Greenshields Capacity

Differentiate the flow–density parabola with respect to density, set the result to zero, and the answer arrives in one line: the maximum sits at $k = k_j/2$, where the speed is $v_f/2$, and the flow there is $q_{max} = v_f k_j / 4$. Three halves and a quarter, all falling out of a straight line. It is the tidiest result in the classical theory, and it is worth being clear that the tidiness belongs to the model rather than to traffic. On a real concave speed–density curve, capacity occurs at rather more than half the free-flow speed and rather less than half the jam density, and observed capacity generally comes out above what this formula predicts. Greenshields is a lower bound dressed up as an identity.

Still, the structure it shows is right. Capacity rises with the free-flow speed and with the jam density, and jam density is set by how tightly vehicles pack — which is why a lane of passenger cars has a higher capacity than the same lane carrying buses and articulated trucks, and why the passenger-car-equivalent adjustments in every capacity method exist at all.

Now the mistake that matters more than any arithmetic on this page: capacity is not a design target. It is the point at which the stream breaks down. It is the apex of the parabola — the top of a hill with a cliff on the far side — and a facility operated at its apex has no margin whatever. One heavy vehicle climbing a grade, one lane change taken a little sharply, one driver braking for no reason, and the stream slips onto the congested branch. Once there it carries less than capacity, and it will keep carrying less until demand falls away, which on a peak-period facility can mean an hour or more. The breakdown is quick and the recovery is slow, and the asymmetry is made worse by capacity drop: the discharge rate after breakdown is measurably below the flow the same bottleneck sustained just before it.

This is why capacity analysis is done against a service volume well below capacity rather than against capacity itself, and why level-of-service frameworks describe operating quality in terms of density and speed rather than in terms of flow. It is also why the flow figure alone is such a poor performance measure: a facility at 95 % of capacity and a facility that has broken down and is delivering 90 % of capacity look almost the same in a volume column and are not remotely the same road.

The corresponding density, $k_j/2$, has its own name — the critical density — and it is the genuinely useful operational variable. It is a single threshold with an unambiguous meaning: below it the facility is on the good branch, above it the facility has broken down. Unlike flow, it does not take the same value on both sides. Freeway management systems are built around watching occupancy against a critical threshold for exactly this reason, and the control action — metering, variable limits, lane control — is triggered by the density crossing, not by the volume.

One final caution about borrowing numbers. Because $q_{max}$ here depends only on $v_f$ and $k_j$, it is tempting to look those two up and treat the result as the capacity of a facility. Do not. Both parameters are calibration outputs for a specific road under specific conditions, and this site takes them as inputs precisely so that the number you get is your road's rather than someone else's. Published capacity values with all their adjustment factors live in the Highway Capacity Manual or your own agency's manual, both copyrighted, and neither is reproduced here.