Equilibrium Strategies in M/M/1 Priority Queues with Balking

We consider an M/M/1 queueing system with a pay-for-priority option, and study customers' joint decisions between joining/balking and pay-for-priority. The equilibrium strategies are thus two-dimensional. First, we fully characterize the equilibrium structure and identify the Pareto-dominant strategies of such a game analytically, under both the observable and unobservable settings. Interestingly, the equilibrium structure, the system throughput, and the service provider's optimal price for priority premium can all be non-monotone in the service reward, which departs from the existing models of priority queues without balking. In particular, we find that an increase in service reward can actually hurt the firm's revenue (everything else being equal). Second, we compare the server's revenue between the observable and the unobservable settings. We find that the service provider is better off with the observable setting when the system load is either low or high, but benefits more from the unobservable setting when the system load is medium. The fact that the optimal setting switches twice as the system load increases, is rather interesting; we explain the intuitions behind it in this study. Finally, we demonstrate the implications of these findings by applying our model framework to Papa John's Pizza, based on publicly available information. Our analysis suggests that Papa John's could benefit from prviding customers with wait information while slightly decreasing its fee for Papa Priority.