Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T21:41:11.370Z Has data issue: false hasContentIssue false
  • English
  • Français

Birth-and-death processes on the integers with phases and general boundaries

Published online by Cambridge University Press:  14 July 2016

Bruce Hajek
Bruce Hajek*
Affiliation:
University of Illinois at Urbana-Champaign
*
∗Postal address: Coordinated Science Laboratory, 1101 W. Springfield, Urbana, IL 61801, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.

Keywords

MARKOV PROCESS WITH PHASESPROBABILITY DISTRIBUTIONS OF PHASE TYPECOMPUTATIONAL PROBABILITYMARKOV PROCESSES WATCHED IN A SETQUASI-BIRTH-AND-DEATH PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 488 - 499
Copyright
Copyright © Applied Probability Trust 1982 

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

This research was supported by the Naval Research Laboratory under Contract U.S. NAVY N00014-80-C-0802.

References

Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York.Google Scholar
Dynkin, E. B. (1965) Markov Processes, Vol. I. Springer-Verlag, New York.Google Scholar
Freedman, D. (1971) Approximating Countable State Markov Chains. Holden-Day, San Francisco.Google Scholar
Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976) Denumerable Markov Chains. Springer-Verlag, New York.CrossRefGoogle Scholar
Neuts, M. F. (1978a) The M/M/l queue with randomly varying arrival and service rates. Opsearch 15, 139–157.Google Scholar
Neuts, M. F. (1978b) Further results on the M/M/1 queue with randomly varying rates. Opsearch 15, 158–168.Google Scholar
Neuts, M. F. (1978C) Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 185–212.Google Scholar
Neuts, M. F. (1980) The probabilistic significance of the rate matrix in matrix-geometric invariant vectors. J. Appl. Prob. 17, 291–296.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models — An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
Nummelin, E. and Tweedie, R. L. (1978) Geometric ergodicity and R -positivity for general Markov chains. Ann. Prob. 6, 404–420.Google Scholar
Tweedie, R. L. (1982) Operator-geometric stationary distributions for Markov chains with application to queueing models. Adv. Appl. Prob. 14, 368–391.Google Scholar