Modeling and Cost Optimization for M/M/1/K Queue with Dual-Phase Service and Repair Under Threshold Recovery

In this paper, we investigate an M/M/1/K queue with dual-phase service and repair under threshold recovery. This system is characterized by a single server that provides customers with two consecutive phases of service. Furthermore, the system may experience the unpredictability of sudden server breakdowns, necessitating repair. A dual-phase repair process is implemented to address server failures, but repair begins only when the number of customers in the system reaches a certain threshold value of Q \((1 \le Q \le K)\) . To thoroughly understand the system, we mathematically construct steady-state equations for the model and then use a recursive method to solve them, thereby yielding steady-state probabilities. We also derive several performance measures, with the primary objective of decision-makers being to maximize profit while minimizing the inconvenience caused by service delays for customers. In pursuit of this goal, we construct a cost function that incorporates the threshold parameter and the service rate as decision variables and define an optimization problem with the aim of attaining the optimal cost. A population-based particle swarm optimization (PSO) is implemented to minimize the cost by identifying the optimal threshold parameter and service rate. After developing a model, the objective of the study is twofold: Firstly, to conduct a thorough analysis of system performance using measures such as system size and waiting time, providing decision-makers with valuable insights into operational efficiency and customer satisfaction levels. Secondly, the study aims to minimize the cost function by using PSO, thereby optimizing operational processes to achieve the most cost-effective solution while maintaining high service quality.