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Birth-death processes with disaster and instantaneous resurrection

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Anyue Chen ,
Hanjun Zhang ,
Kai Liu  and
Keith Rennolls
Anyue Chen*
Affiliation:
University of Greenwich
Hanjun Zhang*
Affiliation:
University of Queensland
Kai Liu*
Affiliation:
University of Liverpool
Keith Rennolls*
Affiliation:
University of Greenwich
*
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, 30 Park Row, Greenwich, London SE10 9LS, UK.
∗∗∗ Postal address: Department of Mathematics, School of Physical Sciences, University of Queensland, QLD 4072, Australia.
∗∗∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK.
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, 30 Park Row, Greenwich, London SE10 9LS, UK.
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Abstract

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Keywords

Birth-death processdisasterinstantaneous resurrectionunstable continuous-time Markov chainexistenceuniquenessrecurrenceergodicityequilibrium distributionsymmetryreversibility

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 267 - 292
Copyright
Copyright © Applied Probability Trust 2004 

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