Autocorrelations in infinite server batch arrival queues

This paper studies an important aspect of queueing theory, autocorrelation properties of system processes. A general infinite server queue with batch arrivals is considered. There areM different types of customers and their arrivals are regulated by a Markov renewal input process. Batch sizes and service times depend on the relevant customer types. With a conditional approach, closed form expressions are obtained for the autocovariance of the continuous time and prearrival system sizes. Some special models are also discussed, giving insights into steady state system behaviour. Autocorrelation functions have a wide range of applications. We highlight one area of application by using autocovariances to derive variances of sample means for a number of special models.This work has been supported by the Natural Sciences and Engineering Council of Canada through Grant A5639 and by the National Natural Science Foundation of China through Grant 19001015.