Stochastic stability of dynamic user equilibrium in unidirectional networks: Weakly acyclic game approach

The aim of this study is to analyze the stability of the dynamic user equilibrium (DUE) with fixed departure times in unidirectional networks. Specifically, the stochastic stability of the equilibrium, which is the concept of stability in a day-to-day dynamics subjected to stochastic effects, is herein considered. To achieve this, a new approach is developed by synthesizing the three concepts: the decomposition technique of DUE assignments, the weakly acyclic game, and the asymptotic analysis of the stationary distribution of the dynamics. Specifically, we first formulate the DUE assignment as a strategic game (DUE game), which deals with atomic users. We then prove that there exists an appropriate order of assigning users one by one to the network for ensuring an equilibrium. With this property, we prove that the DUE game is a weakly acyclic game, which is a generalization of potential games. The convergence and stochastic stability of the DUE game are then established based on the theory of weakly acyclic games. Finally, numerical experiments are conducted to validate these theoretical results.