Second moment relationships for waiting times in queueing systems with Poisson input (Q809490)

The author generalizes for the M/G/1 queue the well-known and simple relationships among the second moments of waiting time of an arbitrary customer under the three service disciplines FIFO (first-in-first-out), nonpreemptive LIFO (last-in-first-out) and ROS (random-order-of-service), namely, denoting the corresponding waiting times by \(W_ F\), \(W_ L\), \(W_ R\), respectively, \[ E(W^ 2_ F)=(1-\rho)E(W^ 2_ L),\quad E(W^ 2_ F)=(1-\rho /2)E(W^ 2_ R). \] The same relationships hold for M/G/1 queues with exceptional first service, M/G/1 queues with server vacation, and M/G/1 queues with static priorities. The method of proof is the use of a generalization of Little's law and the handling of completion times. A significant role is played by the assumption of exhaustive service, i.e. once the server starts working it works continuously until the system is empty.