Infinite-server Queues with Batch Arrivals and Dependent Service Times

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time-varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall-Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall-Olkin multivariate hyperexponential distributions to model dependent hyperexponential service times within a batch.