A queueing model with two input streams of different service priorities is studied. Specifically, IBP+BP/D/1 with head-of-line priority is analyzed. IBP and BP stand for Interrupted Bernoulli Process and Bernoulli Process respectively. The BP-stream customers have the higher service priority over the IBP-stream customers. An exact analysis of this priority queue is presented to derive the distributions of the state of the system at steady state, the waiting time distributions for each class of customers, and the interdeparture time distributions. The numerical results of the analysis are presented to show how the various parameters of the low and high priority arrival processes affect the performance of the system.