Socially and Individually Optimal Control of Arrivals to a GI/M/1 Queue

We study the problem of controlling the arrivals to a single-server queue with general interarrival-time distribution and exponential service-time distribution (a GI/M/1 system). Control is exercised by accepting or rejecting arriving customers or by charging entering customers a toll or entrance fee. An entering customer receives a reward, representing his utility of service. The rewards vary randomly from customer to customer. There is a holding cost, which is a convex function of the number of customers in the system. We compare socially and individually optimal entering policies when the decision horizon is finite, specifically, when the system must close after n future arrivals. There may be a terminal reward or cost, incurred as a function of the number of customers present at the end of the horizon, representing, for example, the holding cost incurred until all of them are served. We show that a socially optimal policy is less likely to accept a customer than an individually optimal policy, and both policies are less likely to accept as the number of customers already present increases, the horizon length increases, or the discount rate decreases. It is shown that, contrary to intuition, the optimal congestion toll cannot be monotonic in the number of customers present. These properties carry over to the infinite horizon, both with and without discounting, except that in the latter case, in which the criterion is long-run average net benefit, the toll can be monotonic. Implications for the study of rush-hour or peak-load phenomena in various congestion systems are briefly discussed.