Mathematical Modeling of Vehicular Traffic: A Discrete Kinetic Theory Approach

Abstract

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.

On the Modelling Crowd Dynamics From Scaling to Hyperbolic Macroscopic Models

“…In modelling dynamics of crowds motivation is also related to the control of panic situations or engineering structural safety as documented in the paper by Venuti et al 30 The contents of this paper in some ways related to traffic flow modelling documented in the review papers, by Helbing, 20 Bellomo et al, 5 Klar et al, 24 which is far more developed than that of crowds and swarms dynamics. Some recent papers on vehicular traffic flow modelling by kinetic theory with discrete velocities, 8,11 provide useful modelling of microscopic interactions that can be used also at the macroscopic level as well as to model pedestrian flows.…”

“…On the other hand, even in high density conditions the number of individuals in the crowd is never large enough to justify the continuous approximation of the distribution function. Two recent papers devoted to vehicular traffic flow modelling, 8,11 suggest to discretize the velocity space so that the assumption of continuity over the velocity variable is technically relaxed. Of course, the same reasoning should be applied to the space and activity variables.…”

From generalized kinetic theory to discrete velocity modeling of vehicular traffic. A stochastic game approach

“…The discrete velocity kinetic model of vehicular traffic [5] can be derived from the framework depicted by Eq. (6) up to a suitable microscopic modeling of the interactions among the vehicles.…”

“…Then in Section 3 specific modeling assumptions for vehicular traffic are outlined, resorting to some ideas of stochastic game theory introduced in [10] for closed systems (but see also [11] for recent advances of the theory for open systems), and a modeling framework is obtained. Finally, the formal derivation of the discrete velocity kinetic traffic model [5] from that framework is discussed and critically analyzed.…”

From the discrete kinetic theory to modelling open systems of active particles

“…Specifically, we mean actions addressed to controlling the system, for instance therapeutical actions in the case of biological systems [4], actions devoted to optimizing the flow conditions in the case of traffic flow models [8,9], and actions devoted to controlling economic systems [3]. Consider the specific actions expressed by the following terms: k (t) cause a proliferation or destruction in the state u h of the test particle.…”