Nagel-Schreckenberg Traffic Simulation

The most annoying thing about driving for some people are certainly traffic jams. I get even more upset when I leave the traffic jam and there was no reason for it in the first place: so-called “phantom traffic”. A simple, yet good model of traffic is the Nagel-Schreckenberg model. It explained, for the first time, phantom congestion as a result of dawdling and tailgating.

Observations

Cars move along a road with a periodic boundary condition (ring), accelerate when they can and brake before they crash into the car in front of them. This boundary condition is often used in theoretical solid state physics and is a good approximation for large distances. With some probability $p_1$ drivers dawdle and slow down. The road is one pixel wide and goes along the $x$ axis. Along the $y$-axis the time evolution of the road is plotted. You can see that slower stripes in yellow appear and disappear quickly if the drivers do not dawdle much. If the drivers are not so attentive, traffic jams form (red stripes), which dissolve again by themselves.

Of course, drivers (hopefully) don’t just slow down out of the blue. One important correction is the velocity-dependent randomization (VDR). It is assumed that when starting $v=0$, the dawdling probability $p_2$ is much GREATER than otherwise. The effect due to dawdling is even more pronounced here, and once a jam has formed it dissipates more slowly. Also, more congestion forms when the maximum speed is higher because drivers tailgate more closely and have to brake more. Even if the maximum speed allowed doubles, the average speed ${\mu }_v$ increases by much less than double.