Consistency and Inconsistency Between the Fundamental Relationships on Which Different Traffic Assignment Models Are Based

We compare the forms and properties of different macroscopic behavioral relationships on which different static and dynamic traffic assignment models have been based, namely, travel time-flow functions, travel time-occupancy functions, and flow-occupancy functions or flow-density functions. Given any one of these three relationships we can derive the other two, so that they appear mathematically equivalent. These three behavioral relationships focus on different variables, but they are meant to be observing the same traffic and are not meant to be inconsistent or incompatible with each other. However, we see that the forms and properties usually assumed for any one of these three relationships are often inconsistent or incompatible with the forms or properties that are usually assumed for the other two, or that are considered acceptable for the other two. It might be thought that any inconsistencies between these relationships are because of modeling different phenomena, however that does not explain some of the surprising differences or inconsistencies, for example, those we find implied by linear or piecewise linear functions or by the best known travel time-flow function, the US Bureau of Public Roads function. Such differences or inconsistencies are seldom mentioned in the literature, though they are important because these models are so central in network traffic assignment. They are important when considering or comparing the solutions or predictions from the various traffic assignment models. Yet even if not comparing solutions, it seems important to know whether the model being used is inconsistent with another model that is widely accepted and widely used for similar scenarios. The results do not suggest discontinuing the use of any of the three forms of relationships or the corresponding assignment models referred to above. Rather, they suggest that more attention should be given to noting such potential inconsistencies, incompatibilities, or limitations when considering, reporting, or comparing results or predictions from traffic assignment models. In some cases the results indicate the direction in which the model biases the predictions compared to those from an alternative model that could have been used. In some cases, alternative properties or functional forms could or should be developed or used.