G/Ma,b/1 queues with server vacations

Abstract

In this paper we analyze a G/M(a,b)/1 queue with multiple vacation discipline. Customers are served in batches according to the following bulk service rule in which at least 'a' customers are needed to start a service and maximum capacity of the server at a time is 'b'. When the server either finishes a service or returns from a vacation, if he finds less than 'a' customers in the system, he takes a vacation with exponential distribution. When the server either finishes a service or returns from a vacation, if he hinds more than 'a' customers in the system, he serves a bulk of maximum of 'b' customers at a time. With the supplementary variable method, we explicitly obtain the queue length probabilities at arrival time points and arbitrary time points simultaneously. The shift operator method is used to solve simultaneous non-homogeneous difference equations. The results for our model in the special case of a = b = 1 coincide with known results for G/M/1 queue with multiple vacation obtained by imbedded Markov chain method.