Non-preemptive priority M/M/m queue with servers’ vacations

We consider a non-preemptive priority M/M/m queue with two classes of customers and multiple vacations. Service times for all customers are exponentially distributed with the same mean, and vacation times follow an exponential distribution. We obtain the vector probability generating function for the stationary distribution of the number of customers in the queue for each class. This is established by deriving a matrix equation for the vector probability generating function of the stationary distribution of the censored Markov process and then studying the analytical properties of the matrix generating function. We also obtain exact expressions for the first two moments of the number of customers in the queue for each class. Finally, as an application, we investigate a customer's equilibrium strategy and the optimal priority fee associated with social cost minimization for an unobservable M/M/m queue with two priority classes and multiple vacations.