On the uniqueness of equilibrated dynamic traffic flow patterns in unidirectional networks

Uniqueness of the dynamic user-equilibrium assignment is still an important issue. This paper proves uniqueness with a milder condition compared to past studies and shows another counterexample to the uniqueness. A unidirectional network, in which any node on any shortest route has a unique node potential, is introduced. Orders of vehicles are determined by this node potential so that, given any two vehicles passing through the same node, the lower potential vehicle arrives at the node before the higher potential vehicle. It is shown that, for a unidirectional network in equilibrium, the link travel times and traffic volumes of congested links are uniquely determined. Moreover, a simple non-unidirectional network having multiple equilibria is introduced. This example exhibits importance of unidirectional-network structure to prove uniqueness.