(The) analysis of the $M^X$/G/1 queue by a martingale method

Abstract

In this paper, we derive several generating function for the M$^x$/G/1 queue, such as the generating function of the number of customers served in a busy period, the generating function of the number of customers at steady state and the transient generating function of the number of customers at departure epochs by a martingale argument. In addition to these, we calculate a new explicit formula for the Laplace transform of the busy period.