An invariance in the priority queue with generalized server vacations and structured batch arrivals

Shanthikumar (Shanthikumar, J.G. Level crossing analysis of priority queues and a conservation identity for vacation models. Nav. Res. Log. 1989, 36, 797-806) studied the priority M/G/1 queue with server vacations and found that the difference between the waiting time distribution under the non-preemptive priority (NPP) and that under the preemptive-resume priority (PRP) is independent of the vacation policy. We extend this interesting property: (i) to the generalized vacations which includes the two vacation policies considered by Shanthikumar; (ii) to the structured batch Poisson arrival process; and (iii) to the discrete-time queues.