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Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process

Published online by Cambridge University Press:  01 July 2016

Erik A. Van Doorn
Erik A. Van Doorn*
Affiliation:
Centre for Mathematics and Computer Science, Amsterdam
*
∗Present address: Department of Applied Mathematics, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

Keywords

DUAL BIRTH-DEATH PROCESSESORTHOGONAL POLYNOMIALSSPECTRAL REPRESENTATIONTRANSITION PROBABILITIES
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 3 , September 1985 , pp. 514 - 530
Copyright
Copyright © Applied Probability Trust 1985 

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