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Aggregated semi-Markov processes incorporating time interval omission

Published online by Cambridge University Press:  01 July 2016

Frank Ball ,
Robin K. Milne  and
Geoffrey F. Yeo
Frank Ball*
Affiliation:
University of Nottingham
Robin K. Milne*
Affiliation:
University of Western Australia
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
∗Postal address: Department of Mathematics, University of Nottingham, Nottingham, NG7 2RD, UK.
∗∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
∗∗∗Postal address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150, Australia.
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Abstract

We consider a semi-Markov process with finite state space, partitioned into two classes termed ‘open' and ‘closed'. It is possible to observe only which class the process is in. We show that complete information concerning the aggregated process is contained in an embedded Markov renewal process, whose parameters, moments and equilibrium behaviour are determined. Such processes have found considerable application in stochastic modelling of single ion channels. In that setting there is time interval omission, i.e. brief sojourns in either class failed to be detected. Complete information on the aggregated process incorporating time interval omission is contained in a Markov renewal process, whose properties are derived, obtained from the above Markov renewal process by a further embedding. The embedded Markov renewal framework is natural, and its invariance to time interval omission leads to considerable economy in the derivation of properties of the observed process. The results are specialised to the case when the underlying process is a continuous-time Markov chain.

Keywords

MARKOV RENEWAL PROCESSCONTINUOUS-TIME MARKOV CHAINREVERSED PROCESSEQUILIBRIUM BEHAVIOURSINGLE ION CHANNEL MODELLING
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 772 - 797
Copyright
Copyright © Applied Probability Trust 1991 

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