Flow Density Models
A consistent problem of speed density models identified by Drake (1967) was that such models fit rather poorly to field observation at capacity. The only exception to this was Edie's two-regime model, who used two separate speed-density models for un-congested and congested operations and assumed a discontinuity at the capacity range. Such discontinuities are more apparent when flow and density data are considered. Therefore, a limited number of models have been proposed that describe the relationship between flow and density.
Many speed flow models were developed subsequent to the early Greenshields work, in an attempt to reconcile the differences between his work and field observations. Descriptions of these models are beyond the scope of this module.
Koshi et al. (1983) gave an empirically-based discussion of the flow-density relationship, in which they suggested that a reverse lambda shape was the best description of the data. These authors also investigated the implications of this type of flow density relationship for car-following models, as well as for wave propagation.
Hall et al (1986), based on data collected upstream a primary bottleneck, concluded that an inverted-V shape is a plausible representation of the flow-occupancy relationship. Banks (1989) verified the suggestion of the bilinear inverted-V proposition (Figure 8 below) and he offered a behavioral interpretation of the phenomenon: The inverted-V model implies that drivers maintain a roughly constant average time gap between their front bumper and the back bumper of the vehicle in front of them, provided their speed is less than some critical value. Once their speed reaches this critical value (which is as fast as they want to go), they cease to be sensitive to vehicle spacing.
Figure 8, The Inverted-V Flow-Density Relationship (Banks, 1989)