The Asymptotic Variance of Departures in Critically Loaded Queues - PowerPoint Presentation

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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

OverviewGI/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 Results

Balancing R

educes 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 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 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 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:

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 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.

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 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 challengingMany 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):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.