Existence, uniqueness, and trip cost function properties of user equilibrium in the bottleneck model with multiple user classes

Under relatively general assumptions a unique deterministic departure-time user equilibrium with a finite departure rate exists in the bottleneck model with drivers who differ in their unit costs of travel time, preferred times of arrival, and schedule delay cost functions. Existence requires that schedule delay cost functions be upper semicontinuous with respect to arrival time, and that schedule delay costs decline at a rate smaller than the unit cost of travel time. Uniqueness requires, more restrictively, that schedule delay cost functions be continuous. Several properties of equilibrium trip cost functions are derived for n groups of users with N-i in group i. The trip cost of a user in group i is a nondecreasing function of each N-j, but typically rises more quickly with respect to N-i than N-j, j not equal i. Thus, users experience lower trip costs when they travel with users unlike themselves than with an equal number of users like themselves.