Mathematical modeling of vehicular traffic: a discrete kinetic theory approach

Following some general ideas on the discrete kinetic and stochastic game theory proposed by one of the authors in a previous work, this paper develops a discrete velocity mathematical model for vehicular traffic along a one-way road. The kinetic scale is chosen because, unlike the macroscopic one, it allows to capture the probabilistic essence of the interactions among the vehicles, and offers at the same time, unlike the microscopic one, the opportunity of a profitable analytical investigation of the relevant global features of the system. The discretization of the velocity variable, rather than being a pure mathematical technicality, plays a role in including the intrinsic granular nature of the flow of vehicles in the mathematical theory of traffic. Other important characteristics of the model concern the gain and loss terms of the kinetic equations, namely the construction of a density-dependent table of games to model velocity transitions and the introduction of a visibility length to account for nonlocal interactions among the vehicles.