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

Abstract

a b s t r a c tThis work reports on vehicular traffic modeling by methods of the discrete kinetic theory. The purpose is to detail a reference mathematical framework for some discrete velocity kinetic models recently introduced in the literature, which proved capable of reproducing interesting traffic phenomena without using experimental information as modeling assumptions. To this end, we firstly derive a general discrete velocity kinetic framework with binary nonlocal interactions. Then, resorting to some ideas of stochastic game theory, we outline specific modeling guidelines for vehicular traffic, and finally we discuss the derivation of the above-mentioned vehicular traffic models from these mathematical structures.

Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach

“…Let us introduce the following compact notations: w := (v, u), I = [0, 1] ⊂ R. Then the expression of the gain operator is as follows (see e.g., [29]):…”

A fully-discrete-state kinetic theory approach to traffic flow on road networks

“…Equations (20)-(22) provide a complete set of transmission conditions for the 2-1 junction up to determining the parameter p. To this end we focus on the case q 1 m1 + q 2 m2 ≤ p noticing that, according to Eq. (20), the density at the entrance of road r = 3 is (cf. Eq.…”

“…When q 1 m1 +q 2 m2 > p the 2-1 junction reduces to a 1-1 junction plus a disconnected road. The choice (20) is such that the corresponding solution is, once again, the same as that of a single road without junction, cf. Section 2.2.1.…”

“…Moreover, to fix the ideas and without loss of generality, we also assume that the road with right-ofway is r * = 1. Thus the transmission conditions (20), (21), (22) become:…”

“…These works were primarily concerned with either a Boltzmann or an Enskog-type treatment of vehicular flow inspired by analogies with classical gas dynamics. More recently new contributions have appeared [3,5,9,10,11,20], which use the generalized kinetic and stochastic game theory of active particles [2] together with discrete microscopic states of the vehicles. This way they successfully capture both the implicit stochasticity of human behaviors, hence ultimately of microscopic car interactions, and the intrinsic microscopic granularity of the distribution of vehicles along a road, which indeed do not properly form a continuum even in congested situations.…”