Link Travel Times II: Properties Derived from Traffic-Flow Models

We investigate the properties of travel times when the latter are derived from traffic-flow models. In particular we consider exit-flow models, which have been used to model time-varying flows on road networks, in dynamic traffic assignment (DTA). But we here define the class more widely to include, for example, models based on finite difference approximations to the LWR (Lighthill, Whitham and Richards) model of traffic flow, and ‘large step’ versions of these. For the derived travel times we investigate the properties of existence, uniqueness, continuity, first-in-first-out (FIFO), causality and time-flow consistency (or intertemporal consistency). We assume a single traffic type and assume that time may be treated as continuous or as discrete, and for each case we obtain conditions under which the above properties are satisfied, and interrelations among the properties. For example, we find that FIFO is easily satisfied, but not strict causality, and find that if we redefine travel time to ensure strict causality then we lose time-flow consistency, and that neither of these conditions is strictly necessary or sufficient for FIFO. All of the models can be viewed as an approximation to a model that is continuous in time and space (the LWR model), and it seems that any loss of desirable properties is the price we pay for using such approximations. We also extend the exit-flow models and results to allow ‘inhomogeneity’ over time (link capacity or other parameters changing over time), and show that FIFO is still ensured if the exit-flow function is defined appropriately.