A Study on State Dependent Accessible and Second Optional Service Queue with Impatient and Feedback Customers

In this paper, we study a single server Markovian queue with state dependent accessible services, reneging and feedback of customers. The server can accommodate at the most $d$ customers in the service station, after which the new arrivals have to wait in the primary queue of infinite waiting space. All arrivals demand First Essential Service (FES), after completing FES, customers decide to join the second optional Service (SOS) with a probability. The services occur singly according to first come first served service discipline and service times in FES and SOS are exponentially distributed. A customer waiting in the primary queue may get impatient and renege from the queue. However, after completing SOS, if the customer is not satisfied with the service quality, he may join the queue again (feedback). The system is analyzed by a quasi birth-death process and the steady state probabilities of the model are obtained using matrix geometric method. Some performance measures and numerical illustrations are also provided. An optimization of the cost function is performed to find the optimal service rate that minimizes the total cost.