A Macroscopic traffic flow model is a

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

traffic model A traffic model is a mathematical model of real-world traffic, usually, but not restricted to, road traffic. Traffic modeling draws heavily on theoretical foundations like network theory and certain theories from physics like the kinematic wave mode ...

that formulates the relationships among

traffic flow In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devi ...

characteristics like

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

, flow, mean speed of a traffic stream, etc.. Such models are conventionally arrived at by integrating microscopic traffic flow models and converting the single-entity level characteristics to comparable system level characteristics. An example is the

Two-fluid model Two-fluid model is a macroscopic traffic flow model to represent traffic in a town/city or metropolitan area, put forward in the 1970s by Ilya Prigogine and Robert Herman. There is also a two-fluid model which helps explain the behavior of superfl ...

. The method of modeling traffic flow at macroscopic level originated under an assumption that traffic streams as a whole are comparable to fluid streams. The first major step in macroscopic modeling of traffic was taken by Lighthill and Whitham in 1955, when they indexed the comparability of ‘traffic flow on long crowded roads’ with ‘flood movements in long rivers’. A year later, Richards (1956) complemented the idea with the introduction of ‘

shock-wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...

s on the highway’, completing the so-called LWR model. Macroscopic modeling may be primarily classified according to the type of traffic as homogeneous and heterogeneous, and further with respect to the order of the mathematical model.

References

* M.J.Lighthill, G.B.Whitham, On kinematic waves II: A theory of traffic flow on long, crowded roads. Proceedings of the Royal Society of London Series A 229, 317–345, 1955 * P.I.Richards, Shock waves on the highway, Operations Research 4, 42–51., 1956 * M.Papageorgiou, Some remarks on macroscopic traffic flow modeling,Elsevier Science Ltd., Vol. 32, No. 5, pp. 323 to 329, 1998 * C.F.Daganzo, Fundamentals of transportation and traffic operations, Elsevier Science Ltd., 1997 * M.Di Francesco, M.D.Rosini, Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit, Archive for Rational Mechanics and Analysis, 2015{{Cite arXiv , eprint = 1404.7062v1, author1 = Marco Di Francesco, title = Rigorous derivation of the Lighthill-Whitham-Richards~model from the follow-the-leader model as many particle limit, last2 = Rosini, first2 = Massimiliano D., class = math.AP, year = 2014 Road traffic management Mathematical modeling Traffic flow