Properties of the Equilibrium State in Transportation Networks

Abstract

The Wardrop equilibrium problem for urban road networks is considered. It is shown that the (unique) equilibrium total flow vector is a continuous function of the input traffic flows. Under fairly weak conditions it is proven that the total origin to destination travel costs are also continuous functions of the input traffic flows. It is then shown that each origin to destination cost is a monotonically nondecreasing function of its own input flow when other inputs are held fixed. Finally, it is demonstrated by means of simple examples that the equilibrium total flow vector and origin to destination travel cost functions are not differentiable at certain (possibly difficult to predict) points in the set of feasible input flow vectors and that the cost functions do not in general possess such other potentially useful properties as convexity or concavity. These results are important to an understanding of the sensitivity of the equilibrium state to variations in input data.