Joining strategies under two kinds of games for a multiple vacations retrial queue with $ N $-policy and breakdowns

Motivated by cost control and information guidance, in this work, we study a multiple vacations retrial queue with N-policy and breakdowns. This service system has the characteristics that there is no waiting space in front of the server and the waiting list is virtual. If the arriving customer finds that the system is available, he immediately receives the complete service. Otherwise, the customer leaves the system or joins the orbit (virtual waiting list). For cost control, the system is activated only when the current vacation is completed and at least N customers are waiting in the system, otherwise, the server continues to the next vacation until the number of customers in the system is not less than N. Two types of customer joining cases apply to this paper, i.e., non-cooperative customers aim to optimize individual interests, and the social planner in the cooperative case considers the profit of the whole service system. The equilibrium joining strategy for the non-cooperative case and the socially optimal joining strategy for the cooperative case are determined. Since it is difficult to obtain analytical characterization, an improved particle swarm optimization (PSO) algorithm is used to explore the impact of system parameters on the profit of the service provider. At the same time, a large number of numerical experiments visualize the influence of parameters on the system.