A Review of Properties of Flow--Density Functions

Flow--density functions are often described as the fundamental relationship of traffic theory and are the basis of kinematic wave models or hydrodynamic models which are used to describe, predict or analyse traffic behaviour. We set out several different properties that it is generally agreed that flow--density functions should satisfy. We then take the many forms of flow--density functions that have been proposed or used over the past several decades, derive the properties of each one and consider whether, or to what extent, they satisfy each of the desirable properties. We find that few, if any, flow--density functions satisfy all of the desirable properties. The main reasons for this are that, in almost all cases, there are not enough independent parameters in the flow--density functions to capture all of the desirable properties and that, in almost all cases, the flow--density functions are specified as a single analytic function rather than a different function for each range of traffic behaviours, such as free-flow, congested and heavily congested traffic.