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,Fact: date=December 2008 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.Fact: date=January 2009 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 (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 $f(t,x,V)$ 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.