Three-Dimensional Models
The idea of considering all three fundamental variables (q, k, v) simultaneously first appeared in the TRB SR-165. The notion of a three-dimensional model appeared in the form of Figure 11, where "v=q/k represents the surface of admissible traffic stream models." The surface shown in Figure 9 (below) is a continuous one, thus by accepting that the u = q/k relationship holds in the entire range of traffic operations one can reasonably conclude that it suffices to study traffic modeling as a two-dimensional problem. However, empirical observations are rarely in accord with the relationship q=v k, especially when the observations are taken during congested conditions. Hence focusing only on the two-dimensional relationships will not often provide even implicitly a valid three-dimensional relationship.
The recognized discontinuities of the speed, density, flow surface lead in the adoption of the mathematics of catastrophe theory in traffic flow modeling. Catastrophe theory deals with models where while most of the variables being change in a continuous fashion, at least one of them can make sudden discontinuous changes, referred to as catastrophes. Navin (1986) and Gilchrist and Hall (1989) suggested that the three-dimensional 'cusp' catastrophe model was appropriate for the three traffic variables.
Figure 9: 3-D surface of traffic stream admissible models