Continuity of the path delay operator for dynamic network loading with spillback

TL;DR

This paper proves the continuity of path delay operators in dynamic network loading models based on the Lighthill-Whitham-Richards framework, accounting for vehicle spillback and ensuring no gridlock occurs within finite time.

Contribution

It establishes the mathematical continuity of path delay operators in DNL models with spillback using PDAEs and wave-front tracking, a novel theoretical result.

Findings

Proves continuity of path delay operators in DNL models with spillback.

Shows gridlock cannot occur in finite time in the model.

Provides estimation of minimum network supply without numerical methods.

Abstract

This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill-Whitham-Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations (PDAEs). We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model.