Convergence of a discretised travel-time model

In network models for dynamic traffic assignment (DTA), the travel time on a link is often treated as a function of the number of vehicles on the link. Instead of applying this model to the whole link, we divide the link into segments, apply the model (suitably adjusted) sequentially to these segments, and investigate how the solution is affected by various levels of discretisation (as the discretisation is refined, the solution converges to the solution of the Lighthill–Whitham–Richards (LWR) model). We also restrict the link (and segment) travel-time function to ensure that it satisfies a first-in-first-out (FIFO) property and explore how this affects and restricts the form of the flow-density functions used in the LWR model. We numerically illustrate the solution of the discretised model for various travel-time functions and patterns of inflows, for both homogeneous and inhomogeneous links. Subject to the above restriction on the flow-density function, the numerical results suggest that dividing “long” links into even a few segments can make the model solution closely approximate the LWR solution, while retaining tractability in the network. We also observe, for example, that the whole-link (undiscretised) travel-time model has a “flattening” effect on the profiles of flows and travel times (this effect can be reduced to any desired extent by using discretisation); and that the travel time and outflow for an inhomogeneous link can be approximated very closely by treating the link as homogeneous, with capacity parameters set equal to the average capacity from the inhomogeneous link.