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Exponential ergodicity in Markov renewal processes

Published online by Cambridge University Press:  14 July 2016

Jozef L. Teugels
Jozef L. Teugels*
Affiliation:
University of Louvain, Belgium
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Extract

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].

We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.

An application to the M/G/1 queue is included in the last section.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 2 , August 1968 , pp. 387 - 400
Copyright
Copyright © Applied Probability Trust 1968 

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References

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