Evidence for Speed Flow Relationships

Evidence for Speed Flow Relationships


N Taylor, N Bourne, S Notley, TRL, UK; G Skrobanski, Highways Agency, UK


New large data sources offer the potential for thorough analysis and definitive explanation of traffic speed/flow/density data, but the right questions about the underlying processes need to be asked.


The English Highways Agency?s network of MIDAS (Motorway Incident Detection and Automatic Signalling) loop detectors furnishes a large resource of data on traffic speeds and flows on the motorway network. Combined with other sources, this contributes to a dataset known as MADJ (Merged All Data for Journeys), within the HATRIS (HA Traffic Information System) database. MADJ records speeds and flows down to 15-minute resolution across approximately 2500 links and 13000 km of trunk roads. Accepting that there are issues with aggregation, these data can be used to study speed/flow relationships after filtering for factors like road works and seasonal differences.

Empirical speed/flow relationships include for example the UK?s COBA relationships, which take account of road geometry, heavy vehicle percentage and local development. Although these may include various types of delay, though not normally heavy congestion, they always predict decreasing average speed as flow increases, and thus are able to serve as supply functions. Recently it has been confirmed that a modification proposed some years ago to cover congested peaks is valid under certain assumptions.

In congested traffic it is observed that speed and flow decrease together. Under certain definitions, flow (q), speed (v) and density (k) satisfy a ?fundamental relationship? q=vk. To explain congested behaviour in particular, contingent relationships between pairs of variables have been proposed. Greenshields? speed/density model is the earliest example, initially hypothesised to apply over the whole range of speeds. The Gazis, Herman and Rothery (GHR) family of microscopic car-following models relate individual vehicle acceleration to distance and relative speed from the vehicle in front. By varying parameters they yield several average steady-state speed/density relationships. These models typically predict maximum flow at some speed, although a two-regime model may be preferred to cover uncongested traffic. However, their form appears to lack direct physical explanation, giving no theoretical reason to prefer one over another. Also, as pointed out by Kenneth A Small, a causal positive relationship between flow and speed is illogical because it implies that vehicles confer a positive externality on each other. While uncongested relationships can be considered endogenous, determined mainly by local interactions between vehicles, congested flows are usually exogenous, depending on downstream capacity, the exception being that endogenous relationships may apply between local values of speed and density, and in the generation and propagation of ?shock waves?. Macroscopic speed/flow/density models should reflect these observed properties.

Our study has looked at speed/flow data in both uncongested and congested regimes. However, while uncongested traffic data are generally available to cover the full range of theoretically achievable states, this is not the case for congested traffic. Uncongested data suggest that pure speed/flow relationships on motorways at least are much ?flatter? than is sometimes supposed, and that prediction of rapidly falling speeds as capacity is approached may owe to the inclusion of data points which do not truly represent uncongested flow. We have also found that maximum flow can depend on the time over which it is measured, so ?capacity? may not be a simple quantity.

In the congested regime, we have looked at several macroscopic models. However, the greatest differences between them tend to occur at the highest flows or densities, where the data are least clear. We have confirmed that the speed/density relationship is non-linear, but find the relationship between speed and the logarithm of density to be quite linear over most of the range. This appears to favour Greenberg?s logarithmic model, except at the highest densities, or possibly Edie?s/Underwood?s exponential model. However, questions remain about how model parameters relate to physical measurements.

Analysing aggregate data in sufficient quantity should eventually make it possible to distinguish between models. The paper will present the results obtained so far, with the aim of stimulating discussion and opening the way to more intensive study of the large data sets now becoming available, in the hope of finally resolving these questions.


Association for European Transport