A jam with nothing in front of it

The moving lines behind every page on this site are a simulation. Specifically they are the Nagel–Schreckenberg model, published in 1992, which is the smallest set of rules that produces real traffic behaviour.

The four rules

Each vehicle sits in a cell and has an integer speed. Every tick: accelerate by one if you can; slow to the gap ahead of you if it is smaller; with probability 0.3, dawdle by one; then move forward by your speed.

That is the whole model. There is no accident in it, no lane change, no road works, no bad driver.

What comes out of it

Jams. Specifically, dense clusters of stopped vehicles that travel backwards along the road while every individual vehicle moves forwards. Watch a busy row in the background for a few seconds and you will see one.

Delete the third rule — the random dawdle — and the traffic settles into a perfectly regular crystal that never jams at any density. The entire phenomenon comes from that one line.

Why a freight operator cares

Because the same thing happens on a rail corridor and in a terminal gate queue, and it means capacity does not degrade gracefully. Below a critical density, throughput rises with traffic. Above it, throughput falls — you put more in and get less out.

This is the argument for a timetable. A carrier who sells every path they own is operating above the critical density and their published transit times become fiction, because the variance explodes long before the average does. We sell slots on a schedule and leave paths unsold, which looks like leaving money on the table right up until the moment it isn’t.

Check it yourself

The corridors section on the home page samples the live simulation and plots its measured flow against its measured density. If the background were an ordinary particle effect the points would land anywhere; because it is the real model, they land on the model’s characteristic triangle.