The Asymptotic Variance of Departures in Critically Loaded Queues

Balancing Reduces Asymptotic Variance of Outputs

The Asymptotic Variance of Departures in Critically Loaded QueuesYoni 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 LefeberOverviewGI/G/1 Queue with number of served customers duringAsymptotic variance:Balancing Reduces Asymptotic Variance of OutputsMain 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 ResultsBalancing Reduces Asymptotic Variance of OutputsTheorem (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 outlinesthe proof and conditions for:

Theorem 1: Assume that is UI, then , with

Theorem 2:

Theorem 3: Assume finite 4th 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 Outlinefor Theorems 1,2,3

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

Theorem 1 (cont.)We now show:

is UI since A(.) is renewal

is UI by assumption

Theorem 2Theorem 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 uNow compute the variance.

Theorem 3: Proving is UI for some cases

After some manipulation

So Q is UIAssumeNow some questions:What is the relation between Q(t) and Q(t)?When does (*) hold?

(*)Some answers:Well known for GI/M/1: Q(.) and Q(.) have the same distributionFor M/M/1 use Doobs maximum inequality:

Lemma:For renewal processes with finite fourth moment, (*) holds.

Ideas of proof: Find related martingale, relate it to a stopped martingale, thenUse Walds identity to look at the order of growth of the moments.

Going beyond the GI/M/1 queueProposition: (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 tQuestion: 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

SummaryCritically loaded GI/G/1 Queue:

UI of in critical case is challenging

Many open questions related to BRAVO,both technical and practical

ReferencesYoni 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.