Daamen, Hoogendoorn e Bovy, 2005

Daamen, W, Hoogendoorn, SP, & Bovy, PHL(2005). First-order Pedestrian Traffic Flow Theory. In Transportation Research Board Annual Meeting 2005 (pp. 1-14). Washington DC: National Academy Press.

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First-order Pedestrian Traffic Flow Theory

Winnie Daamen (corresponding author) Transport & Planning Department Faculty of Civil Engineering and Geosciences Delft University of Technology Stevinweg 1, PO Box 5048, 2600 GA Delft – The Netherlands phone +31 15 278 40 31 fax +31 15 278 31 79 [email protected] Serge P. Hoogendoorn Transport & Planning Department Faculty of Civil Engineering and Geosciences Delft University of Technology Stevinweg 1, PO Box 5048, 2600 GA Delft – The Netherlands phone +31 15 278 54 75 fax +31 15 278 31 79 [email protected] Piet H.L. Bovy Transport & Planning Department Faculty of Civil Engineering and Geosciences Delft University of Technology Stevinweg 1, PO Box 5048, 2600 GA Delft – The Netherlands phone +31 15 278 46 11 fax +31 15 278 31 79 [email protected] Abstract. This paper discusses the validity of first-order traffic flow theory to describe two-dimensional pedestrian flow operations in case of an oversaturated bottleneck in front of which a large high-density region has formed. We show how observations of density, speed and flow that have been collected from laboratory walking experiments can be interpreted from the viewpoint of first-order theory.

It is observed that pedestrians present at the same cross-section inside of the congested region may encounter different flow conditions. This mainly depends on the lateral position of the pedestrian with respect to the centre of the congested region. In the lateral centre, high densities and low speeds are observed. However, on the boundary of the congested region, pedestrians may walk in nearly free flow conditions and literally walk around this congested region.

Visualizing these data in the flow-density plane results in a large scatter of points that have similar flows (bottleneck capacity), but different densities. This can be explained by noticing that observations on congestion of pedestrian traffic over the total width of the cross-section do not belong to a single fundamental diagram but to a set of different fundamental diagrams. This observation has consequences for the estimation of the fundamental diagram describing pedestrian traffic. Word count Abstract 206 Main text 3672 Figures (12 x 250) 3000 Total 6878


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INTRODUCTION

Insight into pedestrian behavior is essential in the planning and design process of public pedestrian facilities, such as transfer stations, airports, inner cities and shopping malls. Managing pedestrian flows through these facilities requires knowledge of pedestrian flow characteristics as well as of the walking behavior that constitutes the flow. Designers of pedestrian facilities often use flow characteristics to determine levels-of-service on specific parts of the facility.

Given the fact that, on average, pedestrians behave the same under similar average conditions, a statistical relation exists between speed, flow and density – the fundamental diagram. Many researchers have reported their empirical findings on this particular aspect, including the flow-density relation for various types of infrastructure, flow composition, etc. Examples of these fundamental diagrams are shown in FIGURE 1, derived from Fruin [1], Weidmann [2], Virkler [3], Older [4], Sarkar [5] and Tanariboon [6].

0 2 4 60

0.5

1

1.5

Density (P/m )2

Spee

d (m

/s)

0 0.5 1 1.5 20

0.5

1

1.5

Flow (P/ms)

Spee

d (m

/s)

0 2 4 60

0.5

1

1.5

2

Density (P/m )2

Flow

(P/m

s)

FruinWeidmannVirklerOlderSarkarTanariboon

FIGURE 1 Fundamental diagrams from literature

First-order pedestrian traffic flow theory combines the use of the fundamental diagram with the conservation of pedestrians, which holds equally for car traffic. This theory has among other things been applied in the pedestrian simulation tool SimPed [7]. Other continuum theories describing pedestrian traffic have been derived by among others Hughes [8], Helbing [9] and Hoogendoorn [10]. Hughes derived equations of motion governing the two-dimensional flow of pedestrians derived for flows of both single and multiple pedestrian types. He distinguishes between two regimes of flow, a high-density (subcritical) flow regime and a low-density (supercritical) flow regime. Helbing and Hoogendoorn have applied the gas-kinetic theory in which the similarities between pedestrian flows and fluids have been the starting point.

