On Discrete Quening Systems

Queuing systems are considered, in which calls arrive in batches of size $\xi $ and are served in batches of size $\eta$ the period of time between two successive arrivals of batches of calls is equal to $\tau$; the time of a service is equal to $\sigma$. The quantities $\xi$, $\eta$, $\tau$ and $\sigma $ are discrete random variables. In the paper some general statements on the existence of stationary distributions are proved. These distributions are found (in terms of the factorization of functions) for the case when one of the random variables $\xi$, $\eta$, $\tau$, $\sigma$ is equal to unity. In the paper a method for finding the stationary distribution with the help of the stationary distribution for the imbedded Markov chain is described.