Analysis of M/G/c queues with multiple vacations using the supplementary variable technique

Abstract

The queueing models with vacation queues have been applied to many systems in various fields of industry. They also provide the solutions for performance evaluation of systems and we think that their applications are inexhaustible. For these reasons, recently, many researchers have studied on vacation queues with deep interest. In view of the results so far achieved, in case of vacation queues with single server, the distribution of the number of customers and their important performance evaluations have been known well. In addition to that, the fact that there is decomposition property between M/G/1 queues and M/G/1/MV queues have been proved. In spite of their efforts, however, due to model complexities and analytical difficulty, the researches on vacation queues with multiple servers of the exact expression have been not easily found in the literatures. In this paper, we derive and show in priority the result of M/G/c/MV queues by more exact approaches considering the appropriate assumption. So, we analyze these queueing models with unconventional methods, such as the Supplementary Variable Technique and a Modified- Supplementary Variable Technique. In order to apply these approaches, we setup the system equations, supplement some variables and integrate them over supplementary variables. And then, we provide two expressions for the steady-state queue length distribution. With these results, we express the distribution of unfinished work in M/G/c/MV queues. And, we hope that they will be foundations of theoretical/approximated analysis on M/G/c/MV queues.