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On the convergence to stationarity of birth-death processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Pauline Coolen-Schrijner  and
Erik A. Van Doorn
Pauline Coolen-Schrijner*
Affiliation:
University of Durham
Erik A. Van Doorn*
Affiliation:
University of Twente
*
∗ Postal address: Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, UK. Email address: pauline.schrijner@durham.ac.uk
∗∗ Postal address: Faculty of Mathematical Sciences, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dt as a measure of the speed of convergence towards stationarity of the process {X(t), t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t), t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

Keywords

Birth-death processspeed of convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 696 - 706
Copyright
Copyright © by the Applied Probability Trust 2001 

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