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 $f(t,x,V)$ 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, 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 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.