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Stability using fluid limits: Illustration through an example "Push-Pull" queuing network

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. 1. 2. 3. 4.

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Stability using fluid limits: Illustration through an example "Push-Pull" queuing network

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  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. 1 2 3 4 KumarSeidmanRybkoStoylar

  3. Purpose of the talk Part 1: Outline research on Multi-ClassQueueing 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… 1990-2000Recommended Book:- Bramson, Stability of Queueing Networks, 2009

  4. Part 1: Multi-class queueing networks (with infinite supplies)

  5. 1 2 3 4 The Push-Pull Network • Continuous Time, Discrete Jobs • 2 job streams, 4 steps • Queues at pull operations • 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

  6. 1 2 3 4 Processing Times “interesting” Configurations:

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

  8. 1 2 3 4 A Markov Process Queue Residual is strong Markov with state space .

  9. Stability Results Theorem (N., Weiss): Pull-priority, , is PHR Theorem (N., Weiss): Linear thresholds, , is PHR Current work: Generalizing to servers Theorem (Lefeber, N.): , pull-priority, if , 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:

  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 impliesthat is “stable” Nice, since stability of is sometimes easier to establish than directly working

  14. 1 2 3 4 Stochastic Model and Fluid Model

  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 Technical Conditions on Markov Process (Pettiness) Positive Harris Recurrence Associationof Fluid Model To Stochastic System Weakly Stable Fluid Model Rate Stability:

  17. Association of Fluid Model and Stochastic System

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

  19. Questions?

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