2010/02/10 by Nadir Farhi, Farhi, Nadir
Engineering · Mathematics · Social Sciences · #Dynamical Systems (math.DS) #Evacuation and Crowd Dynamics #FOS: Mathematics #Optimization and Control (math.OC) #Traffic control and management #Transportation Planning and Optimization #math.DS #math.OC
paper · pdf · doi:10.48550/arxiv.1002.2105
17 pages,
arxiv created 2010/02/10 · openalex publication_date 2010/02/10 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.