On a preemptive Markovian queue with multiple servers and two priority classes

The authors consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. In a convenient state space representation of the system, the authors obtain the matrix equation for the two-dimensional, vector-valued generating function of the equilibrium probability distribution. They give a rigorous proof that, by successively eliminating variables from the matrix equations, a nonsingular, block tridiagonal system of equations is obtained for the set of (m+1)(m+2)/2 constants that describe the probabilities of the states of the system when no customers are awaiting service. The mean waiting time of the low priority customers is shown to be given by a simple formula in terms of the (known) waiting time of the high priority customers and the expected number of low priority customers in the queue when no high priority customers are waiting.