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On the existence of finite-dimensional filters for Markov-modulated traffic

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

C. Olivier  and
J. Walrand
C. Olivier*
Affiliation:
University of California at Berkeley
J. Walrand*
Affiliation:
University of California at Berkeley
*
∗ Present address: Laboratoire Système de Perception, ETCA/CREA, 16 bis avenue Prieur de la Côte d'Or, 94114 Arcueil Cedex, France.
∗∗ Postal address: Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA 94720, USA.
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Abstract

A Markov-modulated Poisson process (MMPP) is a Poisson process whose rate is a finite Markov chain. The Poisson process is a simple MMPP. An MMPP/M/1 queue is a queue with MMPP arrivals, an infinite capacity, and a single exponential server. We prove that the output of an MMPP/M/1 queue is not an MMPP process unless the input is Poisson. We derive this result by analyzing the structure of the non-linear filter of the state given the departure process of the queue. The practical relevance of the result is that it rules out the existence of simple finite descriptions of queueing networks with MMPP inputs.

Keywords

QUEUESOUTPUT PROCESSESNON-LINEAR FILTERINGMARKOV-MODULATED POISSON PROCESS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 515 - 525
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported by Direction des Recherches, Etudes, et Techniques, France, at the University of California, Berkeley.

References

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