Macroscopic fundamental diagram with volume-delay relationship: model derivation, empirical validation and invariance property

TL;DR

This paper develops a new macroscopic fundamental diagram model incorporating volume-delay relationships using novel data sources, and proves an invariance property that aids in estimating network saturation without full traffic volume observability.

Contribution

It introduces a first-order nonlinear differential equation model for MFD with volume-delay, utilizing LPCs and RCIs, and establishes an invariance property for partial traffic volume observation.

Findings

Empirical MFD fits with R^2 > 0.9 using data from a 266 km^2 urban network.

The invariance property allows estimation of critical volume ratios from partial data.

The approach removes assumptions of full observability and known detection rates in traffic modeling.

Abstract

This paper presents a macroscopic fundamental diagram model with volume-delay relationship (MFD-VD) for road traffic networks, by exploring two new data sources: license plate cameras (LPCs) and road congestion indices (RCIs). We derive a first-order, nonlinear and implicit ordinary differential equation involving the network accumulation (the {\it volume}) and average congestion index (the {\it delay}), and use empirical data from a 266 km$^2$ urban network to fit an accumulation-based MFD with $R^2>0.9$. The issue of incomplete traffic volume observed by the LPCs is addressed with a theoretical derivation of the observability-invariant property: The ratio of traffic volume to the critical value (corresponding to the peak of the MFD) is independent of the (unknown) proportion of those detected vehicles. Conditions for such a property to hold are discussed in theory and verified empirically. This offers a practical way to estimate the ratio-to-critical-value, which is an important indicator of network saturation and efficiency, by simply working with a finite set of LPCs. The significance of our work is the introduction of two new data sources widely available to study empirical MFDs, as well as the removal of the assumptions of full observability, known detection rates, and spatially uniform sensors, which are typically required in conventional approaches based on loop detector and floating car data.