# Stop-and-go waves

Probably everyone has at least once have been stuck in the traffic jam. But most probably not everyone had thought about the possible relationship between the complexity science and traffic. For quite a long time it was thought that traffic jams can be caused by noticeable events on the road - car crashes, road works and etc. But in the recent decades number of cars in the streets grew rapidly and it was noticed that sometimes traffic jams form without any obvious reasons. In this text we present a simple traffic model by Nagel and Schreckenberg [1], which predicts traffic jams occurring due to small errors made by drivers themselves.

## Formulation of the Nagel - Schreckenberg model

Let us assume that certain numbers of agents move on closed path. To simplify let us assume that this path is rectangle. Let us divide the path into a number of equally-sized cells (in our applet 124). Each of these cells may be occupied by a single agent.

Initially occupation of the cells may be generated randomly. Namely the cell can be occupied with a probability $$p_o$$ (thus average number of agents in our applet is $$124 p_o$$). Initially all agents may have a certain speed, which may be either random or fixed (we set it to be 0 cell per second (c/s)).

During the next time step the following algorithmic steps are executed:

• Acceleration. All agents increase their speed by a single step (in our applet a speed step is 5 c/s). No agent may have speed larger than $$v_{max}$$.
• Slowing due to other agents. If agent sees that moving at current speed would cause bumping into other agent, it should decrease the speed to the largest safe one.
• Random error. With probability $$p_e$$ each agent, whose speed is positive, may make a driving error. This error causes agent to slow down by one step (-5 c/s).
• Movement. All agents move at their own speed.

These steps are repeated for every time step beyond the initial setup step.

This simple model predicts that traffic jams may happen without any obvious reason - only due to driver errors being amplified by other drivers. This amplification cause stop-and-go waves - namely, cars slow down due to other drivers slowing down and then can move again once the path becomes clear, after some time they encounter another slow down and so on. Note that "cars" (agents) move clockwise, while traffic jam moves in the opposite direction. In our applet below we have picked a minimal values for the model parameters with which traffic jams should form.

## Applet

We invite our readers to test different parameter values. Note that with $$p_o > \frac{5}{v_{max}}$$ traffic is intense and traffic jams may form even if the probability of error is relatively small. While with larger $$p_e$$ traffic jams may form with less intense traffic (smaller $$p_o$$). Don't forget to test different values of $$v_{max}$$.

## Experiment

Experiment similar to the model above was performed in Japan. Scientists of University of Nagoya asked a number of drivers to drive in circular motion. Note that for a brief initial moments drivers move in orderly (jam-less) manner, but afterwards traffic jam begins to form without any apparent reason. As in the model traffic jams moves in opposite direction to the traffic.

## References

• K. Nagel, M. Schreckenberg. A cellular automaton model for freeway traffic. Journal de Physique I. 2 (12): 2221-2229 (1992). doi: 10.1051/jp1:1992277.