Continuity of the M/G/c queue

Consider an M/G/c queue with homogeneous servers and service time distribution F. It is shown that an approximation of the service time distribution F by stochastically smaller distributions, say F-n, leads to an approximation of the stationary distribution pi of the original M/G/c queue by the stationary distributions pi(n) of the M/G/c queues with service time distributions F-n. Here all approximations are in weak convergence. The argument is based on a representation of M/G/c queues in terms of piecewise deterministic Markov processes as well as some coupling methods.