Module 1, lesson 3 · 50 minutes

The fundamental diagram

Speed falls as density rises, so flow has a maximum. That maximum is the capacity of every corridor, gate and ramp you will ever size.

You’ll be able to

sketch speed and flow against density, and mark the free-flow, congested and jammed parts;
state Weidmann’s parameters and maximum, and compare them with the SFPE linear relation and Fruin’s bands;
explain why the flow maximum is the single most important number in crowd engineering.

An exit ramp is emptying too slowly. The obvious fix is to send more people towards it: more people on the ramp, more people off it per minute. For a while that works. Then, past a certain point, it does the opposite. Each extra person on the ramp slows everyone else down by more than their own body adds, and the number leaving each minute starts to fall.

That point is the peak of one curve, the fundamental diagram, the relation between a crowd’s density and its speed, or equivalently its flow. Engineers in the field draw it from memory. By the end of this lesson, so will you.

Speed falls as density rises

In an empty corridor people walk at their own pace, on average 1.34 m/s according to Weidmann’s survey of 52 published values. As the corridor fills, each person has less floor in front of them. They shorten their steps and slow down so as not to tread on the heels ahead. At some density nobody can step at all.

Ulrich Weidmann fitted that behaviour with a curve written in terms of 1/ρ, the space each person has:

$v(\rho) = v_0 \left[1 - \exp\left(-\gamma\left(\frac{1}{\rho} - \frac{1}{\rho_{max}}\right)\right)\right]$

v(ρ) = v₀ · [1 − exp(−γ · (1/ρ − 1/ρmax))]

v₀: free walking speed, 1.34 m/s
γ: fitted constant, 1.913 p/m²
ρmax: jam density, 5.4 p/m²
ρ: density, p/m²

Read it from the inside out. When the space per person 1/ρ is large, the exponential is tiny and the speed is close to v₀. As the space shrinks towards 1/ρmax, the bracket shrinks to zero, and so does the speed. Jam density, ρmax, is the density at which speed reaches zero: 5.4 p/m² in Weidmann’s relation.

Plug in some densities to get a feel for it (our arithmetic, from Weidmann’s parameters):

Weidmann’s curve at a few densitiesSpeed from the formula; flow = density × speed.
ρ (p/m²)	v (m/s)	Jₛ (p/m/s)	Regime
0.5	1.33	0.66	Free flow
1	1.06	1.06	Free flow, slowing
1.75	0.70	1.23	The peak
3	0.33	0.99	Congested
4	0.16	0.63	Congested, shuffling
5.4	0	0	Jammed

Flow has a peak

Flow per metre of width is density times speed, Jₛ = ρ·v (lesson 2). At low density speed barely changes, so flow rises almost in proportion to density. At high density speed collapses, so flow falls. In between there is a maximum. On Weidmann’s curve it is about 1.23 p/m/s, reached at 1.75 p/m², where people walk at 0.70 m/s.

That one point divides the curve into two branches. Left of it is the free-flow branch: fairly sparse, fast-moving crowds. Right of it is the congested branch: dense, slow ones. Every flow below the maximum can be carried in two ways, once on each branch.

The same flow, two crowds. On Weidmann’s curve a specific flow of 1.0 p/m/s comes from 0.9 p/m² walking at 1.11 m/s, or from 3.0 p/m² shuffling at 0.33 m/s (our arithmetic from his parameters). A flow counter cannot tell them apart; a density measurement can.

Drag across the flow chart (or use the slider) from 0.5 to 3 p/m². Watch the flow readout rise, stop, and fall. Where is the top?
Set density to 4 p/m² and read the speed. Would you call that walking?
Tick “SFPE linear relation”. Find its peak. How far apart are the two models’ capacities, and their jam densities?
Set the width to 3 m and compare the flow through it at 1.75 p/m² and at 3.5 p/m².
Tick “Bottleneck flows measured”. Where do the measured bottleneck flows sit relative to both curves?

Drag past 1.75 p/m² and the flow falls while density keeps rising: more people, less flow. The dashed line shows the density on the other branch that carries the same flow, a crowd two or three times denser doing no better.

Weidmann’s formula with his published parameters peaks at 1.225 p/m/s, which he quotes as about 1.23. The SFPE relation is stated for about 0.54–3.8 p/m²; below 0.54 p/m² we hold its speed at the 0.54 value and draw the line dashed. Fruin’s walkway bands are from FHWA’s metric reproduction; the bottleneck range is from Seyfried and colleagues.

A ramp is operating at 3.5 p/m². Would adding more people increase or decrease the number getting through per minute? What would you do instead?

Decrease. 3.5 p/m² is well past the peak on either curve (1.75 p/m² for Weidmann, about 1.9 for SFPE), so the ramp is on the congested branch: more people means slower walking and less flow. The way to raise throughput is the opposite of what feels natural: hold people back upstream so the density on the ramp falls towards the peak. Weidmann’s curve gives about 0.82 p/m/s at 3.5 p/m² against 1.23 at the peak, so thinning the crowd could raise throughput by half.

