Fundamental Diagrams of 1D-Traffic Flow by Optimal Control Models

TL;DR

This paper derives fundamental diagrams of 1D traffic flow using optimal control models, providing analytical relations between car density and flow, and extending classical models with stochastic and game-theoretic approaches.

Contribution

It introduces a novel analytical framework for deriving traffic fundamental diagrams using min-plus algebra and stochastic control models, extending traditional car-following models.

Findings

Analytical fundamental diagrams relating density and flow.

Extension of models to stochastic and game-theoretic frameworks.

Potential for improved interpretation of real traffic data.

Abstract

Traffic on a circular road is described by dynamic programming equations associated to optimal control problems. By solving the equations analytically, we derive the relation between the average car density and the average car flow, known as the fundamental diagram of traffic. First, we present a model based on min-plus algebra, then we extend it to a stochastic dynamic programming model, then to a stochastic game model. The average car flow is derived as the average cost per time unit of optimal control problems, obtained in terms of the average car density. The models presented in this article can also be seen as developed versions of the car-following model. The derivations proposed here can be used to approximate, understand and interprete fundamental diagrams derived from real measurements.