On Stochastic Decomposition in M/G/1 Type Queues with Generalized Server Vacations

Recently, S. W. Fuhrmann and R. B. Cooper showed that the stationary distribution of the number of customers in an M/G/1 queueing system with generalized server vacation is a convolution of the distribution functions of two independent positive random variables. One of these is the stationary distribution of the number of customers in an ordinary M/G/1 queueing system without server vacations. They use an elegant, intuitive approach to establish this result. In this paper, a mechanistic (analytic) proof of this result is given for systems more general than that discussed in Fuhrmann and Cooper. Such systems allow customer arrivals in bulk and some variations with reneging, balking, and in arrival rate that is dependent on system state.