# Empirical analysis of the variability in the flow-density relationship for smart motorways

## TL;DR

This study empirically analyzes the variability in the flow-density relationship on smart motorways, highlighting the influence of variable speed limits and identifying distinct flow regimes with implications for traffic safety.

## Contribution

It provides an empirical characterization of flow-density variability, assesses the impact of variable speed limits, and identifies distinct flow regimes with different safety implications.

## Key findings

- Flow and density have a strongly bimodal joint distribution.
- Variable speed limits significantly affect high-density traffic regimes.
- Three distinct flow-density clusters relate to flow breakdown and accident rates.

## Abstract

The fundamental diagram is an assumed functional relationship between traffic flow and traffic density. In practice, this relationship is noisy and exhibits significant statistical variability. On smart motorways, this variability is increased by variable speed limits that are not captured by the fundamental diagram. To study this variability, it is appropriate to consider the joint probability distribution function (pdf) of density and flow. We perform an empirical study of the variability in the relationship between flow and density using 74 days of data from 64 sections of London's M25. The objectives are to determine how much of the variability in the flow-density relationship results from variable speed limits and to assess whether particular functional forms of the fundamental diagram are systematically preferred. Empirically, the joint pdf of flow and density is strongly bimodal, illustrating that traffic flows are often found in high-density or low-density regimes but rarely in between. We find that the high-density regime is strongly affected by variable speed limits whereas the low-density regime is not. The Daganzo-Newell (triangular) model of the fundamental diagram systematically fits best to the data. However, the optimal parameters vary with location. Clustering analysis of these parameters suggests three qualitatively different types of flow-density relationships applying to different sections of the M25. These clusters have natural interpretations in terms of the frequency and severity of flow breakdown. Accident rates also depend on cluster type suggesting possible links to other properties of traffic flows beyond the flow-density relationship.