This presentation is the property of its rightful owner.
1 / 12

# B alancing R educes A symptotic V ariance of O utputs PowerPoint PPT Presentation

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.

B alancing R educes A symptotic V ariance of O utputs

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

* 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

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

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.