A note on the response time in M/G/1 queues with service in random order and Bernoulli feedback

We consider M/G/1 queueing systems with random order of service and Bernoulli feedback of output customers. These systems may model the aggregate queues of packets waiting for transmission in a contention-based multiaccess communication channel. We study the customer's response time defined as the time from its arrival to final departure. The mean response time is equivalent to that in a batch arrival system in which the batch size is geometrically distributed. The second moment of the response time is newly obtained explicitly. Numerical comparison shows that the random order of service sometimes yields smaller values of the second moment of the response time than first-come first-served and last-come first-served disciplines in feedback systems. We deal with a system without server vacations as well as one with multiple server vacations.