On the convergence to stationarity of birth-death processes

Taking up a recent proposal by Stadje and Parthasarathy in the \linebreak[4] setting of the many-server Poisson queue, we consider the integral \linebreak[4] $\int_0^{\infty}[\lim_{u\to\infty} E(X(u))-E(X(t))]dt$ as a measure of the speed of convergence towards stationarity of the process $\{X(t), t \geq 0\}$, and evaluate the integral explicitly in terms of the parameters of the process in the case that $\{X(t), t \geq 0\}$ is an ergodic birth-death process on $\{0,1,\ldots\}$ starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.