Side-Constrained Dynamic Traffic Equilibria

TL;DR

This paper investigates dynamic traffic equilibria with side-constraints, challenges existing assumptions with a counter-example, and introduces a new framework and existence results for non-convex volume-constrained models.

Contribution

It provides a counter-example to previous assumptions, proposes a novel framework based on feasible deviations, and establishes initial existence results for non-convex volume-constrained dynamic equilibria.

Findings

Counter-example shows non-convex feasible flow space.

Classical variational inequalities are inadequate for side constraints.

Existence of equilibria is proven under non-convex volume constraints.

Abstract

We study dynamic traffic assignment with side-constraints. We first give a counter-example to a key result from the literature regarding the existence of dynamic equilibria for volume-constrained traffic models in the classical edge-delay model. Our counter-example shows that the feasible flow space need not be convex and it further reveals that classical infinite dimensional variational inequalities are not suited for the definition of side-constrained dynamic equilibria. We propose a new framework for side-constrained dynamic equilibria based on the concept of feasible $\gamma$-deviations of flow particles in space and time. Under natural assumptions, we characterize the resulting equilibria by means of quasi-variational and variational inequalities, respectively. Finally, we establish first existence results for side-constrained dynamic equilibria for the non-convex setting of volume-constraints.