B alancing r educes a symptotic v ariance of o utputs
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B alancing R educes A symptotic V ariance of O utputs. Yoni Nazarathy * EURANDOM, Eindhoven University of Technology, The Netherlands. Based on some joint works with Ahmad Al Hanbali , Michel Mandjes , Gideon Weiss and Ward Whitt. QTNA 2010, Beijing, July 26, 2010.

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B alancing R educes A symptotic V ariance of O utputs

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Balancing Reduces Asymptotic Variance of Outputs

Yoni Nazarathy*

EURANDOM, Eindhoven University of Technology,The Netherlands.

Based on some joint works withAhmad Al Hanbali, Michel Mandjes,

Gideon Weiss and Ward Whitt

QTNA 2010, Beijing,

July 26, 2010.

*Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber


Overview

  • GI/G/1/K Queue (with or )

  • number of customers served during

  • Asymptotic variance

  • Surprising results when

Balancing Reduces Asymptotic Variance of Outputs


The GI/G/1/K Queue

overflows

* Assume

* Load:

* Squared coefficient of variation:


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


Asymptotic Variance for (simple)

After finite time, server busy forever…

is approximately the same as when or


Intermediate Summary

GI/G/1

GI/G/1/K

?

?

M/M/1/K

M/M/1

?

?


Balancing Reduces Asymptotic Varianceof Outputs

Theorem (Al Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with , under some further technical conditions:

  • Theorem (N. , Weiss 2008): For the M/M/1/K queue with :

  • Conjecture (N. , 2009):For the GI/G/1/K queue with , under furthertechnical conditions :


BRAVO Summary for GI/G/1/K

For GI/G/1/K with :

Proven:

  • : M/M/1/K

  • : * M/M/1 * Assuming finite forth moments: *M/G/1 *GI/NWU/1 (includes GI/M/1) *Any GI/G/1 with

Numerically Conjectured: GI/G/1/K with light tails


Numerical Illustration: M/M/1/K


Numerical Illustration: M/M/1(finite T)


K-1

K

0

1

Some (partial) intuition for M/M/1/K

Easy to see:


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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