Traffic flow fundamentals: speed, flow and density

Traffic engineers describe a stream of vehicles with three numbers, linked by one simple equation.

The three variables

• Flow, q: vehicles passing a point per hour (veh/h).
• Density, k: vehicles per km of lane (veh/km).
• Speed, v: space mean speed (km/h).

q = k v

Greenshields' linear model

Greenshields observed that speed falls roughly linearly as density rises:

v = v_f ( 1 − kk_j )

v_f is the free-flow speed (an empty road) and k_j the jam density (bumper to bumper, speed zero).

Multiply by k to get flow:

q = v_f ( k − k²k_j )

Capacity

Flow is a parabola in k. It is zero on an empty road and zero in a jam, and peaks in between:

k_m = k_j2 v_m = v_f2 q_max = v_f k_j4

Worked example

A lane has v_f = 80 km/h and k_j = 150 veh/km.

q_max = 80 × 1504 = 3000 veh/h

That happens at 75 veh/km and 40 km/h. Real lanes carry less, around 1800 to 2200 passenger car units per hour, because Greenshields' straight line is an idealisation.

Time mean and space mean speed

Speeds measured at a point (spot speeds) averaged directly give the time mean speed. The space mean speed is their harmonic mean, and it is the one used in q = k v. The time mean speed is always the larger of the two.

Why it matters

Above k_m, adding cars lowers flow: the road is congested. Ramp metering and signal timing aim to keep density just below k_m so the road carries the most traffic.