Relationship between queue-length and waiting time distributions in a priority queue with batch arrivals

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