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. - PowerPoint PPT Presentation
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
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
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 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:
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:
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.