

23. 23 THANK YOU

24. 24 Extensions (not in talk)

25. 25 Inherently stable network Inherently unstable network Unbalanced network Completely balanced network Configuration 12 3 4

26. 26 Calculation of Rates 12 3 4 Corollary: Under assumption (A1), w.p. 1, every fluid limit satisfies:. - Time proportion server works on k - Rate of inflow, outflow through k Full utilization: Stability:

27. 27 Memoryless Processing (Kopzon et. al.) Inherently stable Inherently unstable Policy: Pull priority Policy: Generalized thresholds 12 3 4 Alternating M/M/1 Busy Periods Results: Explicit steady state: Stability (Foster – Lyapounov) - Diagonal thresholds - Fixed thresholds

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