Modeling and optimization of an M/M/1/K/WV retrial queue with dual-phase service, customer's balking under F-policy

This paper examines an M/M/1/K/W V queue, incorporating features such as retrial queue, working vacation (WV), admission control F -policy, customer balking, and dual-phase service. Upon arrival, customers observing an extended queue length have the option to refrain from joining the queue, a phenomenon known as balking. Alternatively, if a customer decides to join the queue but finds the service facility occupied, they are directed to a virtual space termed the retrial orbit. After a random duration, customers in the retrial orbit make subsequent attempts to avail the service. In our study, dissatisfied customers with the initial phase of service have the option to choose a secondary phase of service. The F -policy adeptly regulates customer congestion by limiting admissions when the system capacity is reached and resuming admissions once the capacity decreases to a predefined threshold value F. Introducing the working vacation concept in our study ensures continuous service provision during the vacation period, albeit at a reduced rate compared to the standard operational pace. This multifaceted approach contributes to a more nuanced and flexible management of customer influx in the system. The model is mathematically formulated by developing steady-state difference equations, which are then solved using the matrix analytical method. Diverse system metrics are subsequently derived, providing valuable insights for system managers to effectively analyze and navigate the queueing management and decision-making processes. We also present a numerical example to illustrate the impact and behavior of input parameters on a range of system metrics. Furthermore, we employ the adaptive neuro-fuzzy inference system (ANFIS) to validate our analytical findings against the ANFIS results. Establishing a bivariate cost function is a noteworthy aspect of our research, and we minimize cost function using both the quasi-Newton method (QNM) and genetic algorithm (GA), enhancing the robustness and reliability of our optimization approach. The practical justification of the model is demonstrated in an automotive service facility (ASF) where malfunctioning bikes (MBs) arrive for servicing.