The Fundamental Diagram

Module 6: Explorables and Discrete Models

The fundamental diagram is the relation between traffic density and traffic flow in the stationary state. It is the main macroscopic observable of a traffic model and the one you compare with road measurements (Nagel and Schreckenberg 1992; Chowdhury, Santen, and Schadschneider 2000).

Density, flow and speed

On a ring of $L$ cells with $N$ cars:

\[ \rho = \frac{N}{L}, \qquad \bar v = \frac{1}{N}\sum_{n=1}^N v_n, \qquad q = \frac{1}{L}\sum_{n=1}^N v_n = \rho\, \bar v . \]

The flow $q$ is the number of cars that pass a fixed point per time step, averaged over the ring. In simulations you average $q$ over time after discarding a burn-in period.

Free flow and congested branches

Free flow. At low density cars rarely interact. Each car moves at about $v_{\max} - p$ on average, so $q \approx \rho\,(v_{\max} - p)$ grows linearly with $\rho$ (Chowdhury, Santen, and Schadschneider 2000).
Congested branch. At high density the gaps limit the speeds and $q$ decreases with $\rho$. At $\rho = 1$ nothing moves.
Maximum flow. Between the two branches the flow reaches a maximum, the capacity of the road. Randomization lowers the capacity and moves it to lower density.

For $p = 0$ the diagram is the exact triangle $q = \min(\rho\, v_{\max},\ 1 - \rho)$ with its maximum at $\rho = 1/(v_{\max} + 1)$ (Chowdhury, Santen, and Schadschneider 2000).

Figure 1: Simulated fundamental diagram (ring of 400 cells, v_max = 5, 400 steps after 200 burn-in steps, random initial positions). The dashed line is the exact result for p = 0. Left: flow. Right: mean speed.

The simulation for $p = 0$ lies on the exact triangle, which validates the code. For $p > 0$ the peak is lower and rounded.

Empirical evidence: jams without bottlenecks

Sugiyama and co-workers asked 22 drivers to drive around a circular track of 230 m, starting from a uniform configuration. A stop-and-go wave formed spontaneously and travelled backwards, with no bottleneck on the track (Sugiyama et al. 2008). They interpret the jam as a collective instability of free flow that appears when the density exceeds a critical value.

This is the phenomenon that the randomization step produces in the Nagel-Schreckenberg model: jams on a homogeneous ring, triggered by fluctuations. The experiment does not test the specific rules of the model. It shows that a model of homogeneous traffic on a ring must be able to produce such jams.

Exercise

Use the free-flow approximation $q \approx \rho (v_{\max} - p)$ and the congested branch $q \approx 1 - \rho$ of the deterministic model to estimate where the maximum flow occurs for $v_{\max} = 5$ and $p = 0.3$. Compare with the figure.

Solution

The two lines cross at $\rho (v_{\max} - p) = 1 - \rho$, so $\rho^* = 1/(v_{\max} - p + 1) = 1/5.7 \approx 0.175$ and $q^* \approx 0.82$. The simulated curve for $p = 0.3$ peaks near $\rho \approx 0.1$ with $q$ below 0.5, so at lower density and much lower flow. Randomization also lowers the congested branch below $1 - \rho$, because stopped cars restart with a delay. The crossing of the two straight lines is only an upper estimate.

References

Chowdhury, Debashish, Ludger Santen, and Andreas Schadschneider. 2000. “Statistical Physics of Vehicular Traffic and Some Related Systems.” Physics Reports 329 (4–6): 199–329. https://doi.org/10.1016/S0370-1573(99)00117-9.

Nagel, Kai, and Michael Schreckenberg. 1992. “A Cellular Automaton Model for Freeway Traffic.” Journal de Physique I 2 (12): 2221–29. https://doi.org/10.1051/jp1:1992277.

Sugiyama, Yuki, Minoru Fukui, Macoto Kikuchi, Katsuya Hasebe, Akihiro Nakayama, Katsuhiro Nishinari, Shin-ichi Tadaki, and Satoshi Yukawa. 2008. “Traffic Jams Without Bottlenecks: Experimental Evidence for the Physical Mechanism of the Formation of a Jam.” New Journal of Physics 10 (3): 033001. https://doi.org/10.1088/1367-2630/10/3/033001.

