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The Asymptotic Variance of Departures in Critically Loaded Queues

The Asymptotic Variance of Departures in Critically Loaded Queues. Yoni Nazarathy * EURANDOM, Eindhoven University of Technology, The Netherlands. (As of Dec 1: Swinburne University of Technology, Melbourne) Joint work with Ahmad Al- Hanbali , Michel Mandjes and Ward Whitt.

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The Asymptotic Variance of Departures in Critically Loaded Queues

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  1. The Asymptotic Variance of Departures in Critically Loaded Queues Yoni Nazarathy* EURANDOM, Eindhoven University of Technology,The Netherlands.(As of Dec 1: Swinburne University of Technology, Melbourne) Joint work with Ahmad Al-Hanbali, Michel Mandjes and Ward Whitt. MASCOS Seminar, Melbourne, July 30, 2010. *Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

  2. Overview • GI/G/1 Queue with • number of served customers during • Asymptotic variance: • Balancing Reduces Asymptotic Variance of Outputs • Main Result:

  3. The GI/G/1/K Queue overflows Assume: Load: Squared coefficients of variation:

  4. Variance of Outputs Asymptotic Variance Simple Examples: * Stationary stable M/M/1, D(t) is PoissonProcess( ): * Stationary M/M/1/1 with , D(t) is RenewalProcess(Erlang(2, )): Notes: * In general, for renewal process with : * The output process of most queueing systems is NOT renewal

  5. Asymptotic Variance for (simple) After finite time, server busy forever… is approximately the same as when or

  6. M/M/1/K: Reduction of Variance when

  7. Summary of known BRAVO Results

  8. Balancing Reduces Asymptotic Varianceof Outputs • Theorem (N. , Weiss 2008): For the M/M/1/K queue with : Focus of this talk • Conjecture (N. 2009):For the GI/G/1/K queue with : Theorem (Al Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with , under some further technical conditions:

  9. BRAVO Effect (illustration for M/M/1)

  10. The remainder of the talks outlinesthe proof and conditions for: Assume GI/G/1 with and finite second moments

  11. 3 Steps for Theorem 1: Assume that is UI, then , with Theorem 2: Theorem 3: Assume finite 4’th moments, then, Q is UI under the following cases: (i) Whenever and L(.) bounded (ii) M/G/1 (iii) GI/NWU/1 (includes GI/M/1) (iv) D/G/1 with services bounded away from 0

  12. Proof Outlinefor Theorems 1,2,3

  13. D.L. Iglehart and W. Whitt. Multiple Channel Queues in Heavy Traffic. I. Advances in Applied Probability, 2(1):150-177, 1970. Proof: so also, If, then, Theorem 1: Assume that is UI, then , with

  14. Theorem 1 (cont.) We now show: is UI since A(.) is renewal is UI by assumption

  15. Theorem 2 Brownian Bridge: Theorem 2: Proof Outline:

  16. Theorem 2 (cont.) Now use (e.g. Mandjes 2007), Manipulate + use symmetry of Brownian bridge and uncondition…. Quadratic expression in u Linear expression in u Now compute the variance.

  17. Theorem 3: Proving is UI for some cases Assume (*) After some manipulation… Now some questions: What is the relation between Q’(t) and Q(t)? When does (*) hold? So Q’ is UI Some answers: Well known for GI/M/1: Q’(.) and Q(.) have the same distribution For M/M/1 use Doob’s maximum inequality: Lemma: For renewal processes with finite fourth moment, (*) holds. Ideas of proof: Find related martingale, relate it to a stopped martingale, then Use Wald’s identity to look at the order of growth of the moments.

  18. Going beyond the GI/M/1 queue Proposition: (i) For the GI/NWU/1 case: (ii) For the general GI/G/1 case: C(t) counts the number of busy cycles up to time t Question: How fast does grow? Lemma (Due to Andreas Lopker): For renewal process with Zwart 2001: For M/G/1: So, Q is UI under the following cases: (i) Whenever and L(.) bounded (ii) M/G/1 (iii) GI/NWU/1 (includes GI/M/1) (iv) D/G/1 with services bounded away from 0

  19. Summary • Critically loaded GI/G/1 Queue: • UI of in critical case is challenging • Many open questions related to BRAVO,both technical and practical

  20. References • Yoni Nazarathy and Gideon Weiss, The asymptotic variance rate of the output process of finite capacity birth-death queues.Queueing Systems, 59(2):135-156, 2008. • Yoni Nazarathy, 2009, The variance of departure processes: Puzzling behavior and open problems. Preprint, EURANDOM Technical Report Series, 2009-045. • Ahmad Al-Hanbali, Michel Mandjes, Yoni Nazarathy and Ward Whitt. Preprint. The asymptotic variance of departures in critically loaded queues. Preprint, EURANDOM Technical Report Series, 2010-001.

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