A two phase service M/G/1 queue with a finite number of immediate Bernoulli feedbacks

Abstract

In this paper, we consider an M/G/1 queueing system with two phases of heterogeneous service and a finite number of immediate Bernoulli feedbacks. All arriving customers are provided with the same type of service in the first phase. In the second phase, the customer has to choose from one of the several optional services which are available in the system. After having completed both phases of service, the customer is allowed to make an immediate feedback. The feedback service also consists of two phases. In the feedback, the first phase of service is of the same type as in the previous service. However, in the second phase, the customer may be permitted to choose an optional service different from the one chosen earlier. This feedback scheme is different from the usual Bernoulli feedback scheme. In the earlier papers on the usual Bernoulli feedback, the customer returns to the tail of the queue and waits patiently for his next round of service. In our system, if the customer desires to make a feedback, the customer immediately proceeds for a second round of service after completion of his first round , that is he is his own successor. The customer is allowed to make a finite number of such feedbacks before departing from the system. Our motivation for considering this model comes from our experience in banking transactions through the ATM. We obtain the probability generating function of the system size distribution in the steady state. Some useful performance measures are also obtained. Numerical examples are presented to illustrate the influence of the various parameters involved on the performance of the system.