The departure process for queueing systems with batch arrival of customers

In the article, a single-server queueing system of GI/G/1 type with batch arrival of customers, individual service and unlimited queue is considered. We denote by h(t) the departure process that at any fixed moment t takes on a random value equal to the number of customers completely served during [0, t). A system of integral equations on a half-axis [0,) is written for probabilities P{h(t)=m} under two different initial conditions. Hence some general results for the departure process in transient state are obtained. The results are written in terms of Laplace-Stieltjes transforms of distribution functions F1 and F2 of interarrival and service times respectively, and using factors of a certain factorization identity of Wiener-Hopf type connected with F1 and F2. In particular, the explicit formula for the expression [image omitted] is obtained.