Profit and Cost Optimization for Single-Server Markovian Retrial Queue with Server Breakdown, Customer Balking and Feedback

This study deals with an M/M/1 retrial queueing model with an unreliable server including balking and feedback from the customers. Whenever a customer joins the system and finds that the server is busy, he has the option to either balk or join the retrial orbit. Additionally, some customers, upon receiving service, may express dissatisfaction and choose to rejoin the queue for another attempt. This behavior is referred to as customer feedback. The probability-generating function method is employed to solve the steady-state Chapman-Kolmogorov equations of the model and establish probability distributions. These probability distributions are then utilized to derive various performance measures, including the expected waiting time of customers and the expected number of customers in the system. The profit function and cost function are formulated for the M/M/1 retrial queueing model. The profit function is based on the joining probability of the customers, which represents the likelihood of customers deciding to join the system. On the other hand, the cost function is formulated with the decision variable being the service rate. The goal is to minimize the cost function using a genetic algorithm, which involves optimizing the service rate to find the most efficient configuration for the system. By minimizing the cost function, the model aims to identify the optimal service rate that minimizes costs associated with the queuing system.