Analysis of a queueing system with autoregressive arrivals

The paper studies a single server queueing system with infinite capacity and with the Poisson batch 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 main object of the study is the queue length at an arbitrary moment. The relations derived make it possible to find Laplace transorm in time of the probability generating function of the transient queue length, and also a number of additional characteristics such as the residual service time and the distribution of the size of the last batch that arrived before time $t$.