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, *Supported by NWO-VIDI Grant of Erjen Lefeber

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

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

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

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

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

Summary of known BRAVO Results

B alancing R educes A symptotic V ariance of O utputs Theorem (N., Weiss 2008): For the M/M/1/K queue with : 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: Focus of this talk

BRAVO Effect (illustration for M/M/1)

Assume GI/G/1 with and finite second moments The remainder of the talks outlines the proof and conditions 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 3 Steps for

Proof Outline for Theorems 1,2,3

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

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

Theorem 2 Theorem 2: Proof Outline: Brownian Bridge:

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.

Theorem 3: Proving is UI for some cases After some manipulation… So Q’ is UI Assume Now some questions: 1)What is the relation between Q’(t) and Q(t)? 2)When does (*) hold? (*) Some answers: 1)Well known for GI/M/1: Q’(.) and Q(.) have the same distribution 2)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.

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

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

References Yoni Nazarathy and Gideon Weiss, The asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing Systems, 59(2): , Yoni Nazarathy, 2009, The variance of departure processes: Puzzling behavior and open problems. Preprint, EURANDOM Technical Report Series, 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,