Customer Joining Behavior and Regulation Strategy in Visible Multi-Server Queues with Synchronous Vacations of Partial Servers

This paper studies customer joining behavior and system regulation strategy in non-exhaustive visible M/M/mqueues with synchronous vacations of a part of the servers. Once this part of the servers are idle, they take multiple vacations simultaneously (vacation period). Until there are customers waiting in the queue, they are reactivated and all servers are busy or idle (busy period). The authors call this part of the servers as “partial servers”. In view of the fully visible queue and the almost visible queue, the authors obtain customers’ equilibrium joining threshold strategies and their socially optimal joining threshold strategies, respectively, and observe that customer joining behavior in equilibrium generally makes the system overcrowded, which makes the equilibrium social welfare lower than the optimal social welfare. After regulation, interestingly, for optimizing social welfare, the system manager hopes not only customers arriving in vacation period pay attention to the number of partial servers, but also customers arriving in busy period should care about it rather than ignore. Moreover, arranging more servers for vacation does not necessarily lead to the decrease of social welfare on condition that the number of partial servers is close to m. As for the information advantage of the fully visible case, it is not obvious for increasing social welfare and even unfavorable to servers’ profit unless the number of partial servers is big enough. Furthermore, given the different composition of social welfare, there exists the optimal number of partial servers and the optimal arrival rate of customers for maximizing social welfare.