We analyze the time-dependent behavior of an $M/M/c$ priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least $c$ high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most $c - 1$ high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami's formula from the theory of $M/G/1$-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution: these results seem to yield the most explicit expressions known to date.