A Poisson limit for the departure process from a queue with many busy servers

We establish a limit theorem supporting a Poisson approximation for the departure process from a multi-server queue that tends to have many busy servers. This limit can support approximating a flow out of such a queue in a complex queueing network by an independent Poisson source. The main ideas are: (i) to scale time so that previous many-server heavy-traffic limits can be applied and (ii) for time-varying arrival-rate functions, to scale (spread out) time by a large factor about each fixed time.