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Birth-death processes with an instantaneous reflection barrier

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Anyue Chen  and
Kai Liu
Anyue Chen*
Affiliation:
University of Greenwich
Kai Liu*
Affiliation:
University of Liverpool
*
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, Maritime Greenwich Campus, Old Royal Naval College, Park Row, Greenwich, London SE10 9LS, UK. Email address: [email protected]
∗∗ Postal address: Division of Statistics and OR, Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK.
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Abstract

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth—death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Keywords

Birth—death processunstable continuous-time Markov chainexistenceuniquenessergodicityequilibrium distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 163 - 179
Copyright
Copyright © Applied Probability Trust 2003 

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