The most widely used delay function for both controlled and uncontrolled road segments is the BPR function:
where X is the volume-to-capacity ratio, to is the free travel time, and a and P are empirical coefficients. Many practitioners recommend that capacity be taken as the design volume for the link, normally LOS C. Other practitioners recommend computing X with ultimate capacity. When X is calculated with ultimate capacity, it is possible to approximate a from the free speed, so, and the speed at capacity, Sc. That is,
thereby effectively reducing this function to one with a single parameter, P.
Spiess (1990) has identified seven standards for speed volume functions:
If these standards are met, then it is assured that an equilibrium can be found with Frank-Wolfe decomposition, that the model is easily calibrated, and that the computational effort will be modest. The BPR function meets the first six standards.
Standard 2 assumes that speed at capacity is always one-half of free speed. Unfortunately, Spiess ignored the rest of the speed/volume function, so standard 2 should be revised to read:
The revised second standard is required to retain realistic assignments and to provide good path travel times for the trip distribution and mode split steps. Spiess' third and seventh standards are unnecessary and would be inhibiting, if accuracy is of paramount importance.
Spiess proposed an alternative to the BPR function,
which may fit the various HCM delay/volume relationships more closely:
and X is the volume-to-capacity ratio. This function always yields a travel time at capacity of twice the free travel time - something which may not always be desirable. This function has the general shape of a hyperbola, and is referred to by Spiess as a conical delay function. It is very similar to a delay function developed by the Traffic Research Corporation in 1966 (Branston, 1976).
Still another alternative function with a single parameter has the form:
Like the BPR function, Equation 5 is assured to exactly fit the delay/volume curve at zero volume and capacity. This equation was proposed by Overgaard (1967). It meets Spiess' first six standards.
Networks originally prepared for Planpac and UTPS largely relied on the default coefficients of the BPR function (α=0.15 and β=4.0). With these coefficients, link capacity was set to design capacity, normally taken to be LOS C in earlier editions of the Highway Capacity Manual. More recent travel forecasting packages have generally retained these traditional coefficients and definition of link capacity. Technically, design capacity should be interpreted as the volume that causes free speed to drop by 15 percent. There are valid reasons for trying to retain this definition of capacity in previously calibrated networks.
Unfortunately, the 1985 Highway Capacity Manual does not provide a similarly simplistic relationship between service flow at LOS C and speed. In order to continue using the "design capacity" definition of link capacity, it would be necessary to establish a set of procedures to (1) find it and (2) assure that it yielded reasonable estimates of speed (or delay) at all feasible volumes.
It is possible to develop new parameters for the BPR curve (or another speed/volume function) using any reasonably consistent definition of capacity. There would be little difference in the quality of fits to speed and volume data. Consequently, the choice of a definition for capacity must be made on the grounds of convenience. There are four important arguments for defining link capacity to be ultimate capacity (LOS E for most facilities).
All three delay functions (Spiess', BPR, Overgaard's) were fit to the speed/volume relationships contained in the Highway Capacity Software, Version 1.5, which closely approximate those in the HCM. The coefficient, α, in the BPR function was determined by forcing the curve to fit the speed/volume data at zero volumes (free speed) and at capacity (LOS E). The second coefficient, β, was found by nonlinear regression. The single coefficients of Spiess' function and of Overgaard's function were also found by nonlinear regression. Table 2 summarizes the best coefficients.
It is seen that all three functions performed well, as judged by the standard deviation of the residuals, σv, and the percent of variance explained, R². The quality of the fit varied with the facility type and design speed. In general, it was easier to fit speed/volume functions when the design speed was 50 miles per hour. Spiess' function produced the most consistent results, explaining about 97% of the variance for all six facilities. It is likely that Spiess' function would yield even better results if the assumption about speed at capacity (Spiess' original standard 2) could be improved. Appendix B shows the HCM speed/volume functions for each facility and the best fitting functions.
The HCM provides three slightly different speed/volume curves for freeways with 70 mph design speeds - one each for 4-lane, 6-lane, and 8-lane segments. The curves for 4-lane and 8-lane segments differ from the one for 6-lane segments (used here) by at most 1 mile per hour. Consequently, there is little advantage to having three separate speed/volume functions for 70 mph segments.
It is possible to estimate delay at traffic controlled intersections with any of the three curves discussed in the previous section. Instead of fitting a speed/volume relationship, it is necessary to fit a travel-time/volume relationship, where travel-time is taken from the HCM signalized intersection delay formula. Examples of some nonlinear least-squares fits to HCM's delay formula are seen in Figure 2. The HCM delays are for an intersection with a 90 second cycle length, a 60 second green time, and a saturation flow rate of 5400 vph. It is seen that the BPR and Overgaard's functions can reasonably approximate the HCM formula, but Spiess' formula performs badly. (The BPR function parameters were a = 5.0 and P = 3.5.)
Although it is possible to fit a BPR curve to the HCM delay function, doing so would be undesirable for the following reasons:
A better approach, but one that requires considerable rewriting of software, is to calculate intersection delay directly from the HCM procedures, as described in previous sections and in Appendix A.