In this paper, we develop an efficient method to approximate the transient behavior of queueing systems. More precisely, it is shown how singularity analysis of a known transform function of a transient sequence leads to an approximation of this sequence. To illustrate our approach, the transient system contents of both classes in a two-class discrete-time priority queue are investigated. By means of some numerical examples, we then validate our approximations and demonstrate the usefulness of our approach.