Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-04T12:21:34.997Z Has data issue: false hasContentIssue false
  • English
  • Français

A closure characterisation of phase-type distributions

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Robert S. Maier  and
Colm Art O'Cinneide
Robert S. Maier*
Affiliation:
University of Arizona
Colm Art O'Cinneide*
Affiliation:
University of Arkansas
*
∗ Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. E-mail address: rsm@math.arizona.edu
∗∗ Postal address: Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, USA. E-mail address: co24225@uafsysb.uark.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.

We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

Keywords

LAPLACE TRANSFORMSGENERATING FUNCTIONSAUTOMATA THEORY

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 92 - 103
Copyright
Copyright © Applied Probability Trust 1992 

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.)

Footnotes

Partially supported by the National Science Foundation under grant NCR-9016211, and by the Air Force Office of Scientific Research under contract 88–0189.

Partially supported by the National Science Foundation under grant DMS-8907630.

References

[1]Assaf, D. and Levikson, B. (1982) Closure of phase-type distributions under operations arising in reliability theory. Ann. Prob. 10, 265–269.Google Scholar
[2]Eilenberg, S. (1974) Automata, Languages, and Machines, Vol. A. Academic Press, New York.Google Scholar
[3]Katayama, T., Okamoto, M. and Enomoto, H. (1978) Generating functions of regular sets and the D0L growth functions. Inf. Control 36, 85–101.Google Scholar
[4]Kuich, W. and Salomaa, A. (1986) Semirings, Automata, Languages. Springer-Verlag, Berlin.Google Scholar
[5]Maier, R. S. (1991) The algebraic construction of phase-type distributions. Comm. Statist.-Stoch. Models. To appear.Google Scholar
[6]Neuts, M. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
[7]O'Cinneide, C. A. (1990) Characterization of phase-type distributions. Comm. Statist.-Stoch. Models 6, 1–57.Google Scholar
[8]Seneta, E. (1973) Non-Negative Matrices. Allen and Unwin, London.Google Scholar
[9]Sengupta, B. (1990) Phase-type representations for matrix-geometric solutions. Comm. Statist.-Stoch. Models 6, 163–167.Google Scholar
[10]Soittola, M. (1976) Positive rational sequences. Theoret. Comput. Sci. 2, 317–322.Google Scholar