Abstract
We study a batch arrival MX/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 MX/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.
Highlights
Vacation queues have been investigated for over two decades as a very useful tool for modeling and analyzing computer systems, communication networks, manufacturing and production systems and many others
We study a batch arrival MX/M/1 queue with multiple working vacation
With the matrix analytic method, they derived the probability generating function (PGF) of the stationary system length distribution, from which they got the stochastic decomposition result for the PGF of the stationary system length which indicates the evident relationship with that of the classical MX/M/1 queue without vacation
Summary
Vacation queues have been investigated for over two decades as a very useful tool for modeling and analyzing computer systems, communication networks, manufacturing and production systems and many others. With the matrix analytic method, they derived the PGF of the stationary system length distribution, from which they got the stochastic decomposition result for the PGF of the stationary system length which indicates the evident relationship with that of the classical MX/M/1 queue without vacation. They found the upper bound and lower bound of the stationary waiting time in the Laplace transform order, from which they got the upper bound and lower bound of the waiting time.
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