Experimental microscopic pedestrian data

The Transport & Planning Department of the Delft University of Technology has performed experimental research by organizing controlled walking behavior experiments. Main advantages of performing experiments is the control of the conditions – both with respect to the observed situation and the system of data collection, such as location of the camera, ambient and weather conditions, etc. – and the flexibility to systematically vary the experimental variables to see effects of these variables on the behavior of individual pedestrians and of the total pedestrian flow. It may be argued that the behavior of pedestrians during the experiments will not be very different from their real-life walking behavior since walking is mostly a skill-based task, thus requiring little or none conscious consideration. For further comparisons and more information see [11].


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The microscopic pedestrian data available enable performing innovative and detailed studies on walkers’ behavior and the spatial-dynamic patterns characterizing it. This paper will focus on the fundamental diagram derived from such experiments and shows the applicability of first-order theory to explain two-dimensional pedestrian flow operations.

In the following, the experimental design will be described briefly. Based on the collected data, we derive fundamental diagrams using cumulative curves for two cross-sections. Next, we discuss the congested state observations in the flow-density diagram. To find an explanation for seemingly confusing results, we focus on the lateral positions where pedestrians pass a specific cross-section and look at the corresponding speeds. We show that the physical form of the congestion influences the observations of the flow on the total cross-section. This indicates an essential difference in the use of the fundamental diagram in pedestrian traffic compared to car traffic. Finally, we end with some conclusions.

PERFORMING LABORATORY WALKING EXPERIMENTS

By performing walking experiments, we can determine the stimuli, the walkers’ responses and the relations between them that determine pedestrian behavior. Apart from the methodological advantages, experiments allow observations of conditions that are not readily available or are very difficult to observe. The process variables are both the input and output variables that are deemed relevant, for details see [12].

Ten walking experiments were conducted in a large hallway. The ambient conditions were favorable. A digital camera was mounted at the ceiling of the hallway, at a height of 10 m, observing an area of approximately 14 m by 12 m. In each of these experiments approximately 75 pedestrians were involved, not only TU Delft students, but representative for the Dutch population. This paper only considers the narrow bottleneck experiment, in which the observed area had a length of 10 m and a width of 4 m. The experiments lasted between 15 to 25 min. The experiments involve pedestrians gradually entering the walking area. This gradual increase in traffic demand was achieved by dividing the participants into heterogeneous groups (consisting of men and women of different ages). Having entered, the participants walked till the end of the area, after which they walked around it and re-entered the area directly. At the end of the experiment, participants were removed group-by-group from the walking area.

The narrow bottleneck experiment is characterized by the presence of a bottleneck having a length of 5 m and a width of 1 m. The bottleneck width is such that pedestrians inside of the bottleneck are not able to pass each other. As pedestrian demand increases, it will exceed the capacity of the bottleneck at some moment. From that time onward, congestion appears just upstream of the bottleneck moving upstream toward the entry of the area. After removing pedestrians from the observed area (see FIGURE 2) group by group, congestion resolved in due time. Hoogendoorn et al. [13] discuss the approach to extract individual pedestrian data from digital video footage, allowing pedestrian trajectories to be determined with high accuracy.

FIGURE 2 shows the view from above on the narrow bottleneck experiment. All pedestrians wear caps and white shirts to facilitate automated detection and tracking of the pedestrians. The bottleneck of 1 m wide is situated on the left, whereas pedestrians walk from the right to the left. The white circles indicate cones marking the observation area.


