Investigating Factors for Existence of Multiple Equilibria in Dynamic Traffic Network

Uniqueness is an important characteristic of network user equilibrium. Uniqueness in dynamic user equilibrium (DUE) has been shown to exist under certain conditions, whereas a case with multiple Wardrop equilibria has been found in a twofold symmetric network with two bottlenecks. This setting is very special, and therefore, it is worthwhile to investigate the factors that are essential to obtain non-unique solutions in this case in order to discuss the implication of this special example for real network problems. The asymmetric structure of the network, stochastic user equilibrium (SUE), and whole-link model were employed to check whether these factors affect the existence of multiple equilibria. It was shown that placing bottlenecks with equivalent capacities causes the singularity of the linear equation system that describes Wardrop's first principle in a particular network structure called as a loopy network. Then, it was shown that this singularity causes the existence of multiple equilibria. On the other hand, this singularity was not found in SUE or the whole-link cases, and therefore, non-unique solutions were not found in them. It was also shown that the singularity does not appear to be the only source of non-uniqueness by presenting another example in which at least three equilibrium solutions exists when either the bottleneck model or the whole-link model is employed.