Whole-link models of traffic flows have been widely used in mathematical programming models for dynamic traffic assignment. In this paper, we consider a well-known whole-link model in which the link travel time, for traffic entering at time t, is a function of the number of vehicles on the link, and may also be a function of the inflow rate or outflow rate at time t. Instead of considering this in a network context, we examine its behaviour for a single link, for given inflow profiles, so as to distinguish behaviour within a link from network behaviour. We consider steady state solutions, for constant inflows and outflows, note that various model forms can yield the same solution, and that under certain conditions the model may admit multiple values for the link travel time. We derive the complete analytic solution for a model where the travel time depends linearly only on the number of vehicles on the link, and show that the solution exhibits pseudo-periodicity, and converges to a steady state solution. The results indicate that the analytic solution is quite complex even for very simple cases, and that care has to be exercised in the choice of parameters. We illustrate the solutions numerically.
