Optimal Service Rate for M/M/1/DV Queues with Interrupted Vacations and Impatient Customers Using Particle Swarm Optimization

This paper investigates an M/M/1 queueing system with differentiated vacations, threshold-based interruptions, and customer impatience in the form of balking and reneging. Using recursive analytical methods, we derive closed-form steady-state probabilities and key performance metrics, including average queue length and customer loss rates. To address the practical need for cost-efficient operation, we formulate an economic cost function and determine the optimal service rate using Particle Swarm Optimization (PSO). Numerical experiments conducted in R show that the optimal service rate ranges between 2.71 and 3.48 across different cost structures, achieving minimum expected total costs between 183.23 and 199.04. The results further reveal that the cost function is convex with a clear global minimum, and that earlier vacation interruptions (smaller n1 and n2) significantly reduce both system congestion and customer loss. The proposed approach provides actionable insights for designing and managing service systems in domains such as healthcare, telecommunications, and cloud computing, where server availability is intermittent and customer patience is limited.