The M^X/M/1 Queue with Multiple Working Vacation

Yutaka Baba

Department of Mathematics Education, College of Education and Human Sciences, Yokohama National University, Yokohama, Japan.

DOI: 10.4236/ajor.2012.22025                

Abstract

We study a batch arrival M^X/M/1 queue with multiple working vacation. The server serves customers at a lower rate rather than completely stopping service during the service period. Using a quasi upper triangular transition probability matrix of two-dimensional Markov chain and matrix analytic method, the probability generating function (PGF) of the stationary system length distribution is obtained, from which we obtain the stochastic decomposition structure of system length which indicates the relationship with that of the M^X/M/1 queue without vacation. Some performance indices are derived by using the PGF of the stationary system length distribution. It is important that we obtain the Laplace Stieltjes transform (LST) of the stationary waiting time distribution. Further, we obtain the mean system length and the mean waiting time. Finally, numerical results for some special cases are presented to show the effects of system parameters.

Keywords

M^X/M/1 Queue; Multiple Working Vacation; Probability Generating Function; Waiting Time Distribution; Stochastic Decomposition

Conflicts of Interest

The authors declare no conflicts of interest.

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