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The mathematical study of traffic flow, and in particular vehicular traffic flow, is done with the aim to get a better understanding of these phenomena and to assist in prevention of traffic congestion problems. The first attempts to give a mathematical theory of traffic flow dated back to the 1950s, but to this day we still do not have a satisfactory and general theory to be applied in real flow conditions.

This is because traffic phenomena are complex and nonlinear, depending on the interactions of a large number of vehicles. Moreover, vehicles do not interact simply following the laws of mechanics, but also due to the reactions of human drivers. In particular, they show phenomena of cluster formation and forward and backward-propagating shock waves of vehicle density. Fluctuations in measured quantities (e.g. mean velocity of vehicles) are often huge, leading to a difficult understanding of experiments.

Vehicular traffic flow analysis is made more complicated by the "sideways parabola" shape of the speed-flow curve. As the total number of vehicles operating on a roadway reaches the maximum flow rate (or rather: flux), at densities beyond a point known as the "optimum density" the traffic flow becomes unstable. At that point even a minor incident can lead to a breakdown in traffic flow, resulting in persistent stop-and-go driving conditions. Estimates of jam density, the density associated with completely stopped traffic flow, are in the range of 185-250 vehicles per mile per lane, while optimum densities for freeways are typically 40-50 vehicles per mile per lane.

Scientists approach the problem in mainly three ways, corresponding to the three main scales of observation in physics.

• microscopic scale: at a first level, every vehicle is considered as an individual, and therefore for everyone is written an equation, that is usually an ODE.
• macroscopic scale: in analogy with fluid dynamics models, it is something more useful to write a system of (PDE) balance laws for some gross quantities of interest, e.g the density of vehicles or their mean velocity.
• mesoscopic (kinetic) scale: a third, intermediate, possibility, is to define a function f(t,x,V) which expresses the probability of having a vehicle at time t in position x which runs with velocity V. This function, following methods of statistical mechanics, can be computed solving an integro-differential equation, like the Boltzmann Equation.

The engineering approach to analysis of highway traffic flow problems is primarily based on empirical analysis (i.e. observation and mathematical curve fitting). One of the most comprehensive references on this topic is the Highway Capacity Manual ^[1] published by the Transportation Research Board (TRB), which is part of the United States National Academy of Sciences.

A survey about the state of art in traffic flow modelling:

• N. Bellomo, V. Coscia, M. Delitala, On the Mathematical Theory of Vehicular Traffic Flow I. Fluid Dynamic and Kinetic Modelling, Math. Mod. Meth. App. Sc., Vol. 12, No. 12 (2002) 1801-1843

A useful book from the physical point of view:

• B. Kerner, The Physics of traffic, Springer Verlag (2004)
• Traffic flow on arxiv.org
• May, Adolf. Traffic Flow Fundamentals. Prentice Hall, Englewood Cliffs, NJ, 1990.

• Fundamental diagram of traffic flow
• Microsimulation
• Microscopic traffic flow model
• Road traffic control
• Rule 184
• Three phase traffic theory
• Traffic wave
• Traffic counter
• Traffic Pulse
• TIRTL

1. ^ http://www.trb.org/news/blurb_detail.asp?id=1166

• Highway Capacity Manual (HCM)

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