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Reward processes for semi-Markov processes: asymptotic behaviour

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

A. Reza Soltani  and
K. Khorshidian
A. Reza Soltani*
Affiliation:
Shiraz University
K. Khorshidian*
Affiliation:
Shiraz University
*
∗Postal address: Department of Statistics, Shiraz University, College of Sciences, Shiraz 71454, Iran.
∗Postal address: Department of Statistics, Shiraz University, College of Sciences, Shiraz 71454, Iran.
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Abstract

The asymptotic behaviour of the cumulative mean of a reward process 𝒵ρ, where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E𝒵ρ(t) = C0 + C1t + o(1), t → ∞, where C0 and C1 are fully specified. A section is devoted to the dual process of a semi-Markov process, and a formula is given for the mean of the first passage time from a state i to a state j of the dual process, in terms of the means of passage times of the original process.

Keywords

Semi-Markov processesreward processesdual processesMarkov renewal theoremfirst passage time

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K05: Renewal theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 833 - 842
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

The research was supported by the Institute for Studies in Theoretical Physics and Mathematics (IPM).

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