Wait Time Based Pricing for Queues with Customer-Chosen Service Times

This paper studies a pricing problem for a single-server queue where customers arrive according to a Poisson process. For each arriving customer, the service provider announces a price rate and a system wait time, and the customer decides whether to join the queue, and, if so, the duration of the service time. The objective is to maximize either the long-run average revenue or social welfare. We formulate this problem as a continuous-time control model whose optimality conditions involve solving a set of delay differential equations. We develop an innovative method to obtain the optimal control policy, whose structure reveals interesting insights. The optimal dynamic price rate policy is not monotonic in wait time. In particular, in addition to the congestion effect often reported in the literature, i.e., the optimal price rate increases in the queue length (measured by the wait time in our setting), we find a compensation effect, meaning that the service provider should lower the price rate when the wait time is longer than a threshold. Compared with the prevalent flat pricing policy, our optimal dynamic pricing policy improves the objective value through admission control, which, in turn, increases the utilization of the server. We use a real data set obtained from a public charging station to calibrate our model with an objective of maximizing the average revenue. We find that our optimal pricing policy outperforms the best flat pricing policy, especially when the arrival rate is low and drivers are impatient. Interestingly, our revenue-maximizing pricing policy also improves social welfare over the flat pricing policy in most of the tested cases.