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

We study (batch arrival) M^X/G/1 queues with/without vacations under random order of service (ROS) discipline. By considering the conditional waiting times given the states of the system when an arbitrary message arrives, we derive the Laplace-Stieltjes transforms of the waiting time distributions and explicitly obtain their first two moments. The relationship for the second mements under ROS and first-come first-served disciplines is shown to be precisely the same as that found by Takács and Fuhrmann for (single arrival) M/G/1 queues.