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Positive Harris Recurrence and Diffusion Scale Analysis of a Push-Pull Queueing Network

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. Full Utilization Without Congestion. 1. 2. 3. 4. The Push-Pull Network.

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Positive Harris Recurrence and Diffusion Scale Analysis of a Push-Pull Queueing Network

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  1. Positive Harris Recurrence and Diffusion Scale Analysis of a Push-Pull QueueingNetwork Yoni NazarathyandGideon WeissUniversity of Haifa ValueTools Conference Athens, 21 – 23 October, 2008

  2. Full Utilization Without Congestion

  3. 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

  4. 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.):

  5. 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:

  6. Similar to KSRS But different

  7. 1 2 3 4 KSRS

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

  9. Results

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

  11. Fluid Stability

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

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

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

  15. Positive Harris Recurrence

  16. 1 2 3 4 A Markov Process Assume (A2), I.I.D. Queue Residual is strong Markov with state space .

  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: • Stability of fluid limit model (Thm 1). • Compact sets are petite (minorization).

  18. DiffusionLimit

  19. Diffusion Scaling

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

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

  23. THANK YOU

  24. Extensions (not in talk)

  25. 1 2 3 4 Configuration • Inherently stable network • Inherently unstable network • Unbalanced network • Completely balanced network

  26. Calculation of Rates 1 2 3 4 - Time proportion server works on k - Rate of inflow, outflow through k Full utilization: Stability: Corollary: Under assumption (A1), w.p. 1, every fluid limit satisfies: .

  27. 1 2 3 4 Memoryless Processing(Kopzon et. al.) Inherently stable Alternating M/M/1 Busy Periods Policy: Pull priority Inherently unstable Policy: Generalized thresholds Results: Stability (Foster – Lyapounov) Explicit steady state: - Fixed thresholds - Diagonal thresholds

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