Mechanisms of jamming in the Nagel-Schreckenberg model for traffic flow

We study the Nagel-Schreckenberg cellular automata model for traffic flow by both simulations and analytical techniques. To better understand the nature of the jamming transition, we analyze the fraction of stopped cars $P(v=0)$ as a function of the mean car density. We present a simple argument that yields an estimate for the free density where jamming occurs, and show satisfying agreement with simulation results. We demonstrate that the fraction of jammed cars $P(v\in \{0,1\left\}\right)$ can be decomposed into the three factors (jamming rate, jam lifetime, and jam size) for which we derive, from random walk arguments, exponents that control their scaling close to the critical density.

Figure 1

A car C that will end up behind the end of the jam E, if it is not dawdling, as the cells in between are empty.

Figure 2

A car C will block cells for the next time step as no car can overtake or arrive at distance $d\le d_{\mathrm{pre}}$.

Figure 3

Only one car C can leave the jam in succeeding time steps if the jam front is considered immovable. In the depicted situation, C moves without dawdling.

Figure 4

Probability $P(v=0)$ to find a standing car for different $v_{\max }$ as a function of mean car density. The horizontal lines show the calculated values for ${\rho }_{\mathrm{f}}$ obtained from Eq. (1).

Figure 5

Normalized number of stopped cars $P(v=0)$ compared with the global flow $J$ for simulations with $v_{\max }=5$. The vertical line shows the calculated ${\rho }_{\mathrm{f}}$. The amount of stopped cars begins to rise before the maximum flow is reached. This is mainly due to cars in jams not having a vanishing average velocity.

Figure 6

Comparison of the directly measured ratio of jammed cars (points) and the product of the individually measured three factors (lines): jamming rate, jam lifetime, and cars in a jam. Two vertical lines represent the calculated free density and the critical density obtained from the fundamental diagram.

Figure 7

The rate at which traffic jams form for different maximum velocities. Two vertical lines represent the calculated free density and the critical density; black lines represent fitting in the low-density region. Here a power-law behavior $\propto {\rho }^{v_{\max }-1}$ is visible as expected and described by Ref. [11]. The rise near the free density for $v_{\max }>3$ is closely related to the cutoff in jam lifetimes at that point and results from the measuring algorithm.

Figure 8

The average lifetime of traffic jams and the average number of cars therein for different maximum velocities. The two vertical lines represent the calculated free density and the critical density. Near the free density a steep rise in the lifetime of jams is observable. For maximum velocities greater than three a sharp cutoff in the lifetime is visible. This is due to the definition of “jammed cars” used for the measurement.

Figure 9

The average lifetime of a traffic jam plotted over a rescaled density $\tilde{\rho }=1-(\rho /{\rho }_c)$. The rise near the critical density can be seen to behave proportional to ${({\rho }_c-\rho )}^{-1}$ (solid lines). An earlier rise for low-density seems to scale with ${({\rho }_c-\rho )}^{-1/2}$ (dashed lines).

Figure 10

The average number of cars in a traffic jam plotted over a rescaled density $\tilde{\rho }=1-(\rho /{\rho }_c)$. The rise near the critical density can be seen to behave proportionally to ${({\rho }_c-\rho )}^{-1}$ (solid line). An earlier rise for low-density seems to scale with ${({\rho }_c-\rho )}^{-1/4}$ (dashed line).