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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B alancing r educes a symptotic v ariance of o utputs

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

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

The GI/G/1/K Queue

overflows

* Assume

* Load:

* Squared coefficient of variation:


Variance of outputs

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

Asymptotic Variance for (simple)

After finite time, server busy forever…

is approximately the same as when or


Intermediate summary

Intermediate Summary

GI/G/1

GI/G/1/K

?

?

M/M/1/K

M/M/1

?

?


B alancing r educes a symptotic v ariance of o utputs1

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

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


B alancing r educes a symptotic v ariance of o utputs

Numerical Illustration: M/M/1/K


B alancing r educes a symptotic v ariance of o utputs

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


Some partial intuition for m m 1 k

K-1

K

0

1

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

Easy to see:


References

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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