Disributions in a Priority Queue with Batch Arrivals

We consider a single-server priority queue with batch arrivals. We treat the head-of-the-line (HL) or preemptive-resume (PR) priority rule. Assuming that the arrival process of batches is renewal for each priority class and using the point process approach, we express the individual class queue-length distribution in terms of the waiting time and the completion time distributions. Assuming further a batch Poisson an-ival for each class, together with the previous result on the Laplace-Stieltjes transforms for the waiting time and completion time distributions, we derive the z-transform for the queue-length distribution in closed fonn.