Modeling and solving continuous‐time instantaneous dynamic user equilibria: A differential complementarity systems approach

This paper is the second of a two‐part research wherein we undertake a mathematically rigorous investigation of the continuous‐time dynamic user equilibrium (DUE) problem using the recently introduced mathematical paradigm of differential complementarity systems (DCSs). Based on the thorough study of continuous‐time single‐destination point‐queue models in the previous part, we first extend this special case to multiple destinations respecting the First‐In–First‐Out property of travel flows. A DCS with constant time delay is then introduced to formulate the continuous‐time model of instantaneous dynamic traffic equilibria (IDUE) with a fixed demand profile. We develop a time decomposition scheme based on link free flow travel times to convert the delay DCS to a series of DCSs without time delays that are solved by a numerical time‐stepping method. We provide rigorous numerical treatment of the time‐decomposed IDUE model, including solvability of the discrete‐time complementarity problems and convergence of the numerical trajectories to a continuous‐time solution. We present numerical results to validate the IDUE on a small network and also on the Sioux Falls network.