Continuous formulations and analytical properties of the link transmission model

TL;DR

This paper develops continuous formulations of the link transmission model (LTM), analyzes its stationary states and stability, and highlights its advantages over existing models for large-scale traffic network simulation.

Contribution

It introduces systematic continuous formulations of LTM using the Hopf-Lax formula and analyzes its analytical properties, including stationary states and stability.

Findings

Continuous LTM derived using Hopf-Lax formula

Stationary states characterized and analyzed

LTM stability assessed via Poincaré map

Abstract

The link transmission model (LTM) has great potential for simulating traffic flow in large-scale networks since it is much more efficient and accurate than the Cell Transmission Model (CTM). However, there lack general continuous formulations of LTM, and there has been no systematic study on its analytical properties such as stationary states and stability of network traffic flow. In this study we attempt to fill the gaps. First we apply the Hopf-Lax formula to derive Newell's simplified kinematic wave model with given boundary cumulative flows and the triangular fundamental diagram. We then apply the Hopf-Lax formula to define link demand and supply functions, as well as link queue and vacancy functions, and present two continuous formulations of LTM, by incorporating boundary demands and supplies as well as invariant macroscopic junction models. With continuous LTM, we define and solve the stationary states in a road network. We also apply LTM to directly derive a Poincar\'e map to analyze the stability of stationary states in a diverge-merge network. Finally we present an example to show that LTM is not well-defined with non-invariant junction models. We can see that Newell's model and LTM complement each other and provide an alternative formulation of the network kinematic wave model. This study paves the way for further extensions, analyses, and applications of LTM in the future.