A queueing system of limited capacity with distributions of phase type depending on the queue state

A single-line feedback service system with a finite number $r(1\leq r<\infty)$ of queueing stations is considered. The arrival and servicing of customers are assumed to be described by phase distributions, $PH$-distributions. Such a ystem is encoded as $PH|PH|1|r$. The parameters of $PH$-distributions are assumed to depend on the queue state. Matrix expressions are obtained for a stationary distribution of the state probabilities. When the dependence of the arriving flow and of service on the system state is neglected the expressions for stationary distribution are provided as a matrix progression. Numerous other examples are given. The results may be useful in estimating the throughput of a two-phase system wilh losses or interlocks and in computing a closed-loop two-server system.