The travel time τ(t) on a link has often been treated in dynamic traffic assignment (DTA) as a function of the number of vehicles x(t) on the link, that is, τ(t) = f(x(t)). In earlier papers, bounds on the gradient of this travel time function f(x) have been introduced to ensure that the model, and in particular the exit times and outflows, have various desirable properties, including a first-in-first-out (FIFO) property. These gradient conditions can be restrictive, because most commonly used travel time functions do not satisfy the conditions for all inflow rates. However, in this paper we extend the earlier results to show that the same properties (including FIFO) can be achieved by instead assuming f(x) is convex, convex about a point, or has certain weaker properties that are satisfied by most travel time functions f(x) proposed or used in practice. These results hold under the conditions in which the travel time function τ(t) = f(x(t)) has generally been applied in the DTA literature, that is, with each link being homogeneous (uniform capacity along the link) and without obstructions or traffic lights. In that case, even if f(x) does not satisfy the above gradient condition, the range in which it is violated is not attainable and hence cannot cause a problem.
