Fundamental Traffic Flow Relation

Three numbers describe a stream of traffic. Flow $q$ is how many vehicles pass a fixed point per unit time — what a counter at the roadside records. Density $k$ is how many vehicles occupy a unit length of road at one instant — what you would count from a photograph taken from a helicopter. Speed $v$ is how fast they are going. The identity $q = kv$ ties them together, and it holds for any traffic stream at any time, because it is arithmetic rather than physics: vehicles per kilometre multiplied by kilometres per hour is vehicles per hour, and the kilometres cancel.

That is worth dwelling on, because the identity is often mistaken for a model. It predicts nothing. It contains no information about drivers, vehicles, weather or road design. All it says is that if you know any two of the three quantities, the third is determined. To predict anything at all you need a second equation — a relation saying how speed responds to density — and that is exactly what Greenshields proposed in 1935 and what everyone since has been arguing about.

The practical value of the identity is that density is difficult to measure and the other two are easy. Nobody hires a helicopter. What a modern detector station has is a pair of inductive loops a metre or two apart, which gives a count and, from the time offset between the two loops, a speed. Divide and you have the density, which is the quantity that actually tells you the state of the road. A single loop gives you count and occupancy — the fraction of time the loop is covered by metal — and occupancy is very nearly proportional to density, which is why occupancy is the variable transportation-management centres actually watch.

Now the trap, and it is a real one that costs money in real studies. The $v$ in $q = kv$ is the space mean speed: the average speed of the vehicles occupying a length of road at one moment, which is the same thing as the harmonic mean of the speeds of the vehicles passing a point. A speed trap, a radar gun or a single loop gives you the time mean speed: the plain arithmetic average of the vehicles that went by. The time mean speed is always the larger of the two, and it is larger by roughly the variance of the speed distribution divided by the space mean speed. On a free-flowing motorway with everybody doing much the same speed the difference is a percent or two and nobody notices. In congestion, where a few vehicles are moving briskly and many are crawling, the difference is substantial — and congestion is precisely the condition you were trying to measure. Substituting a time mean speed into this identity understates the density, and it understates it worst exactly where accuracy matters.

The reason for the harmonic mean is intuitive once seen. A slow vehicle spends longer on your kilometre of road than a fast one, so at any given instant the road contains proportionally more slow vehicles than the point counter ever sees. Density is an instantaneous property of a length; flow is a cumulative property of a point. They are different kinds of average, and the space mean speed is the one that reconciles them.

One last piece of vocabulary that this identity makes sense of. The reciprocal of density is spacing, the average distance between vehicle front bumpers, and the reciprocal of flow is headway, the average time between them passing a point. So $q = kv$ can equally be written as spacing equals headway times speed, which is how a driver experiences it: at 100 km/h a two-second headway is about 56 m of spacing, and 56 m of spacing on every vehicle is a density of about 18 veh/km. That single arithmetic step is the whole reason a "two-second rule" and a lane capacity are the same subject.