Threshold-Based Recovery in Queue Theory: Addressing Server Breakdowns, Catastrophic Events, and Customer Reneging

In this research article, we present a comprehensive analysis of a repairable M/M/1/K queueing system that incorporates a threshold-based recovery policy to address server breakdowns, catastrophic events, and customer behaviors such as reneging and balking. In this model, the server fails only when at least one customer is present, and recovery is initiated once the number of customers in the queue reaches a specified threshold T (1≤T<K). We derive closed-form expressions for the system’s steady-state solutions using successive over-relaxation. The study develops critical system characteristics, including the number of customers in the system, the probability of the server being busy, the effective arrival rate, and the expected waiting time. We formulate a cost model to determine the optimal threshold value, system capacity, and service rate that minimize the total cost, which includes repair costs, downtime expenses, and revenue loss. Optimization is performed using Newton-Quasi method. The findings offer valuable insights into queue system design and management, aiding decision-makers in optimizing cost-effectiveness and enhancing overall system performance.