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# Traffic flow

The mathematical or engineering study of traffic flow, and in particular vehicular traffic flow, is done with the aim of achieving a better understanding of these phenomena and to assist in the reduction 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. Current traffic models use a mixture of empirical and theoretical techniques.

Approaches

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 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 three main ways, corresponding to the three main scales of observation in physics.
* Microscopic scale: At the most basic level, every vehicle is considered as an individual, and therefore an equation is written for each, usually an ODE.
* Macroscopic scale: Similar to models of fluid dynamics, it is considered useful to employ a system of partial differential equations which 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 $f\left(t,x,V\right)$ 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 using 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 major references on this topic used by American planners is the "Highway Capacity Manual" [ [http://www.trb.org/news/blurb_detail.asp?id=1166 Highway Capacity Manual 2000 ] ] published by the Transportation Research Board, which is part of the United States National Academy of Sciences. This recommends modelling traffic flows using the whole travel time across a link using a delay/flow function, including the effects of queuing. This technique is used in many US traffic models and the SATURN model in Europe. [ [http://www.saturnsoftware.co.uk SATURN ITS Transport Software Site] ]

In many parts of Europe a hybrid empirical approach to traffic design is used, combining macro-, micro- and mesoscopic features. Rather than simulating a steady state of flow for a journey, they simulate transient "demand peaks" of congestion which they model by using small "time-slices" across the network throughout the working day or weekend. Typically the origins and destinations for trips are first estimated and a traffic model generated, before being calibrated by comparing the mathematical model with observed counts of actual traffic flows, classified by type of vehicle. "Matrix estimation" is then applied to the model to achieve a better match to observed link counts before any changes and the revised model is used to generate a more realistic traffic forecast for any proposed scheme. The model would be run several times, including a current baseline, an "average day" forecast based on a range of economic parameters, and supported by sensitivity analysis to understand the implications of temporary blockages or incidents around the network. From the models it is possible to total the time taken for all drivers of different types of vehicle on the network, and thus deduce average fuel consumption and emissions.

Much of the UK, Scandinavian and Dutch authority practice is to use the modelling program CONTRAM for large schemes, which has been developed over several decades under the auspices of the UK's Transport Research Laboratory, and more recently with the support of the Swedish Road Administration. [ [http://www.contram.com/about/introduction.shtml Introduction to Contram] ] By modelling forecasts of the road network for several decades into the future the economic benefits of changes to the road network can be calculated, using estimates for value of time and other parameters. The output of these models can then be fed into a cost benefit analysis program. [ [http://www.webtag.org.uk/overview/modelling.htm UK Department for Transport's WebTag guidance on the conduct of transport studies] ]

A major consideration in road capacity relates to the design of junctions. By allowing long "weaving sections" on gently curving roads at graded intersections vehicles can often move across lanes without causing significant interference to the flow. However this is expensive and takes up a large amount of land so other patterns are often used, particularly in urban or very rural areas. Most large models use crude simulations for intersections, but computer simulations are available to model specific sets of traffic lights, roundabouts, and other scenarios where flow is interrupted or shared with other types of road users or pedestrians. A well-designed junction can pass through significantly more traffic at a range of traffic densities during the day. By matching such a model to an "Intelligent Transport System", traffic can be sent in uninterrupted "packets" of vehicles at predetermined speeds through a series of phased traffic lights.The UK's TRL has developed junction modelling programs for small scale local schemes that can take account of detailed geometry and sight-lines; ARCADY for roundabouts, PICADY for priority intersections and OSCADY and TRANSYT for signals.

A common failing of road traffic models is that they do not take into account the effects of changes in public transport on the demand for road traffic; thus a new generation of traffic modelling software can now compare public transport with private road traffic, and thus help inform demand forecasts. [ [http://www.english.ptv.de/cgi-bin/traffic/traf_visum.pl VISUM overview] ]

*Dijkstra's algorithm
*Fundamental diagram of traffic flow
*Microscopic traffic flow model
*Microsimulation
*Rule 184
*Three phase traffic theory
*TIRTL
*Traffic wave
*Traffic counter

References

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) 1801-1843A useful book from the physical point of view:
* B. Kerner, "The Physics of traffic", Springer Verlag (2004)
* [http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=cond-mat&level=2&index1=28 Traffic flow on arxiv.org]
*May, Adolf. "Traffic Flow Fundamentals". Prentice Hall, Englewood Cliffs, NJ, 1990.
* Taylor, Nicholas. " [http://www.contram.com/download/NETS_CONTRAM_DTA.pdf The Contram dynamic traffic assignment model] " TRL 2003

* [http://trb.org/news/blurb_detail.asp?id=1166 Highway Capacity Manual (HCM)]

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