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Information about Traffic Flow

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 which expresses the probability of having a vehicle at time in position which runs with velocity . 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.

See also

External links

References

1. ^ [1]
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.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Traffic is the movement of motorized vehicles, unmotorized vehicles and pedestrians on roads. Traffic laws are the laws which govern traffic and regulate vehicles, while rules of the road
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Flow may refer to:
  • Flow (physics) is the flux times the area. This is the rate at which something travels through a given cross section.

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Traffic congestion is a condition on any network as use increases and is characterized by slower speeds, longer trip times, and increased queueing. The most common example is for physical use of roads by vehicles.
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Vehicles are non-living means of transport. They are most often man-made (e.g. bicycles, cars, motorcycles, trains, ships, and aircraft), although some other means of transport which are not made by man can also be called vehicles; examples include icebergs and floating tree trunks.
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A cluster is a small group or bunch of something.

In science:
  • Cluster (physics), a small group of atoms or molecules
  • In astrophysics:

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Structure formation refers to a fundamental problem in physical cosmology. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago.
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shock wave (or simply "shock") is a type of propagating disturbance. Like an ordinary wave, a shock wave carries energy and can propagate through a medium (solid, liquid or gas), or, in special cases, through a field such as the electromagnetic field in the absence of a
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In physics, density is mass m per unit volume V—how heavy something is compared to its size. A small, heavy object, such as a rock or a lump of lead, is denser than a lighter object of the same size or a larger object of the same weight, such as pieces of
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In fluid dynamics and hydrometry, the volumetric flow rate, also volume flow rate and rate of fluid flow, is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second [m3 s-1
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flux, both with rigorous mathematical frameworks.
  • In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the amount that flows through a unit area per unit time, the volumetric flow rate.

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freeway — also known as a highway, superhighway, autoroute, autobahn, autopista, autovía, autostrada, dual carriageway, expressway, or motorway — is a type of road designed for safer high-speed operation of motor vehicles through the elimination of at-grade
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Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion).
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In mathematics, a partial differential equation (PDE) is a type of differential equation, i. e. a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables.
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Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.
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The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of particles in a fluid.
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Empirical studies in social sciences are when the research ends are based on evidence and not just theory. This is done to comply with the scientific method that asserts the objective discovery of knowledge based on verifiable facts of evidence.
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The Transportation Research Board (TRB) is a division of the National Research Council, which serves as an independent adviser to the President, the Congress and federal agencies on scientific and technical questions of national importance.
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National Academy of Sciences (NAS) is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine."

Overview

Origin

The Civil War caused a need for a national academy.
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The Fundamental Diagram of traffic flow is a diagram that gives a relation between the traffic flux (cars/hour) and the traffic density (cars/km). A macroscopic traffic model involving traffic flux, traffic density and velocity forms the basis of the fundamental diagram.
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Microsimulation is a research area in applied econometrics. It tries to simulate the behaviour of individuals over time. Microsimulation can either be dynamic or static. If it is dynamic the behaviour of people changes over time, whereas in the static case a constant behaviour is
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Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics.

In contrast to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, thus the dynamic variables of the models represent microscopic properties
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Road traffic control involves directing vehicular and pedestrian traffic around a construction zone, accident or other road disruption, thus ensuring the safety of emergency response teams, construction workers and the general public.
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Rule 184 is a one-dimensional binary cellular automaton rule, notable for solving the majority problem as well as for its ability to simultaneously describe several, seemingly quite different, particle systems:

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The three phase traffic theory is an alternative traffic theory developed by Boris Kerner. It is mainly concerned with the physics of traffic congestion at freeways. Kerner describes three phases, opposed to the classical theories based on the fundamental diagram of traffic flow,
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Traffic waves, also called stop waves or traffic shocks, are travelling disturbances in the distribution of cars on a highway. Traffic waves usually travel backwards in relation to the motion of the cars themselves, or "upstream.
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traffic counter is a device, often electronic in nature, used to count, classify, and/or, measure the speed of vehicular traffic passing along a given roadway. The device is usually deployed in near proximity to the roadway and uses an intrusive medium, such as pneumatic road tubes
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Traffic Pulse, also known as Traffic.com and Mobility Technologies, is a nationwide provider of traffic information via a number of media, including the Internet, cell phones, radio, satellite radio and television.
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The TIRTL (The Infra-Red Traffic Logger) is a type of traffic counter. It has the ability to record volume, speed, and classification on a bi-directional, multi-lane roadway. This product is produced by CEOS Pty Ltd and marketed by CEOS Industrial Pty Ltd.
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