A second curve: the SFPE relation

The SFPE Handbook (Nelson and Mowrer) uses a simpler, straight-line relation. The handbook is paywalled, so we read it here through a thesis and evacuation-software documentation that reproduce it:

$S = k - a\,k\,D \qquad F_s = S\,D = kD - a\,k\,D^2$

S = k − a·k·D Fₛ = S·D = kD − a·k·D²

S: speed, m/s
D: density, p/m²
k: 1.40 m/s for level corridors and doors
a: 0.266 m²/p
Fₛ: specific flow, p/m/s

It is stated for densities between about 0.54 p/m² (below which movement is free) and 3.8 p/m² (above which there is none). Because the flow is a downward parabola, you can find its peak with one line of calculus. Set dFₛ/dD = k − 2akD = 0, so the peak is at D = 1/(2a) ≈ 1.88 p/m², and the maximum flow is k/(4a) ≈ 1.32 p/m/s. Seyfried and colleagues quote the SFPE capacity as 1.30 p/m/s for a 1 m width, with the maximum near 1.8 p/m². For stairs the handbook gives lower k values, from 1.00 to 1.23 m/s depending on riser and tread.

Two curves, side by sideLevel walkways. SFPE values derived from k = 1.40 m/s and a = 0.266 m²/p.
	Weidmann (1993)	SFPE
Speed when uncrowded	1.34 m/s	about 1.20 m/s below 0.54 p/m²
Density at peak flow	1.75 p/m²	1.88 p/m²
Peak specific flow	1.23 p/m/s	1.32 p/m/s
Speed reaches zero at	5.4 p/m²	about 3.8 p/m²

The two agree well near the peak, within about 7% on capacity, and disagree badly in the congested branch. At 3.5 p/m² Weidmann’s crowd still shuffles forward at 0.23 m/s; the SFPE crowd has almost stopped. That matters whenever you need to estimate how a queue moves, which is lesson 5.

Weidmann himself notes that other authors put the maximum higher, at 1.4–1.5 p/m/s near 1.8–2.0 p/m². A capacity is a model choice, and a careful engineer says which curve they used.

What experiments add

Armin Seyfried’s group at Forschungszentrum Jülich and their colleagues have measured pedestrian flow in controlled experiments with volunteers, from single-file runs to bottlenecks. Three of their findings change how you should use the curves above.

Direction matters. When people walk in both directions at once, the fundamental diagram departs from the one-way version above about 1.0 p/m², and the one-way maximum is higher. A concourse with counterflow has less capacity than the same concourse with everyone going one way.

Specific flow has limits. The idea that flow scales with width, so that you can quote one number per metre, holds for densities below about 3.5 p/m². Above that, width-scaling becomes unreliable.

Bottlenecks flow faster than corridors. In bottleneck experiments with widths from 0.8 to 1.2 m, Seyfried and colleagues measured specific flows of 1.61 to 2.31 p/m/s, well above either corridor peak. Across several datasets the total flow fitted a straight line, J ≈ 1.9·b, with b the width in metres. People squeeze through a short gap in ways they cannot sustain along a corridor.

Contested. Does bottleneck capacity rise smoothly with width, or in steps as extra lanes fit? Seyfried and colleagues found it grows linearly and continuously, with no steps apart from a small edge effect near 0.7 m. Hoogendoorn and Daamen, in their own experiments, found capacity rising stepwise below 3 m. People formed lanes about 45 cm apart, interleaved like a zip, so a 1 m gap held two lanes and a 2 m gap four or five short-lived ones. The question is not settled. It matters most for modelling, where it decides whether a model needs to represent lanes.

Go deeper: where the data come from, and why they scatter

Every curve in this lesson is a fit to noisy measurements. Older data, the kind Weidmann synthesised, came from counting and timing people in streets and stations. Modern data come from overhead video, with software tracking each head frame by frame; the Jülich group publishes many of its trajectory datasets in an open archive.

The scatter has several sources. Density and speed depend on how and over what area you measure them. The people differ: age, fitness, culture, what they are carrying and where they are going all shift the curve, and studies in different countries report different curves. The setting differs too: corridor or bottleneck, one direction or two, a relaxed walk or a hurry. Treat any single curve as a reasonable centre line, not a law, and expect a site you measure yourself to differ.

Why the maximum matters

The peak of the fundamental diagram is the most a metre of width can pass, so it sets the capacity of an opening: the largest flow it can sustain, which is the peak specific flow times the width. Every exit-width calculation, every egress time and every queue estimate is built on it.

A worked example shows how much the choice of number matters. A stand empties through a corridor 3 m wide.

Weidmann’s peak: 1.23 p/m/s × 3 m × 60 = about 221 p/min.
SFPE peak: 1.32 × 3 × 60 = about 238 p/min.
The Green Guide’s level rate of passage, 82 p/m/min: 82 × 3 = 246 p/min.

For 5,000 people that is between about 20 and 23 minutes, a spread of more than two minutes from the choice of curve alone. Notice also that the Green Guide’s 82 p/m/min (1.37 p/m/s) sits slightly above both corridor peaks, though well below the bottleneck flows measured at Jülich. Rates come from different sources with different conventions: when you use one, know where it came from.

Capacity is the peak of the flow curve times the width. Below the peak, more people means more flow. Above it, more people means less flow and more density, which is where people get hurt. Crowd management is largely the art of keeping crowds on the left-hand side of this curve.

Using Weidmann’s formula, what speed and specific flow do you expect at 1 p/m²? Is that crowd on the free-flow or the congested branch?

1/ρ − 1/ρmax = 1 − 0.185 = 0.815; × 1.913 = 1.559; exp(−1.559) ≈ 0.210; v = 1.34 × (1 − 0.210) ≈ 1.06 m/s. The specific flow is 1 × 1.06 ≈ 1.06 p/m/s. Since 1 p/m² is below the peak density of 1.75, this crowd is on the free-flow branch.