The M/M/1 Queue witch Bernoulli-Schedule-Controlled Vacation and Vacation Interruption

In this paper, we study an M/M/1 queue with multiple vacation policies described as follows: when the server becomes empty, it either goes for an ordinary vacation with probability p or takes a working vacation with probability 1 ― p. In the working vacation, a customer is serviced at a lower rate (called the sever is in an non-regular busy period), and at the instants of a service completion, the vacation is interrupted and the server is resumed to a regular busy period with probability 1 ― q (if there are customers in the queue), or continues the vacation with probability q. We give an analytic expression for the stationary distribution of queue length and demonstrate the stochastic decomposition structures of the stationary queue length and waiting-time. Also we obtain the additional queue length and the additional delay.