Waiting time analysis for M^X/G/1 Priority queues with/without vacations under random order of service discipline

We study M^X/G/1 nonpreemptive and preemptive-resume priority queues with/without vacations under random order of service (ROS) discipline within each class. By considering the conditional waiting times given the states of the system, which an arbitrary message observes upon arrival, we derive the Laplace-Stieltjes transforms of the waiting time distributions and explicitly obtain the first two moments. The relationship for the second moments under ROS and first-come first-served disciplines extends the one found previously by Takács and Fuhrmann for non-priority single arrival queues.