On an M/G/1 queue in random environment with Min(N, V) policy

In this paper, we analyze an M∕G∕1 queue operating in multi-phase random environment with Min(N, V) vacation policy. In operative phase i, i = 1, 2, …, n, customers are served according to the discipline of First Come First Served (FCFS). When the system becomes empty, the server takes a vacation under the Min(N, V) policy, causing the system to move to vacation phase 0. At the end of a vacation, if the server finds no customer waiting, another vacation begins. Otherwise, the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n. And whenever the number of the waiting customers in the system reaches N, the server interrupts its vacation immediately and the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n, too. Using the method of supplementary variable, we derive the distribution for the stationary system size at arbitrary epoch. We also obtain mean system size, the results of the cycle analysis and the sojourn time distribution. In addition, some special cases and numerical examples are presented.