Analysis of the steady-state behavior of a queueing system with autoregressive arrivals

The paper studies a single server queueing system with infinite capacity and with batch Poisson arrival process. A feature of the system under study is autoregressive dependence of the arriving batch sizes: the size of the $n$-th batch is equal to the size of the $(n-1)$-st batch with a fixed probability, and is an independent random variable with complementary probability. Service times are supposed to be independent random variables with a specified distribution. The steady-state behaviour is studied; expression for the probability generating function of the queue length is derived, as well as the mean queue length for a special case.