Abstract: In this paper, we study an M/M/1 queue with multiple vacation policy under the assumption that the decisions whether or not to take a new vacation and take what kind of vacation depend on certain probability, when the server becomes empty. For the queue model, by the quasi-birth-and-death (QBD) process and matrix geometric method, we derive the analytic expression of the stationary queue length, and demonstrate stochastic decomposition structures of the stationary queue length and sojourn time. Moreover, we also obtain the additional queue length and the additional delay. The results show that the classical M/M/1 queue, the M/M/1 queue with vacation or with working vacation are special cases of our model.

Key words: M/M/1 queue, vacation, working vacation, matrix-geometric solution, stochastic decomposition