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 lambda, mean service time 1 and lambda/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 p(n)(t) = P(X(t) = n) converges to the stationary probability pi(n) = P(X = n). The integrals integral(0)(infinity)[E(X) - E(X(t))] dt and integral(0)(infinity)[P(X less than or equal to n) - P(X(t) less than or equal to n)] dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of lambda and c.