Joining strategies of noncooperative and cooperative in a single server retrial queue with N-policy and multiple server vacations

Motivated by service price and cost control, in this work, we study a single-server queueing system with a constant retrial rate, N-policy, and multiple server vacations. There is no waiting space in front of the server in this system. If the server is unavailable upon a customer’s arrival, the arriving customer decides whether to join an orbit (waiting list). The server goes on vacation as soon as the system becomes empty. The server is activated again upon completing a vacation, only when the number of customers in the system is at least N. Otherwise, the server goes on another vacation. The joining strategies for the customers, respectively, in noncooperative and cooperative cases, are derived after extensive analysis. On the other hand, we explore how a social planner can choose an appropriate service price to induce selfish customers to join the system with the best strategy. We also consider how to set up a price to obtain the optimal profit from the service provider’s perspective. Finally, we present some numerical examples to illustrate some system parameters’ effect on optimal joining probabilities and the maximum of social welfare.