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 Q (e) (t,y) and Q (r) (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 W (r) (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,a),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.