A queueing system with heterogeneous arrivals, two kinds of vacations and impatient customers

We consider an M/ M/1 queueing system with two kinds of vacations and impatient customers. The server begins the 1st kind of vacation each time the system becomes empty after serving all the customers. If the system is empty, when returning from this vacation it takes the 2nd kind of vacation. When the server comes back from either 1st kind or 2nd kind of vacation, if there is at least one customer in the system, the server immediately starts the service. If the server is busy or on 1st kind of vacation when a customer arrives at the system, it activates an impatience timer. If the customer's service has not been finished before the customer's timer perishes, the customer leaves the system and never returns. The balance equations of the model are all derived using a state-transition diagram. Probability generating functions (PGFs) technique has been used to derive the steady-state probabilities. Various measures of performance are presented and numerical illustrations are also provided.