Two-parameter heavy-traffic limits for infinite-server queues with dependent service times

This paper is a sequel to our 2010 paper in this journal in which we established heavy-traffic limits for two-parameter processes in infinite-server queues with an arrival process that satisfies an FCLT and i. i. d. service times with a general distribution. The arrival process can have a time-varying arrival rate. In particular, an FWLLN and an FCLT were established for the two-parameter process describing the number of customers in the system at time t that have been so for a duration y. The present paper extends the previous results to cover the case in which the successive service times are weakly dependent. The deterministic fluid limit obtained from the new FWLLN is unaffected by the dependence, whereas the Gaussian process limit (random field) obtained from the FCLT has a term resulting from the dependence. Explicit expressions are derived for the time-dependent means, variances, and covariances for the common case in which the limit process for the arrival process is a (possibly time scaled) Brownian motion.