What is the Greenshields traffic flow model?

The Greenshields model is a simple macroscopic traffic flow model that assumes a linear relationship between speed and density. As density increases from zero to the jam density, speed decreases linearly from the free-flow speed to zero. From this linear relation, you can derive flow–density and speed–flow relationships and estimate capacity and critical density.

How do you calculate flow using the Greenshields model?

In the Greenshields model, speed v and density k are related by v = vf (1 − k/kj), where vf is the free-flow speed and kj is the jam density. Flow q is q = k · v. Substituting v gives q = vf k (1 − k/kj). The maximum flow (capacity) occurs at k = kj/2 and equals qmax = vf kj / 4.

When should I use the Greenshields model?

Use the Greenshields model for quick, first-approximation analysis of uninterrupted flow facilities such as freeways and expressways. It is useful for teaching, conceptual design, and rough capacity checks. For detailed operational analysis, calibration with field data or more advanced models (e.g., Greenberg, Underwood, or simulation) is recommended.

What are free-flow speed, jam density, and critical density?

Free-flow speed vf is the average speed vehicles would travel at very low density with no interactions. Jam density kj is the maximum possible density when vehicles are bumper-to-bumper and speed is effectively zero. Critical density kc is the density at which flow is maximized (capacity). In the Greenshields model, kc = kj / 2 and the corresponding speed is vf / 2.

Greenshields Model Traffic Flow Calculator

Compute speed, density, and flow using the Greenshields traffic flow model, find capacity and critical density, and visualize the fundamental diagrams for a highway or arterial.

Greenshields Model Calculator

Units:

Free-flow speed \(v_f\)

km/h

Typical freeway: 80–120 km/h (50–75 mph).

Jam density \(k_j\)

veh/km/lane

Bumper-to-bumper density per lane (e.g., 120–180 veh/km/lane).

Known variable

Density \(k\) Speed \(v\) Flow \(q\)

Density \(k\)

veh/km/lane

Fundamental Diagrams

These plots are generated from your chosen free-flow speed and jam density. Capacity and critical density are highlighted.

Speed–Density diagram \(v(k)\)

Flow–Density diagram \(q(k)\)

Speed–Flow diagram \(v(q)\)

Greenshields traffic flow model – theory and formulas

The Greenshields model is one of the earliest and simplest macroscopic traffic flow models. It assumes a linear relationship between speed and density on an uninterrupted facility such as a freeway:

Speed–density relationship:

\[ v(k) = v_f \left(1 - \frac{k}{k_j}\right) \]

\(v\) – mean speed (km/h or mph)
\(k\) – density (veh/km/lane or veh/mi/lane)
\(v_f\) – free-flow speed
\(k_j\) – jam density

Flow \(q\) is defined as the product of density and speed:

Flow–density relationship:

\[ q(k) = k \cdot v(k) = v_f k \left(1 - \frac{k}{k_j}\right) \]

Critical density and capacity

The critical density \(k_c\) is the density at which flow is maximized (capacity). To find it, differentiate \(q(k)\) with respect to \(k\) and set the derivative to zero:

\[ q(k) = v_f k \left(1 - \frac{k}{k_j}\right) \]

\[ \frac{dq}{dk} = v_f \left(1 - 2\frac{k}{k_j}\right) = 0 \quad\Rightarrow\quad k_c = \frac{k_j}{2} \]

The corresponding speed at capacity is:

\[ v_c = v(k_c) = v_f \left(1 - \frac{k_c}{k_j}\right) = v_f \left(1 - \frac{1}{2}\right) = \frac{v_f}{2} \]

The maximum flow (capacity) is then:

\[ q_{\max} = q(k_c) = v_f \cdot \frac{k_j}{2} \left(1 - \frac{1}{2}\right) = \frac{v_f k_j}{4} \]

Headway and spacing

Density is the inverse of average spacing between vehicles. If \(k\) is in veh/km/lane, the average spacing \(s\) in meters is:

\[ s = \frac{1000}{k} \quad [\text{m/veh}] \]

The space mean headway (time between vehicles at a point) is:

\[ h = \frac{1}{q} \quad [\text{h/veh}] \quad\Rightarrow\quad h_s = \frac{3600}{q} \quad [\text{s/veh}] \]

How to interpret the diagrams

Speed–density diagram

The speed–density line starts at \((k=0, v=v_f)\) and decreases linearly to \((k=k_j, v=0)\). At low densities, vehicles travel close to free-flow speed. As density increases, interactions grow and speed drops.

Flow–density diagram

The flow–density curve is a concave parabola opening downward. It:

Starts at \(q=0\) when \(k=0\) (no vehicles, no flow).
Reaches a maximum at \(k_c = k_j/2\) (capacity).
Returns to \(q=0\) at \(k=k_j\) (jammed traffic, zero speed).

Densities below \(k_c\) correspond to under-saturated conditions (stable flow), while densities above \(k_c\) indicate congested conditions.

Speed–flow diagram

The speed–flow diagram shows two possible speeds for a given flow (except at capacity):

A higher speed, lower density operating point (stable, free-flow regime).
A lower speed, higher density operating point (congested regime).

In practice, traffic tends to operate on the stable branch until disturbances push it into congestion.

Typical parameter values

Free-flow speed \(v_f\): 90–120 km/h (55–75 mph) on freeways; 60–80 km/h (35–50 mph) on arterials.
Jam density \(k_j\): 120–180 veh/km/lane (190–290 veh/mi/lane), depending on lane width and vehicle mix.
Capacity \(q_{\max}\): often 1800–2400 veh/h/lane for freeways, which you can check with \(q_{\max} = v_f k_j / 4\).

Limitations of the Greenshields model

While the Greenshields model is widely used for teaching and quick estimates, it has several limitations:

The linear speed–density assumption may not fit real data well, especially near jam density.
It is a macroscopic model and does not capture individual vehicle behavior or lane-changing.
It assumes homogeneous traffic and roadway conditions (no bottlenecks, signals, or ramps).

For detailed design or operational analysis, engineers often calibrate more flexible models (e.g., Greenberg, Underwood, Drake) or use microscopic simulation tools.

Greenshields model – frequently asked questions

Is the Greenshields model realistic enough for design?

It is adequate for conceptual design, teaching, and quick checks, but not for final design of critical facilities. For design-level work, you should calibrate the model with local data or use more advanced models and simulation that reflect driver behavior, lane distribution, and control devices.

How do I choose jam density \(k_j\)?

Jam density depends on lane width, vehicle size, and driver behavior. A common rule of thumb is:

Passenger-car only freeway: 140–180 veh/km/lane (225–290 veh/mi/lane).
Mixed traffic or narrow lanes: slightly lower values.

If you have field data, you can estimate \(k_j\) from observed maximum densities during severe congestion.

Can this calculator handle multiple lanes?

Yes. All calculations are done per lane. To get total flow for a multi-lane facility, multiply the per-lane flow \(q\) or capacity \(q_{\max}\) by the number of lanes, assuming lanes are used evenly.

What is the difference between time mean speed and space mean speed?

The Greenshields model uses space mean speed, which is the harmonic mean of individual speeds over a road segment. Time mean speed is the arithmetic mean of speeds observed at a point. In uncongested conditions they are similar, but they diverge as variability increases.