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

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.