Guarantees of convergence to a dynamic user equilibrium for a single arc network

Abstract

While steady state traffic equilibrium problems have been successfully studied, the analysis complicates when the dynamics of the vehicles is taken into account. The literature on the topic presents a variety of models, but usually the algorithms suggested do not possess convergence guarantees. We propose a simple game where all the vehicles travel along the same arc and choose the starting time of their trip. Based on the first Wardrop principle, we formulate the traffic user equilibrium as a variational inequality. Since monotonicity of the corresponding operator guarantees convergence of gradient-based algorithms, we provide theoretical guarantees for short time horizons, and analyze it numerically for longer ones. We conclude with simulations showing that convergence can be achieved also in more general setups, for example with multiple origins or destinations.