ON THE CONVERGENCE TO STATIONARITY OF BIRTH-DEATH PROCESSES

PAULINE COOLEN-SCHRIJNER^{⋆}, University of Durham

ERIK A. VAN DOORN^{†}, University of Twente

Abstract

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫_{0}^{∞}[lim_{u→∞} 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.

BIRTH-DEATH PROCESS, SPEED OF CONVERGENCE

AMS 2000 SUBJECT CLASSIFICATION: PRIMARY 60J80

⋆ Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK. E-mail address: pauline.schrijner@durham.ac.uk † Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. E-mail address: doorn@math.utwente.nl