On the Convergence to Stationarity of Birth-death Processes

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. BIRTH-DEATH PROCESS, SPEED OF CONVERGENCE AMS 2000 SUBJECT CLASSIFICATION: PRIMARY 60J80