Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T14:48:26.450Z Has data issue: false hasContentIssue false
  • English
  • Français

The output process of an MMPP/M/1 queue

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Nigel Bean ,
David Green  and
Peter Taylor
Nigel Bean*
Affiliation:
University of Adelaide
David Green*
Affiliation:
University of Adelaide
Peter Taylor*
Affiliation:
University of Adelaide
*
∗Postal address: Department of Applied Mathematics, The University of Adelaide, Adelaide 5005, Australia. Email address: {nbean dgreen ptaylor}@maths.adelaide.edu.au.
∗Postal address: Department of Applied Mathematics, The University of Adelaide, Adelaide 5005, Australia. Email address: {nbean dgreen ptaylor}@maths.adelaide.edu.au.
∗Postal address: Department of Applied Mathematics, The University of Adelaide, Adelaide 5005, Australia. Email address: {nbean dgreen ptaylor}@maths.adelaide.edu.au.
Get access Rights & Permissions [Opens in a new window]

Abstract

Olivier and Walrand (1994) claimed that the departure process of an MMPP/M/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. They also conjectured that the departure process of an MAP/PH/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. We show that their proof of the first result has an algebraic error, which leaves open the above question of whether the departure process of an MMPP/M/1 can be an MAP.

Keywords

Markov arrival process (MAP)Markov modulated Poisson process (MMPP)(PH) phase-renewal processnonlinear filtering

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 998 - 1002
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bean, N. G., and Green, D. A. (1999). When is an MAP Poisson? To appear in Stochastic Models in Engineering, Technology and Management (a special issue of Mathematical and Computer Modelling).Google Scholar
Burke, P. J. (1956). The output of a queueing system. Operat. Res. 4, 135–165.Google Scholar
Neuts, M. F. (1981). Matrix-geometric Solutions in Stochastic Models. The John Hopkins University Press, Baltimore, MD.Google Scholar
Olivier, C., and Walrand, J. (1994). On the existence of finite-dimensional filters for Markov-modulated traffic. J. Appl. Prob. 31, 515–525.CrossRefGoogle Scholar
Pollett, P. K., and Taylor, P. G. (1993). On the problem of establishing the existence of stationary distributions of continuous-time Markov chains. Prob. Eng. Inf. Sci. 7, 529–543.Google Scholar
Walrand, J. (1988). An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar