Speed-Flow-Density Relationships

Speed-Density Model

This subsection deals with mathematical models for the

speed-density relationship, going back to as early as 1935. Greenshields’ (1935) linear model of speed and density was mentioned in the previous section. . . . The most interesting aspect of this particular model is that its empirical basis consisted of half a dozen points in one cluster near free-flow speed, and a single observation under congested conditions. . . . The linear relationship comes from connecting the cluster with the single point. . . . What is surprising is not that such simple analytical methods were used in 1935, but that their results (the linear speed-density model) have continued to be so widely accepted for so long. While there have been studies that claimed to have confirmed this model ¼ they tended to have similarly sparse portions of the full range of data, usually omitting both the lowest flows and flow in the range near capacity. . . .

A second early model was that put forward by Greenberg (1959), showing a logarithmic relationship:

His paper showed the fit of the model to two data sets, both of which visually looked very reasonable. However, the first data set was derived from speed and headway data on individual vehicles, which "was then separated into speed classes and the average headway was calculated for each speed class". In other words, the vehicles that appear in one data point(speed class) may not even have been traveling together! While a density can always be calculated as the reciprocal of average headway, when that average is taken over vehicles that may well not have been traveling together, it is not clear what that density is meant to represent. . . .

Duncan (1976, 1979) showed that the tree step procedure of (1) calculating density from speed and flow data, (2) fitting a speed-density function to that data, and then (3) transforming the speed-density function into a speed-flow function results in a curve that does not fit the original speed-flow data particularly well. . . . Duncan’s 1979 paper expanded on the difficulties to show that minor changes in the speed-density function led to major changes in the speed-flow function. This result suggests the need for further caution in using this method of double transformations to calibrate a speed-flow curve. . . .

The car-following models gave rise to four of the speed-density models tested by Drake et al. The results of their testing suggest that the speed-density models are not particularly good. Logic says that if the consequences of a set of premises are shown to be false, then one (at least) of the premises is not valid. It is possible, then, that the car-following models are not valid for freeways. This is not surprising, as they were not developed for this context.

Flow-Concentration Model

Although Gerlough and Huber did not give the topic of flow-concentration models such extensive treatment as they gave the speed-concentration models, they nonetheless thought this topic to be very important. . . .

Edie was perhaps the first to point out that empirical flow-concentration data frequently have discontinuities in the vicinity of what would be maximum flow, and to suggest that therefore discontinuous curves might be needed for this relationship. . . .

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 phenomenon for car-following models, as well as for wave propagation.

. . . there appears to be strong evidence that traffic operations on a freeway can move from one branch of the curve to the other without going all the way around the capacity point. This is an aspect of traffic behavior that none of the mathematical models . . . either explain or lead one to expect. Nonetheless, the phenomenon has been at least implicitly recognized since Lighthill and Witham’s (1955) discussion of shock waves in traffic, which assumes instantaneous jumps from one branch to the other on a speed-flow or flow-occupancy curve. As well, queuing models (e.g. Newell 1982) imply that immediately upstream from the back end of a queue there must be points where the speed is changing rapidly from the uncongested branch of the speed-flow curve to that of the congested branch. It would be beneficial if flow-concentration (and speed-flow) models explicitly took this possibility into account.

One of the conclusions of the paper by Hall et al. (1986), . . . is that an inverted ‘V’ shape is a plausible representation of the flow-occupancy relationship. Although that conclusion was based on limited data from near Toronto, Hall and Gunter (1986) supported it with data from a larger number of stations. Banks (1989) tested their proposition using data from the San Diego area, and confirmed the suggestion of the inverted ‘V’. He also offered a mathematical statement of this proposition and a behavioral interpretation of it (p. 58): The inverted-V model implies that drivers maintain a roughly constant average time gap between their front bumper and 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. . . .