A diffusion-based analysis of a multi-class road traffic network

TL;DR

This paper introduces a stochastic, multi-class traffic flow model on arbitrary networks, deriving fluid and diffusion limits to accurately approximate vehicle densities and travel times, validated through numerical experiments.

Contribution

It develops a novel multi-class stochastic model with Gaussian approximations for vehicle densities and travel times on complex networks, extending classical deterministic models.

Findings

Gaussian process approximations are highly accurate for vehicle densities.

Efficient computation of means and variances for traffic metrics.

Numerical experiments confirm the model's practical usefulness.

Abstract

This paper studies a stochastic model that describes the evolution of vehicle densities in a road network. It is consistent with the class of (deterministic) kinematic wave models, which describe traffic flows on the basis of conservation laws that incorporate the macroscopic fundamental diagram (a functional relationship between vehicle density and flow). Our setup is capable of handling multiple types of vehicle densities, with general macroscopic fundamental diagrams, on a network with arbitrary topology. Interpreting our system as a spatial population process, we derive, under a natural scaling, fluid and diffusion limits. More specifically, the vehicle density process can be approximated with a suitable Gaussian process, which yield accurate normal approximations to the joint (in the spatial and temporal sense) vehicle density process. The corresponding means and variances can be computed efficiently. Along the same lines, we develop an approximation to the vehicles' travel-time distribution between any given origin and destination pair. Finally, we present a series of numerical experiments that demonstrate the accuracy of the approximations and illustrate the usefulness of the results.