Stationary analysis for the bulk input Geom([X])/Geom/1 queue with working vacation

In this paper, we analyze a bulk input Geom[X]/Geom/1 queue with single working vacation. Using the matrix analysis method, highly complicated PGF of the stationary queue size is firstly derived, from which we got the stochastic decomposition result for the PGF of the stationary queue size. It is important that we find the biparameter addition theorems of the conditional negative binomial distribution, with which we find the upper bound and the lower bound of the stationary waiting time in the moment generating function order. Furthermore, we gain the mean queue size and the upper bound and the lower bound of the mean waiting time. Finally, several numerical examples are presented