Heuristic interpretation of the mean performance measures of the $M^x$/G/1 queue

Abstract

Most results on the analysis of M/G/1 vacation models so far has been expressed in the form of Laplace-Stieltjes Transform (LST), but it is not easy to invert transforms and to retrieve the probability distributions. For some complex systems, it is not even easy to differentiate the transform to obtain the mean performance measures. So we provide a supplementary tool to the formal procedure of queueing system analysis. The supplementary tool we present is a transform free method to find and interpret major performance measures. With this supplementary tool, one can not only verify one``s results but also obtain insights for further extensions of the formal results. As a demonstration, we interpret the mean queue waiting time of the M$^X$/G/1 queue with server vacations combined with N-policy. These interpretations are mainly based on the residual life analysis. Besides the mean queue waiting time, some other performance measures are also interpreted in relation to the mean queue waiting time. For some specific models which Takagi [1992] dealt with, the mean depletion time is interpreted in detail. Furthermore, we extend the idea to some more complex systems.