The name of the game in traffic modeling is congestion and delay. More specifically, the task of the modeler is to reasonably quantify these notions into the driver's perceptions of travel times on links (streets) and delays at nodes (intersections).

TMODEL2 uses link and node capacity constraint functions which relate the travel times and delays to the volume to capacity (V/C) ratio. The link function used is taken from the Bureau of Public Roads and the node function is derived using data from Transportation Research Circular 212.

Perhaps you have noticed on occasion that TMODEL2 (and other models) will asign traffic in such as way as to cause V/C to exceed 1.0 and sometimes reach even 2.0 or 3.0. While this can occasionally be attributed to distribution and assignment of increments which are too large, it is more often due to the failure of the model functions to produce adequate delay.

There is some theoretical satisfaction in high V/Cs in that it is a recognition of some serious travel demands. However, operationally, there is no sense in those high values.

If we define capacity as the maximum traffic flow a facility can achieve, then exceeding capacity should not be possible. More to the point for our purposes, however, is the need to be able to calculate operational speeds or travel times over streets and delays encountered at intersections according to some function of V/C ratios.

We have long felt that the functions TMODEL2 (and other models) used were not adequate. In the March 1992 issue of TMODEL Reports we printed an article by Steve Schooley detailing some research he had done to develop better node delay parameters for signalized intersections. Even with increased parameter values, however, these functions failed to develop much delay at V/Cs under 1.0. This is exacerbated by the fact that V/C is raised to an exponent and this further diminishes the influence of V/Cs under 1.0. Several users have also commented that these functions provided more delay at very low V/C ratios than desired.

While we've had discussions on this matter with many people, the impetus for change finally came from Murray Totland in Saskatoon, Saskatchewan, and Matt Whitney from IMC Consulting in Cambridge, Ontario.

The core of the functions used in TMODEL2 for both street and intersections times or delays is K*(V/C)^E. The problem with this exponential function is that until V/C starts exceeding a value of 1.0, the rise in its calculated value is suppressed by the exponent rather than increased by it. Its effectiveness in that form depends mostly on K for V/Cs up to 1.0. To correct this constant, K₂ is added to V/C within the parentheses.

Next it was recognized that while the values used in this function can work very well for V/Cs up to the .80 to .95 area, the delays produced for V/Cs above that were not sufficiently draconian.

Therefore, it was suggested that an upper limit be instituted for that function and that a more severe set of function parameters be applied above that line.

The result is a combined function in which the "A" paramters [K_1A, K_2A, and E_A] apply at or below the upper limit (UL) and the "B" paramters [K_1B, K_2B, and E_B] apply above the UL.

The accompanying graph shows a combined function (labeled New Function) which very closely approximates the travel time increase which can be derived from the Speed-flow relationships under ideal conditions shown in Figure 3-4 in SR-209, the 1985 Highway Capacity Manual.

That can be compared with the BPR function using the default values shown on the graph.

The function can be effectively continuous if the UL breakpoint is set where the functions cross over or provide equal values. One easy way to do this is to find functions which adequately describe your curve on both sides of the UL in which

In other words, the exponents, E_A and E_B are the only differences.

While this still will not adequately describe BOTH stable AND unstable flows, it should work well for the goals of planning roadways at levels of service which are not dysfunctional.

TMODEL2 allows the modeler to set up different sets of link delay coefficients and ULs for different classes of roads for up to 100 classes.

We subscribe to the KISS philosophy. "Keep It Simple, Smarty!" Keep in mind, if you are modeling for 10 or 20 years in the future that if you go into extreme detail in the calibration (e.g., using 37 classes for link delay coefficients), you should be able to assume future knowledge to that level of detail as well.

The new node delay coefficients area provides an expanded set of parameters which are analogous to those in the link delay coefficients. The only difference is that it also includes base delay terms, BD_A and BD_B for each side of the UL values.

Since determining intersection capacities is somewhat arbitrary (not defined by any easily applied standards) you may wish to experiment and determine reasonable node delay coefficients for your model. In other words, we're copping out on providing any values at this time until more modelers have worked with these.

In conclusion, we've expanded the options and control in determining link travel times and intersection delays. We think these can help you build still better models and look forward to hearing from any of you using these expanded functions for your reactions and your experience in determining coefficients.