Fundamental theorems of traffic flow

TL;DR

This paper establishes four fundamental theorems to mathematically describe the kinematics and properties of traffic flow, providing a theoretical framework for understanding vehicle interactions and flow stability.

Contribution

It introduces four new theorems that define critical distances, spacing margins, and conditions for constant headway, forming a foundational theoretical framework for traffic flow analysis.

Findings

Derived critical vehicle spacing distances

Established conditions for constant time headway

Outlined the governing equations of vehicle spacing and flow stability

Abstract

To gain essential understandings of traffic flow, four theorems are derived to establish the kinematics of the basic unit of traffic flow, namely two consecutive vehicles. The first is to determine the two critical distances of the vehicle spacing, the second is to derive the spacing margin and the critical distance difference as well as the relationship they must satisfy, the third is to complete the mathematical definition and give the necessary and sufficient condition for the time headway to remain constant, and the fourth is to derive the governing equation of the spacing margin and determine the two key constraint spaces of the motion of consecutive vehicles. On this basis, the composition of the traffic flow phenomenon and that of the traffic flow property are determined, and the inherent regularity of traffic flow is briefly elucidated. Thus, a basic theoretical framework specific to the study of traffic flow is obtained. Based on the framework, three important issues, namely traffic flow instability, phase, and empirical datasets, are briefly reviewed.