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Analysis of a counting process associated with a semi-Markov process: number of entries into a subset of state space

Published online by Cambridge University Press:  01 July 2016

Yasushi Masuda  and
Ushio Sumita
Yasushi Masuda
Affiliation:
University of Rochester
Ushio Sumita*
Affiliation:
University of Rochester
*
∗Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.
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Abstract

Let N(t) be a finite semi-Markov process on 𝒩 and let X(t) be the associated age process. Of interest is the counting process M(t) for transitions of the semi-Markov process from a subset G of 𝒩 to another subset B where 𝒩 = B ∪ G and B ∩ G = ∅. By studying the trivariate process Y(t) =[N(t), M(t), X(t)] in its state space, new transform results are derived. By taking M(t) as a marginal process of Y(t), the Laplace transform generating function of M(t) is then obtained. Furthermore, this result is recaptured in the context of first-passage times of the semi-Markov process, providing a simple probabilistic interpretation. The asymptotic behavior of the moments of M(t) as t → ∞ is also discussed. In particular, an asymptotic expansion for E[M(t)] and the limit for Var [M(t)]/t as t → ∞ are given explicitly.

Keywords

LAPLACE TRANSFORM GENERATING FUNCTIONS: ASYMPTOTIC ANALYSIS
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 767 - 783
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

This paper has been partially supported by the IBM Program of Support for Education in the Management of Information Systems.

The second author was partially supported by the National Science Foundation under grant No. ECS-8404071. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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