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The bulk input M[X] /M/1 queue with working vacations
Authors:Xiu-li Xu  Ming-xin Liu  Xiao-hua Zhao
Institution:1. College of Science, Yanshan University, Qinhuangdao 066004, Hebei, China
2. College of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, Hebei, China
Abstract:In this paper, we analyze a bulk input MX] /M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matrix analysis method, highly complicated probability generating function(PGF) of the stationary queue length is firstly derived, from which we got the stochastic decomposition result for the stationary queue length which indicates the evident relationship with that of the classical MX] /M/1 queue without vacation. It is important that we find the upper and the lower bounds of the stationary waiting time in the Laplace transform order using the properties of the conditional Erlang distribution. Furthermore, we gain the mean queue length and the upper and the lower bounds of the mean waiting time.
Keywords:M[X]/M/1 queue  bulk input  multiple working vacations  stochastic decomposition  the Laplace transform order
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