Model Validation and Strategy Analysis in Retrial Queues with Delayed Vacations and Feedback Based on Monte Carlo Simulation

Inspired by call centers, this paper models them as a constant retrial queue, with feedback and delayed vacations to balance high efficiency and low cost for service agents. After completing the service, the server randomly waits for an idle period. If customers arrive during this period, the service is provided immediately, otherwise, the server will take a vacation. We first derive steady-state probabilities and key performance measures. Then, the system cost is modeled. Particle Swarm Optimization (PSO), Ant Colony Algorithm (ACA) and Sparrow Search Algorithm (SSA) are applied to obtain the minimum system cost, respectively. To verify the correctness of the theoretical results of the system model, we simulate the model using Monte Carlo simulation to obtain the probabilities of different server states and the expected number of customers in the system, and then compare them with the theoretical values. At the same time, the sensitivity of the performance measures obtained by Monte Carlo simulation to the system parameters is also analyzed. Finally, customer behavior is analyzed, and equilibrium and socially optimal arrival rates are derived. In addition, the efficiency of the system is evaluated by examining efficiency indicators such as throughput and price of anarchy.