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FIGURE 2 Overview of the narrow bottleneck experiment

DERIVATION OF THE FUNDAMENTAL DIAGRAM

In this section, fundamental diagrams are derived from cumulative flow plots (see FIGURE 3). A cumulative plot of pedestrians is a function N(x,t) that represents the counted number of pedestrians that pass a cross-section x from an arbitrary starting moment. The flow measured at a cross-section x during a time period from t1 to t2 equals:

( ) ( ) ( )2 11 2

2 1

, ,,

N x t N x tq x t to t

t t

−=

− (1)

At each time instant t when a pedestrian passes cross-section x, the speed of this pedestrian is measured as well. The density at spot x and instant t is then derived from the fundamental relation between speed, flow and density:

( ) ( )( )

,,

,

q x tk x t

u x t= (2)

This fundamental relation mainly exists during homogeneous conditions, while during congestion steady state conditions do not hold and dynamic changes might violate the fundamental equation. In the following, it is shown that the calculated densities correspond to the observed densities, using Edie’s definitions [14].

We consider a small, three-dimensional cell C with dimensions X × Y × T. For all pedestrian trajectories passing through the cell, we determine three quantities, namely:

1. The travel time 0 < TTi ≤ T defined by the duration pedestrian i is in cell X × Y × T; 2. The traveled distance 0 < Di ≤ X in the x-direction, defined by the distance pedestrian i walks in the x

direction during his stay in cell X × Y × T; 3. The traveled distance 0 < Zi ≤ X in the y-direction, defined by the distance pedestrian i walks in the y

direction during his stay in cell X × Y × T

From these quantities, the generalized definition of density and flow in the x and y direction are given by the following equations:

Generalized definition of density k (in P/m2):

ii CTT

kXYT

∈= ∑ (3)

Generalized definition of flow in x and y direction respectively (in P/ms):

and i ii C i Cx y

D Zq q

XYT XYT∈ ∈= =∑ ∑ (4)


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Similar to the one-dimensional definition of Edie, it can be easily shown that upon taking the limit T ↓ 0 (implying TTi = T), the generalized definition of the density k yields the classical (instantaneous) definition of the density:

iTT T

ii CTT nT n

kXYT XYT XY

=∈= = =∑ (5)

where n equals the number of pedestrians on area X × Y at a certain time instant; the same holds for the generalized flow definitions qx and qy for taking the limit X ↓ 0 and Y ↓ 0 respectively.

Furthermore, the pedestrian speeds in x and y directions can be determined from the definitions easily as follows:

and i iyx i C i Cx y

i ii C i C

D Zqqv v

k TT k TT∈ ∈

∈ ∈

= = = =∑ ∑∑ ∑

(6)

These data have been aggregated for a fixed number of pedestrians (here N = 30) passing cross-section x. Usually, an aggregation is performed on a fixed period of time, but this may result in a very low number of observations when flows are low. Rather than fixed interval lengths, we consider fixed sample sizes. That is, starting at some time instant tn, the number of observed pedestrians is accumulated until a fixed number N (here 30) of observations are collected. Then, the averages of the relevant traffic flow variable can be determined. From a statistical point of view, this method offers important merits, as described in [15]. Among other things, the averages have comparable statistical accuracy, independent of the occurring flow and pedestrian flow composition.

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

Passage time (s)

Cum

ulat

ive

num

ber o

f pas

sing

ped

estr

ians

FIGURE 3 Cumulative curve at x = 7.0m

Fundamental diagrams have been constructed, based on the data of the narrow bottleneck experiment (total duration 15 minutes), since this is the only experiment for which the congestion part of the fundamental diagram could be estimated. The cross-sections on which speed and flow are derived are situated both inside (x = 4 m) and in front (x = 7 m) of the bottleneck (see FIGURE 4d). FIGURE 4 also shows observations in the three phase-spaces (speed-density phase-space in FIGURE 4a; speed-flow phase-space in FIGURE 4b; flow-density phase-space in FIGURE 4c).


