A Discrete-Time Geo/G/1 Retrial Queue with Balking Customer, Second Optional Service, Bernoulli Vacation and General Retrial Time

In this paper, we study a discrete-timeGeo/G/1 retrial queue with a balking customer and a second optional service where retrial time follows a general distribution. If an arriving customer finds the server free, he begins the service immediately. If an arriving customer finds the server busy or to be on vacation, then he will consider whether he enters into the system. After a customer accepts his first essential service, he may leave the system or ask for a second service. In this model, we assume that the server, after each service completion, may begin a vacation or wait to serve the following customer. This paper studies the Markov chain underlying this model, we establish the probability generating functions of the orbit size and system size. Finally, some performance measures and some numerical examples are presented.