On the $M^x$/G/1 queue with vacation time

Abstract

This paper studies the batch arrival $M^x$/G/1 queue with vacation time under nonpreemptive last-come first-served (LCFS) discipline. As usual, the server is busy as long as there are units in the main system. However, as soon as the server becomes idle he leaves for a "vacation". The duration of a vacation is a random variable with a known distribution function. Two models are considered. In the first one, upon termination of a vacation the server returns to the main queue and begins to serve those units, if any, that have arrived during the vacation. If no units have arrived the server waits for the first arrival when an ordinary $M^x$/G/l busy period is initiated. In the second model, if the server finds the system empty at the end of a vacation, he immediately takes another vacation, etc. For both models Laplace-Stieltjes transforms (LSTs) of the busy period and waiting time are derived and probability generating function (p.g.f.) of the number of units in the system are calculated. The two models are then compared to each other.