The authors study the queueing process in a single-server queueing system with state dependent service and with the input modulated by a semi-Markov process embedded in the queueing process. It is also assumed that the server capacity is r ≥ 1 and that any service act will not begin until the queue accumulates at least r units. In this model, therefore, idle periods also depend upon the queue length.

The authors establish an ergodicity criterion for the queueing process and evaluate explicitly its stationary distribution and other characteristics of the system, such as the mean service cycle, intensity of the system, intensity of the input stream, distribution of the idle period, and the mean busy period. Various special cases are treated.