On the convergence to stationarity of the many-server Poisson queue

Abstract

We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then pn(t) = P(X(t) = n) converges to the stationary probability πn = P(X = n). The integrals ∫0∞[E(X)-E(X(t))]dt and ∫0∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

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