Analysis for the M[X]/M/1 Working Vacation Queue

In this paper, we analyze a bulk input M[X]/M/1 queue with single working vacation. An quasi upper triangle transition probability matrix of this two-dimensional Markov chain is obtained. With the matrix analysis method, highly complicated PGF of the stationary queue length distribution is firstly derived, from which we got the stochastic decomposition result for the PGF of the stationary queue length which indicates the evident relationship with that of the classical M[X]/M/1 queue without vacation. It is important that we find the upper bound and lower bound 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 bound and lower bound of the mean waiting time. Finally, some special cases and numerical examples are presented