On a Queueing System with Differentiated Vacations and General Service Time

We consider a vacation queueing model with Poisson arrivals, general service distribution and differentiating vacation. The server begins a type I vacation of random length each time the system (and the server) becomes empty. If the server returns from a vacation to find an empty queue, a new random vacation of type II initiates. On returning from either type I or II vacation, if there are some customers in the system, the server immediately start servicing customers until the system is empty again. Using generating function and Laplace transform we obtain the stationary distribution of the joint distribution of the server state and the queue size. Next, we derive several mean performance measures of interest.