# Speed-flow relations and cost functions for congested traffic: Theory and empirical analysis

*Erik Verhoef* ()

*Transportation Research Part A: Policy and Practice*, 2005, vol. 39, issue 7-9, pages 792-812

**Abstract:**
A dynamic 'car-following' extension of the conventional economic model of traffic congestion is presented, which predicts the average cost function for trips in stationary states to be significantly different from the conventional average cost function derived from the speed-flow function. When applied to a homogeneous road, the model reproduces the same stationary state equilibria as the conventional model, including the hypercongested ones. However, stability analysis shows that the latter are dynamically unstable. The average cost function for stationary state traffic coincides with the conventional function for non-hypercongested traffic, but rises vertically at the road's capacity due to queuing, instead of bending backwards. When extending the model to include an upstream road segment, it predicts that such queuing will occur under hypercongested conditions, while the general shape of the average cost function for full trips does not change, implying that hypercongestion will not occur on the downstream road segment. These qualitative predictions are verified empirically using traffic data from a Dutch bottleneck. Finally, it is shown that reduced-form average cost functions, that relate the sum of average travel cost and average schedule delay costs to the number of users in a dynamic equilibrium, certainly need not have the intuitive convex shape, but may very well be concave - despite the fact that the underlying speed-flow function may be convex.

**Date:** 2005

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**Related works:**

Working Paper: Speed-Flow Relations and Cost Functions for Congested Traffic: Theory and Empirical Analysis (2003)

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