Analysis on queueing systems with synchronous vacations of partial servers

We study an M/M/c queue with server vacations. In this queueing system, d (≤c) of c servers take synchronous vacations when these d servers become idle at a service completion instant. This type of queueing model captures the major characteristics of a stochastic service system which processes both random arriving jobs and constantly available jobs. In this paper, the multi-server vacation queueing system has been analyzed as a quasi-birth and death process. Using matrix geometric method, we obtain the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.