On the mathematical theory of vehicular traffic flow II: Discrete velocity kinetic models

Abstract

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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.…”

Intelligent Transportation System Based Traffic Congestion Modeling for Urban Corridor

“…Conventionally, O-D matrices are used for trip planning, traffic management and operation studies [11]. Additionally, even though at macroscopic level traffic flow theory is studied in analogy with fluid dynamics [6], the essence of entry and exit traffic parameters in typical traffic streams has not been appreciated. This may be attributed to the lack of apparent prominent relationship of these factors with congestion as opposed to the more basic characteristics pertinent to its explicit definition.…”

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.…”