On the convergence to stationarity of birth-death processes

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.