Two-parameter process limits for an infinite-server queue with arrival dependent service times

We study an infinite-server queue with a general arrival process and a large class of general time-varying service time distributions. Specifically, customers’ service times are conditionally independent given their arrival times, and each customer’s service time, conditional on her arrival time, has a general distribution function. We prove functional limit theorems for the two-parameter processes Xe(t,y)Xe(t,y) and Xr(t,y)Xr(t,y) that represent the numbers of customers in the system at time tt that have received an amount of service less than or equal to yy, and that have a residual amount of service strictly greater than yy, respectively. When the arrival process and the initial content process both have continuous Gaussian limits, we show that the two-parameter limit processes are continuous Gaussian random fields. In the proofs, we introduce a new class of sequential empirical processes with conditionally independent variables of non-stationary distributions, and employ the moment bounds resulting from the method of chaining for the two-parameter stochastic processes.