(A) new approach for the system size of queueing systems and a quick approximation for the M/G/1 and M/G/1/K

Abstract

In this study, we present a new approach for the system size distribution of the queueing systems. The main idea of our approach is to regard each waiting space in the queue, rather than entire queue, as a target of analysis. From this approach, we derive simple transform-free expressions for the M/G/1 and M/G/1/K in terms of certain variables. From these expressions, we reconstruct, with great ease, the proportionality relationships between the values of probabilities of the system size for the M/G/1 and M/G/1/K. With some simplification of these results, we develop a remarkably simple and quick approximation for the system size distribution. The input parameters required for this approximation are arrival rate and, first and second moment of the service time. Numerical aspects have been tested for a variety of service-time distributions and our approximation has worked fairly well in most practical cases considered. A sample of numerical computations is included in the form of tables and graphs. It is hoped that our new approach could give a new viewpoint into the queueing analysis. It is also hoped that our exact and approximate results obtained would prove to be beneficial to both theorists and practitioners dealing with approximations and other aspects.