Malachy Carey

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.

In dynamic network loading and dynamic traffic assignment for networks, the link travel time is often taken as a function of the number of vehicles x(t) on the link at time t of entry to the link, that is, τ(t) = f(x(t)), which implies that the performance of the link is invariant (homogeneous) over time. Here we let this relationship vary over time, letting the travel time depend directly on the time of day, thus τ(t) = f(x(t), t). Various authors have investigated the properties of the previous (homogeneous) model, including conditions sufficient to ensure that it satisfies first-in-first-out (FIFO). Here we extend these results to the inhomogeneous model, and find that the new sufficient conditions have a natural interpretation. We find that the results derived by several previous authors continue to hold if we introduce one additional condition, namely that the rate of change of f(x(t), t) with respect to the second parameter has a certain (negative) lower bound. As a prelude, we discuss the equivalen...

The traffic assignment problem aims to calculate an equilibrium route flow vector, generally by seeking a zero of an appropriate objective function. If a continuous dynamical system follows a descent direction for this objective function at each nonequilibrium route flow vector, the system converges to equilibrium. It is shown that when this dynamical system is discretized with a fixed step length, the system eventually approaches close to equilibrium provided that the objective function is continuously differentiable and that the rate of descent is bounded below. The method of successive averages is widely used in traffic assignment; it has a decreasing step size at each iteration. With the same conditions as above, it is shown that the resulting dynamical system converges to equilibrium. In the steady-state model, the necessary conditions are shown to be satisfied, provided that the route cost vector is a continuously differentiable monotone function of the route flow vector. However, continuous differentiability of the cost function is shown not to hold in the dynamic queueing model.

Flow–density functions are often described as the fundamental relationship of traffic theory and are the basis of kinematic wave models or hydrodynamic models which are used to describe, predict or analyse traffic behaviour. We set out several different properties that it is generally agreed that flow–density functions should satisfy. We then take the many forms of flow–density functions that have been proposed or used over the past several decades, derive the properties of each one and consider whether, or to what extent, they satisfy each of the desirable properties. We find that few, if any, flow–density functions satisfy all of the desirable properties. The main reasons for this are that, in almost all cases, there are not enough independent parameters in the flow–density functions to capture all of the desirable properties and that, in almost all cases, the flow–density functions are specified as a single analytic function rather than a different function for each range of traffic behaviours, such as free-flow, congested and heavily congested traffic.