1. Stability using fluid limits: Illustration through an example "Push-Pull" queuing network Yoni Nazarathy* EURANDOM Contains Joint work with Gideon Weiss and Erjen Lefeber Universiteit Gent October 14, 2010 * Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

2. K umar S eidman R ybko S toylar 12 3 4

3. Purpose of the talk Part 1: Outline research on Multi-Class Queueing Networks (with Infinite Supplies) - N., Weiss, 2009 - Ongoing work with Lefeber Part 2: An overview of “the fluid limit” method for stability of queueing networks Key papers: - Rybko, Stolyar 1992 - Dai 1995 - Bramson/Mandelbaum/Dai/Meyn… 1990-2000 Recommended Book: - Bramson, Stability of Queueing Networks, 2009

4. PART 1: MULTI-CLASS QUEUEING NETWORKS (WITH INFINITE SUPPLIES)

5. Continuous Time, Discrete Jobs 2 job streams, 4 steps Queues at pull operations 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

6. “interesting” Configurations: Processing Times 12 3 4

7. Policies 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

8. 8 is strong Markov with state space. A Markov Process 12 3 4 Queue Residual

9. Stability Results Theorem (N., Weiss): Pull-priority,, is PHR Theorem (N., Weiss): Linear thresholds,, is PHR Theorem (in progress) (Lefeber, N.):, pull-priority, is PHR if More generally, when there is a matrix such that is PHR when e.g: Theorem (Lefeber, N.):, pull-priority, if, is PHR Current work: Generalizing to servers

10. Heuristic Modes Graph for M=3 Pull-Priority

11. Heuristic Stable Fluid Trajectory of M=3 Pull-Priority Case

12. PART 2: THE “FLUID LIMIT METHOD” FOR STABILITY

13. Main Idea Establish that an “associated” deterministic system is “stable” The “framework” then implies that is “stable” Nice, since stability of is sometimes easier to establish than directly working

14. Stochastic Model and Fluid Model 12 3 4

15. Comments on the Fluid Model T is Lipschitz and thus has derivative almost everywhere Any Y=(Q,T) that satisfies the fluid model is called a solution In general (for arbitrary networks) a solution can be non-unique

16. Stability of Fluid Model Definition: A fluid model is stable, if when ever, there exists T, such that for all solutions, Definition: A fluid model is weakly stable, if when ever Main Results of “Fluid Limit Method” Stable Fluid Model Positive Harris Recurrence Weakly Stable Fluid Model Technical Conditions on Markov Process (Pettiness) Rate Stability: Association of Fluid Model To Stochastic System

17. Association of Fluid Model and Stochastic System

18. Lyapounov Proofs for Fluid Stability When, it stays at 0. When, at regular points of t,. Need: for every solution of fluid model:

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21. QUESTIONS?