Two-parameter heavy-traffic limits for infinite-server queues

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables Qe(t,y) and Qr(t,y) representing the number of customers in the system at time t that have elapsed service times less than or equal to time y, or residual service times strictly greater than y. We also consider Wr(t,y) representing the total amount of work in service time remaining to be done at time t+y for customers in the system at time t. The two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (Adv. Appl. Probab. 23, 188–209, 1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (Queueing Syst. 25, 235–280, 1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.