Vacation Queues With Impatient Customers and a Waiting Server

queueing models with impatient customers, server vacations and a waiting server. The arrival process is assumed to be Poisson. The server is allowed to take a vacation whenever the system is empty after waiting for a random period of time. Whenever the server is on vacation, each customer in the queue independently of the other customers sets up an impatience timer. If the server does not return from the vacation before the expiry of the customer impatience time, the customer abandons the system forever. We first take up the case where the service times, customer impatience times, waiting times of the server in the empty system and the duration of the server vacations are all exponentially distributed. We then generalize our results to the case where the service times and the server waiting times have general distributions. We derive the steady state probability generating functions (PGF) of the system size and the queue size. We obtain some useful performance measures for the systems. We discuss some particular cases.