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We analyze the time-dependent behavior of an $M/M/c$ priority queue havingtwo customer classes, class-dependent service rates, and preemptive prioritybetween classes. More particularly, we develop a method that determines theLaplace transforms of the transition functions when the system is initiallyempty. The Laplace transforms corresponding to states with at least $c$high-priority customers are expressed explicitly in terms of the Laplacetransforms corresponding to states with at most $c - 1$ high-prioritycustomers. We then show how to compute the remaining Laplace transformsrecursively, by making use of a variant of Ramaswami's formula from the theoryof $M/G/1$-type Markov processes. While the primary focus of our work is onderiving Laplace transforms of transition functions, analogous results can bederived for the stationary distribution: these results seem to yield the mostexplicit expressions known to date.

Time-dependent analysis of an $M/M/c$ preemptive priority system with two priority classes