Summary: We consider a discrete-time queueing system with head-of-line priority. First, we will give some general results on a $GI$-1-1 queue with priority scheduling. In particular, we will derive expressions for the probability generating function of the system contents and the cell delay. Some performance measures (such as mean, variance and approximate tail distributions) of these quantities will be derived, and used to illustrate the impact and significance of priority scheduling in an ATM output queueing switch.