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The fundamental diagram is an assumed functional relationship between trafficflow and traffic density. In practice, this relationship is noisy and exhibitssignificant statistical variability. On smart motorways, this variability isincreased by variable speed limits that are not captured by the fundamentaldiagram. To study this variability, it is appropriate to consider the jointprobability distribution function (pdf) of density and flow. We perform anempirical study of the variability in the relationship between flow and densityusing 74 days of data from 64 sections of London's M25. The objectives are todetermine how much of the variability in the flow-density relationship resultsfrom variable speed limits and to assess whether particular functional forms ofthe fundamental diagram are systematically preferred. Empirically, the jointpdf of flow and density is strongly bimodal, illustrating that traffic flowsare often found in high-density or low-density regimes but rarely in between.We find that the high-density regime is strongly affected by variable speedlimits whereas the low-density regime is not. The Daganzo-Newell (triangular)model of the fundamental diagram systematically fits best to the data. However,the optimal parameters vary with location. Clustering analysis of theseparameters suggests three qualitatively different types of flow-densityrelationships applying to different sections of the M25. These clusters havenatural interpretations in terms of the frequency and severity of flowbreakdown. Accident rates also depend on cluster type suggesting possible linksto other properties of traffic flows beyond the flow-density relationship.

Empirical analysis of the variability in the flow-density relationship for smart motorways