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Superposition of Interacting Aggregated Continuous-Time Markov Chains

Part of: Markov processes Physiological, cellular and medical topics Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Frank Ball ,
Robin K. Milne ,
Ian D. Tame  and
Geoffrey F. Yeo
Frank Ball*
Affiliation:
The University of Nottingham
Robin K. Milne*
Affiliation:
The University of Western Australia
Ian D. Tame*
Affiliation:
The University of Nottingham
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
∗ Postal address: Department of Mathematics, The University of Nottingham, Nottingham, NG7 2RD, UK. E-mail address: fgb@maths.nott.ac.uk
∗∗ Postal address: Department of Mathematics, The University of Western Australia, Nedlands, 6907, Australia. E-mail address: milne@maths.uwa.edu.au
∗∗∗ Postal address: Area of Humanities and Science, Bridgwater College, Bath Road, Bridgwater, Somerset, TA6 4PZ, UK.
∗∗∗∗ Postal address: School of Physical Sciences, Engineering and Technology, Murdoch University, Murdoch, 6150, Australia. E-mail address: yeo@prodigal.murdoch.edu.au
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Abstract

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

Keywords

AGGREGATED MARKOV CHAINCONTINUOUS-TIME MARKOV CHAINEQUILIBRIUM BEHAVIOURINTERACTING SUBSYSTEMSION CHANNEL MODELLINGPHASE-TYPE DISTRIBUTIONSRELIABILITY MODELLINGREVERSIBILITYSOJOURN-TIME DISTRIBUTIONSSUPERPOSITION PROCESS

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 92C05: Biophysics

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 56 - 91
Copyright
Copyright © Applied Probability Trust 1997 

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