Equilibria on a Congested Transportation Network

Abstract

Network equilibrium models arise in applied contexts as varied as urban transportation, energy distribution, spatially separated economic markets, electrical networks and water resource planning. In this paper, we propose and study an equilibrium model for one of these applications, namely for predicting traffic flow on a congested transportation network. The model is quite similar to those that arise in most contexts of network equilibria, however, and the methods that we use are applicable in these other settings as well. Our transportation model includes such features as (i) multiple modes of transit, (ii) link interactions and their effect on congestion, (iii) limited choices (or perceptions) of paths for flow between any origin-destination pair, (iv) generalized cost or disutility for travel, and (v) demand relationships for travel between origin-destination pairs that depend upon the travel time (cost) between all other origin-destination pairs. Using Brouwer’s fixed-point theorem, we establish existence of an equilibrium solution to the model. By imposing monotonicity conditions on the delay and demand functions, we also show that travel times (costs) are unique and, in certain instances, that link flows are unique.