Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T16:43:02.705Z Has data issue: false hasContentIssue false
  • English
  • Français

First passage times and lumpability of semi-Markov processes

Published online by Cambridge University Press:  14 July 2016

Ushio Sumita  and
Maria Rieders
Ushio Sumita
Affiliation:
University of Rochester
Maria Rieders
Affiliation:
University of Rochester
Get access Rights & Permissions [Opens in a new window]

Abstract

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

Keywords

PARTITION OF STATE SPACELUMPED PROCESSESFIRST-EXIT TIMESCHARACTERIZATION OF LUMPABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 4 , December 1988 , pp. 675 - 687
Copyright
Copyright © Applied Probability Trust 1988 

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 paper has been partially supported by the IBM Program of Support for Education in the Management of Information Systems and by NSF Grant ECS-8600992.

References

[1] Abdel-Moneim, A. M. and Leysieffer, F. W. (1982) Weak lumpability in finite Markov chains. J. Appl. Prob. 19, 685–691.Google Scholar
[2] Barr, D. R. and Thomas, M. U. (1977) An eigenvector condition for Markov chain lumpability. Operat. Res. 25, 1028–1031.Google Scholar
[3] Burke, C. and Rosenblatt, M. (1958) A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122.Google Scholar
[4] Cinlar., E. (1966) Decomposition of a semi-Markov process under a Markovian rule. Austral. J. Statist. 8, 163–170.Google Scholar
[5] Cinlar, E. (1967) Decomposition of a semi-Markov process under a state dependent rule. SIAM J. Appl. Math. 15, 252–263.Google Scholar
[6] Cinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
[7] Çinlar, E. (1975) Markov renewal theory: A survey. Management Sci. 21, 727–752.Google Scholar
[8] Dynkin, E. (1965) Markov Processes I. Springer-Verlag, Berlin.Google Scholar
[9] Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality . Springer-Verlag, New York.CrossRefGoogle Scholar
[10] Kemeny, J. and Snell, L. (1969) Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
[11] Rosenblatt, M. (1959) Functions of a Markov process that are markovian. J. Math. Mech. 8, 585–596.Google Scholar
[12] Rosenblatt, M. (1962) Random Processes. Oxford University Press, New York.Google Scholar
[13] Serfozo, R. F. (1969) Time and Space Transformations of Semi-Markov Processes. Doctoral Thesis in Applied Mathematics, Northwestern University.Google Scholar
[14] Serfozo, R. F. (1971) Functions of semi-Markov processes. SIAM J. Appl. Math. 20, 530–535.CrossRefGoogle Scholar
[15] Sumita, U. and Masuda, Y. (1987) An alternative approach to the analysis of finite semi-Markov and related processes. Stoch. Models 3, 67–87.Google Scholar
[16] Thomas, M. U. and Barr, D. R. (1977) An approximate test of Markov chain lumpability. J. Amer. Statist. Assoc. 72, 175–179.CrossRefGoogle Scholar