The Nagel-Schreckenberg cellular automaton for highway traffic. Watch phantom jams emerge spontaneously from random braking, even with no bottleneck. The space-time diagram reveals backward-moving jam waves.
Four simple rules per time step — acceleration, speed limiting, random braking (with probability p), and motion — are sufficient to produce realistic traffic phenomena including phantom jams, stop-and-go waves, and a fundamental diagram matching real highway data.
Adjust vehicle density (0–100%) and braking probability (0–1). At a critical density (~30%), phantom jams appear as backward-moving waves. The space-time diagram shows jam trajectories — diagonal lines moving upstream.
Phantom traffic jams — where everyone brakes for no apparent reason — are caused by exactly the mechanism in this model: small random perturbations amplify in dense traffic. Japanese researchers reproduced this on a circular test track in 2008.