Equilibrium Joining Strategies in the Retrial Queue with Two Classes of Customers and Delayed Vacations

We consider a non-preemptive priority M/M/1 retrial queue with two classes of customers (low-priority and high-priority customers) and delayed vacations. When the server is unavailable, an arriving high-priority customer can wait in line, whereas an arriving low-priority customer needs to enter a virtual queue and retry later. After completing a service, the server will remain idle for a reserved idle time if it finds no high-priority customers in the system. Arrivals during the reserved idle period will be served immediately. Otherwise, if no customers arrive during this interval, the server will switch to the vacation state. By constructing a three-dimensional Markov chain, we successively obtain the stability condition of the system and some main performance measures. Then depending on a linear reward-cost structure, we derive customers' two-dimensional equilibrium joining strategies in the fully unobservable case. Due to the complexity of the social welfare function, we explore the socially optimal joining strategies through the Particle Swarm Optimization (PSO) algorithm. Finally, we illustrate the impact of system parameters on the two types of joining strategies via numerical experiments.