Stationary distribution of a finite queue with recurrent input and Markov service

A single-server queue system with a limited waiting room and recurrent input is investigated. Phase-type service is a Markov process and is defined by the intensity matrices of phase transitions with and without output. A matrix algorithm is designed for computing the stationary distribution of the Markov process that describes the queueing process for an arbitrary instant as well as for the customer arrival times and customer service times. A Laplace-Stieltjes transformation is derived for waiting limes under the FCFS discipline.