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Simple derivations of properties of counting processes associated with Markov renewal processes

Part of: Stochastic processes Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Robin K. Milne
Frank Ball*
Affiliation:
The University of Nottingham
Robin K. Milne*
Affiliation:
The University of Western Australia
*
∗Postal address: School of Mathematical Sciences, The University of Nottingham, Nottingham NG7 2RD, UK. Email address: frank.ball@nottingham.ac.uk
∗∗Postal address: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia. Email address: milne@maths.uwa.edu.au
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Abstract

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A simple, widely applicable method is described for determining factorial moments of N̂t, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂t, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.

Keywords

Continuous-time Markov chaincounting processfactorial momentfactorial moment measureion channel modellingMarkov renewal processpoint processsemi-Markov processtime interval omission

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60G55: Point processes 60J27: Continuous-time Markov processes on discrete state spaces 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1031 - 1043
Copyright
© Applied Probability Trust 2005 

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