A matrix-multiplicative solution for a single-line system with server vacations, a finite, retrial queue, and phase-type distributions

Studies are made of a single-line queueing system without a waiting space (buffer) and with a recurrent Pow of requests that is specified by the phase-type distribution function. A request that finds the server busy providing service joins the queue of retrial requests, which is called an orbit. The requests moving from the orbit attempt to engage the server again, in which case the time intervals between repeated attempts of each request are distributed in an exponential manner and the rate of the flow of requests from the orbit is directly proportional to the number of requests in the orbit. The service times of both the primary requests and the requests coming from the orbit also display phase-type distributions. The orbit size is limited by the number s, and so the request arriving at the orbit at the instant when the number of requests in the orbit is equal to s is lost. It is also assumed that the server can disengage itself (stay vacant) in the case of complete emptying of the system for a random time with phase-type distribution function. ii stationary distribution is found for the Markov process describing the given system, the distribution being represented in the matrix-multiplicative form. Some generalizations of the Markov model are studied. Stationary probabilities are obtained for the states of the system considered at the instants of both the arrival of requests and the completion of their service. Numerical examples are given.