A One-Line Queueing System with a Limited Number of Waiting Places and Priorities

A one-line queueing system with two incoming Poisson flows of demands is analyzed. The service times of the demands in both flows are arbitrarily distributed. The system has $r<\infty$ waiting places. The case is considered in which, when demands are serviced out of turn, a relative priority is established in the system, and when a demand is placed in the queue, an absolute priority is established. The system is analyzed by using line Markov processes. For a given system an algorithm is obtained for the stationary probability distribution of the system; it amounts to the solution of a nonhomogeneous system of $r+1$ linear algebraic equations.