Customer equilibrium in a single-server system with virtual and system queues

Consider a non-preemptive M/M/1 system with two first-come first-served queues, virtual (VQ) and system (SQ). An arriving customer who finds the server busy decides which queue to join. Customers in the SQ have non-preemptive priority over those in the VQ, but waiting in the SQ is more costly. We study two information models of the system. In the unobservable model customers are notified only whether the server is busy, and in the observable model they are also informed about the number of customers currently waiting in the SQ. We characterize the Nash equilibrium of joining strategies in the two models and demonstrate a surprising similarity of the solutions.