M/M/c Queue with Two Priority Classes

This paper provides the first exact analysis of a preemptive M/M/c queue with two priority classes having different service rates. To perform our analysis, we introduce a new technique to reduce the 2-dimensionally (2D) infinite Markov Chain (MC), representing the two class state space, into a 1-dimensionally (1D) infinite MC, from which the Generating Function (GF) of the number of low-priority jobs can be derived in closed form. (The high-priority jobs form a simple M/M/c system, and are thus easy to solve.) We demonstrate our methodology for the c= 1,2 cases; when c > 2, the closed-form expression of the GF becomes cumbersome. We thus develop an exact algorithm to calculate the moments of the number of low-priority jobs for any c ≥ 2. Numerical examples demonstrate the accuracy of our algorithm, and generate insights on: (i) the relative effect of improving the service rate of either priority class on the mean sojourn time of low-priority jobs; (ii) the performance of a system having many slow servers compared with one having fewer fast servers; and (iii) the validity of the square root staffing rule in maintaining a fixed service level for the low priority class. Finally, we demonstrate the potential of our methodology to solve other problems such as an M/M/c queue with two priority classes, where the high-priority class is completely impatient.