Individual versus Social Optimization in Exponential Congestion Systems

We consider a stochastic congestion system modeled as a birth-death process. Customers arrive from a Poisson process. The departure rate when i customers are in the system is non-decreasing, concave, and bounded above in i. The cost structure consists of a linear holding cost and a random reward received when a customer enters the system. The system can be controlled by deciding which customers will enter. Our main result (extending those of Naor, Yechiali, and Knudsen) is that, with or without discounting and for a finite or infinite time horizon, the individually optimal rule calls for the customer to enter the system whenever the socially optimal rule does. We also study the properties of the optimal congestion toll, which induces customers acting in their own interest to follow a socially optimal rule.