On the decomposition property of the M/G/1 queue with generalized vacations

Abstract

In this study, we consider a class of M/G/1 queue with server vacations. As has been noted by other researchers, most models of this class have an interesting property. That is, the system size (waiting time/unfinshed work) distribution is convolution of two distributions, one of which is the system size (waiting time/unfinished work) distribution of the standard M/G/1 queue without vacations, and the other is a distribution due to the effect of vacation policy. This property is called the stochastic decomposition. The main purpose of this study is to present general decomposition results which simplify the analysis of various vacation models. For this purpose, we take an efficient approach. By solving two equations which relate the PGF(Probability Generating Funtion) of the system size at certain epochs in time simultaneously, we express the PGF of the second variable in the decomposition in the following three ways.:(1)The equation of the PGF of the system size at epochs of a busy period-beginning and a busy period-completion (2)The equation of the PGF of the system size at epochs of a vacation-beginning and a vacation-completion (3)The equation of the PGF of the system size at epochs of a grand vacation-beginning and a grand vacation-completion From this result, we derive the decomposition for the waiting time and unfinshed work respectively. We illustrate several examples to show that these results simplify the analysis of various vacation models. We hope that the results and approach presented in this study provide a guideline for the analysis of complex vacation models.