Equilibrium joining strategies in the single server queues with negative customers

We consider an M/M/1 queue with negative customers. An arriving negative customer will break the server down and the positive customer being served (if any) is forced to leave the system. Once a breakdown occurs, the server is sent immediately for repair while positive customers are not allowed to join the system during the repair process. When the server is available, positive arrivals decide whether to join or balk the system based on a common reward-cost structure. We consider an observable case that the positive arrivals are informed about the number of customers in the system and an unobservable case without any information. The corresponding Nash equilibrium strategies and the socially optimal joining strategies are explored. We get a socially optimal threshold in the observable case and a mixed joining strategy in the unobservable case. The profit maximization issue is studied, and we derive optimal strategies in two information cases. Finally, numerical examples are provided to show the influence of different parameters on the strategies and social benefit.