--- title: "The Fundamental Diagram" subtitle: "Module 6: Explorables and Discrete Models" format: html --- The fundamental diagram is the relation between traffic density and traffic flow in the stationary state. It is the main macroscopic observable of a traffic model and the one you compare with road measurements [@nagel1992cellular; @chowdhury2000statistical]. ## Density, flow and speed On a ring of $L$ cells with $N$ cars: $$ \rho = \frac{N}{L}, \qquad \bar v = \frac{1}{N}\sum_{n=1}^N v_n, \qquad q = \frac{1}{L}\sum_{n=1}^N v_n = \rho\, \bar v . $$ The flow $q$ is the number of cars that pass a fixed point per time step, averaged over the ring. In simulations you average $q$ over time after discarding a burn-in period. ## Free flow and congested branches - **Free flow.** At low density cars rarely interact. Each car moves at about $v_{\max} - p$ on average, so $q \approx \rho\,(v_{\max} - p)$ grows linearly with $\rho$ [@chowdhury2000statistical]. - **Congested branch.** At high density the gaps limit the speeds and $q$ decreases with $\rho$. At $\rho = 1$ nothing moves. - **Maximum flow.** Between the two branches the flow reaches a maximum, the capacity of the road. Randomization lowers the capacity and moves it to lower density. For $p = 0$ the diagram is the exact triangle $q = \min(\rho\, v_{\max},\ 1 - \rho)$ with its maximum at $\rho = 1/(v_{\max} + 1)$ [@chowdhury2000statistical]. ```{python} #| echo: false #| label: fig-fundamental-diagram #| fig-cap: 'Simulated fundamental diagram (ring of 400 cells, v_max = 5, 400 steps after 200 burn-in steps, random initial positions). The dashed line is the exact result for p = 0. Left: flow. Right: mean speed.' #| fig-width: 8 #| fig-height: 3.5 import numpy as np import matplotlib.pyplot as plt from amlab.cellular_automata.traffic_models import fundamental_diagram, deterministic_flow densities = np.linspace(0.02, 0.9, 30) fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 3.5)) for p in (0.0, 0.1, 0.3, 0.5): q = fundamental_diagram(densities, length=400, steps=600, burn_in=200, p=p, seed=1) ax1.plot(densities, q, marker=".", label=f"p = {p}") ax2.plot(densities, q / densities, marker=".") ax1.plot(densities, deterministic_flow(densities), "k--", linewidth=1) ax1.set_xlabel(r"Density $\rho$") ax1.set_ylabel(r"Flow $q$") ax1.legend() ax2.set_xlabel(r"Density $\rho$") ax2.set_ylabel(r"Mean speed $\bar v$") plt.tight_layout() plt.show() ``` The simulation for $p = 0$ lies on the exact triangle, which validates the code. For $p > 0$ the peak is lower and rounded. ## Empirical evidence: jams without bottlenecks Sugiyama and co-workers asked 22 drivers to drive around a circular track of 230 m, starting from a uniform configuration. A stop-and-go wave formed spontaneously and travelled backwards, with no bottleneck on the track [@sugiyama2008traffic]. They interpret the jam as a collective instability of free flow that appears when the density exceeds a critical value. This is the phenomenon that the randomization step produces in the Nagel-Schreckenberg model: jams on a homogeneous ring, triggered by fluctuations. The experiment does not test the specific rules of the model. It shows that a model of homogeneous traffic on a ring must be able to produce such jams. ::: {.callout-note} ## Exercise Use the free-flow approximation $q \approx \rho (v_{\max} - p)$ and the congested branch $q \approx 1 - \rho$ of the deterministic model to estimate where the maximum flow occurs for $v_{\max} = 5$ and $p = 0.3$. Compare with the figure. ::: ::: {.callout-tip collapse="true"} ## Solution The two lines cross at $\rho (v_{\max} - p) = 1 - \rho$, so $\rho^* = 1/(v_{\max} - p + 1) = 1/5.7 \approx 0.175$ and $q^* \approx 0.82$. The simulated curve for $p = 0.3$ peaks near $\rho \approx 0.1$ with $q$ below 0.5, so at lower density and much lower flow. Randomization also lowers the congested branch below $1 - \rho$, because stopped cars restart with a delay. The crossing of the two straight lines is only an upper estimate. :::