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; 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?).

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 (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.

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.

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.

--- 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. ::: ## Next