The analysis of MX/M/1 queue with two-stage vacations policy

In this article, we consider a batch arrival M-X/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by M-X/M/1/SWV + MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace-Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples.