On Microscopic, Macroscopic, and Kinetic Modeling of Vehicular Traffic

Abstract

A system of vehicular traffic can be modeled in various ways, such as microscopic, macroscopic, and kinetic models. Microscopic models focus on the modeling of individual cars with their deterministic or stochastic interactions. On the other hand, with macroscopic models we can determine the relation between some observable values of a vehicular system, such as density and flux, which usually have the form of partial differential equations of conservation type. In kinetic models, the traffic is resembled as a system of interacting gas particles described by a distribution function with a time evolution. This mesoscopic model is formulated in a Boltzmann-like equation, based on corresponding gain and loss terms of microscopic interactions. These above mentioned modeling approaches have actually interrelations with one another, such that microsopic behaviour of vehicles will imply certain macroscopic state of the traffic system and, by using kinetic model, the detailed transition from microscopic to macroscopic state can be explained. This paper is intended to present a study on this interrelations between microscopic, macroscopic, and kinetic models based on the optimal velocity model. Some numerical simulation results are also presented. Keywords: Vehicular Traffic; Microscopic Interaction; Macroscopic Model; Kinetic Model 2010 Mathematics Subject Classification: 90B20