On Vlasov-Fokker-Planck Type Kinetic Models for Multilane Traffic Flow

Abstract

Abstract. We discuss rationales for kinetic descriptions of traffic dynamics and present a class of new models of VlasovFokker-Planck type. These models incorporate (nonlocal and time-delayed) braking and acceleration terms which are consistent with realistic time scales. Correlation assumptions are made such that braking and acceleration terms depend only on macroscopic densities and the relative speeds with respect to the average speed; the braking term includes lane change probabilities, and reasonable assumptions on the dependencies of these probabilities on the traffic situation lead to multivalued fundamental diagrams, consistent with traffic observations.

A Traffic-Flow Model With Constraints for the Modeling of Traffic Jams

“…(i) Microscopic models or Carfollowing models e.g. [11,2]: they are based on supposed mechanisms describing the process of one vehicle following another; (ii) Kinetic models [22,20,19,14,18,13]: they describe the dynamics of the velocity distribution of vehicles, in the traffic flow; (iii) Fluid-dynamical models [17,23,21,9,22,1,25,6,8,5,4]:…”

A Vehicular Traffic Flow Model Based on a Stochastic Acceleration Process

“…The velocity part of this density is well known in kinetic traffic flow theory and used there as an argument for the existence of bimodal velocity distributions [31,37]. There, [31], the factor 1/2 is replaced by a more complex function of the distance probability at v = w, but this difference is mainly due differences in the interaction model.…”

“…Let the lower α-value is identified to a more aggressive driver and the higher α-value to a more conservative one, a traffic flow of a mixture of both produce traffic densities in the area between both curves. This could be a first guess to multivalued traffic flows in the framework of this model as discussed in literature [50,37].…”

“…One of them is to introduce a continuous velocity change due to a mean acceleration and a mean velocity scattering. Then the single car state probability density is described by a Vlasov-Fokker-Planck type equation [27,37]. Another possibility is to introduce the acceleration variable into the underlying stochastic process to be the main driver control variable of the interaction, changed in a discontinuous manner.…”

On a Functional-Differential Equation Arising from a Traffic Flow Model

“…The model which leads to Equation (1.1) is of macroscopic type, and earlier versions of macroscopic models and their analysis may be found in [1,8,9,10,11,17,18,23,30]. Kinetic models are a possible bridge between microscopic/ stochastic and macroscopic models, as discussed in [20,5,21].…”