Analysis of a discrete-time repairable queue with disasters and working breakdowns

In this paper, we analyse a discrete-time queue with a primary server of high service capacity and a substitute server of low service capacity. Disasters that only arrive during the busy periods of the primary server remove all customers from the system and make the primary server breakdown. When the primary server fails and is being repaired, the substitute server handles arriving customers. Applying the embedded Markov chain technique and the supplementary variable method, we determine the distribution of the system length at departure epochs and the joint distribution of the queue length and server’s state at an arbitrary instant. Then we derive the sojourn time distribution. We also provide the probability generating function of the time between failures. Some numerical examples are delivered to give an insight into the impact of system parameters on performance measures and a cost function.

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.