Travel-Time Models With and Without Homogeneity Over Time

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 equivalence of equations for flow propagation equations and for intertemporal conservation of flows, and argue that neither these equations nor the travel-time model are physically meaningful if FIFO is not satisfied. In §7 we provide some examples of time-dependent travel times and some numerical illustrations of when these will or will not adhere to FIFO.