Stationary distribution of a finite queue with recursive input flow and Markov service discipline

An one-line queueing system with the limited accumulator and recursive demand input flow is studied. The service process of the phase type is Markov one and specified by the transition intensity matrices by phases with and without the demand output from a device. A matrix algorithm is obtained for the calculation of stationary distribution of the Markov process describing the queue process for both arbitrary time moment and the moments of demands entering into the system and their service completion. Laplace-Stieltjes transformation is obtained for the demand expecting time in the service discipline FCFS.