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Traffic flow, in mathematics and engineering, is the study of interactions between vehicles, drivers, and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal road network with efficient movement of traffic and minimal traffic congestion problems.

Attempts to produce a mathematical theory of traffic flow date back to the 1950s, but have so far failed to produce a satisfactory general theory that can be consistently applied to real flow conditions. Current traffic models use a mixture of empirical and theoretical techniques.

## Overview

Traffic phenomena are complex and nonlinear, depending on the interactions of a large number of vehicles. Due to the individual reactions of human drivers, vehicles do not interact simply following the laws of mechanics, but rather show phenomena of cluster formation and shock wave propagation, both forward and backward, depending on vehicle density in a given area.

In a free flowing network, traffic flow theory refers to the traffic stream variables of speed, flow, and concentration. These relationships are mainly concerned with uninterrupted traffic flow, primarily found on freeways or expressways."Optimum density" for U.S. freeways is sometimes described as 40–50 vehicles per mile per lane. As the density reaches the maximum flow rate (or flux) and exceeds the optimum density, traffic flow becomes unstable, and even a minor incident can result in persistent stop-and-go driving conditions. The term jam density refers to extreme traffic density associated with completely stopped traffic flow, usually in the range of 185–250 vehicles per mile per lane.

However, calculations within congested networks are much more complex and rely more on empirical studies and extrapolations from actual road counts. Because these are often urban or suburban in nature, other factors (such as road-user safety and environmental considerations) also dictate the optimum conditions.

## Methods of analysis

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 ordinary differential equation .
• 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(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 using an integro-differential equation, such as 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, 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 U.S. traffic models and the SATURN model in Europe.

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, transient "demand peaks" of congestion are simulated. These are modeled 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 is 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) in order 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 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. 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.

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 enable significantly more traffic flow 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.

## References

1. Highway Capacity Manual 2000
2. SATURN ITS Transport Software Site
3. Introduction to Contram
4. UK Department for Transport's WebTag guidance on the conduct of transport studies
5. VISUM overview

A survey about the state of art in traffic flow modelling:
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.
• Taylor, Nicholas.