6

0 1 2 3 40

0.5

1

1.5

2

Density (P/m)

Flow

(P/s

)

0.20.40.60.8

11.21.41.6

00

1 2 3 4Density (P/m)

Spee

d (m

/s)

x = 4 mx = 7 m

x = 4 mx = 7 m

x = 4 m x = 7 m

x = 4 mx = 7 m

0.20.40.60.8

11.21.41.6

00

0.5 1 1.5 2Flow (P/s)

Spee

d (m

/s)

(c) (d)

(a) (b)

Narrow bottleneck experiment

FIGURE 4 Observations in the three phase-spaces of the narrow bottleneck experiment

for two cross-sections

Although congestion occurs, pedestrians continue walking with speeds higher than 0.4 m/s. All three phase-spaces indicate high variance during congestion, whereas we would have expected a smaller range of observation. Specifically the large scatter at different densities for similar flows (equal to the bottleneck capacity) is remarkable (see dotted ellipse in FIGURE 4c). Since one might hypothesize that walkers adapt their following and speed choice behavior if confronted with high density and low speed conditions during a longer period of time, we would like to see whether pedestrian behavior changes over time. To that end FIGURE 5 shows speeds, flows and densities during the narrow bottleneck experiment for the complete width of both cross-sections.

First, the flow increases, until after 300 s capacity of the bottleneck is reached. Congestion occurs upstream of the bottleneck, which is reason for the decrease in speeds measured at the cross-section at x = 7 m. This capacity situation is maintained for a period of 400 s. After 600 s, we started to reduce the number of groups participating in the experiment. After 700 s, free flow conditions are restored; flows decrease and speeds increase again. During the capacity situation, both density and speed are only slightly decreasing over time, but not sufficiently to conclude that pedestrian behavior has changed during the experiment.


7

0 100 200 300 400 500 600 700 800 9000

0.5

1

1.5

2

Time (seconds)

Spee

d (m

/s)

x = 4mx = 7m

0 100 200 300 400 500 600 700 800 9000

0.5

1

1.5

2

Time (seconds)

Flow

(P/s

)

0 100 200 300 400 500 600 700 800 9000

1

2

3

4

Time (seconds)

Den

sity

(P/m

)

FIGURE 5 Speed, flow and density over time in and in front of the bottleneck during the experiment

FIRST-ORDER PEDESTRIAN TRAFFIC FLOW THEORY

Let us reconsider FIGURE 4, and focus on the large variance of the congested measurements. This large variance is observed in particular in the flow-density plane, and cannot be contributed to random noise, representing changes in pedestrian behavior during congestion. In the following, an explanation for this phenomenon is presented.

For this explanation, we will go back to traffic flow theory for cars in a similar bottleneck situation. Then, we apply this theory to pedestrian traffic. It appears that the width actually occupied by pedestrians upstream of the bottleneck plays an important role. Therefore, we will look at the lateral positions where pedestrians pass a specific cross-section as well as to their speeds at those positions. Finally, we will show our explanation for this phenomenon.

First, let us consider a related situation in car-traffic. FIGURE 6a shows the flow-density relation common for car traffic [16]. The situation is a lane drop of a two-lane road, being similar to the narrow bottleneck experiment for pedestrians. Two fundamental diagrams are applied: one for the two-lane part of the road upstream of the bottleneck (solid line) and another one (grey stripe-dotted line) inside the bottleneck, behind the lane drop. During small flows, the free flow part of the fundamental diagram is observed (solid ellipse in FIGURE 6a). When the flow increases until the bottleneck capacity is reached, congestion occurs upstream of the bottleneck. Observations are found on the congestion branch of the two lane fundamental diagram for the solid cross-section (dotted ellipse on the right in FIGURE 6a) as well as on the capacity part of the single lane fundamental diagram for the grey stripe-dotted cross-section (dotted ellipse on the left in FIGURE 6a).


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Density (veh/m)

Flow

(veh

/h)

2 lanes

1 lane

Flow

(P/s

)

Density (P/m)

1m wide

2m wide

3m wide

4m wide

b. Pedestrian traffic

a. Car traffic

Free flow observationsCongestion observations

FIGURE 6 Traffic flow theory for both car traffic and pedestrian traffic respectively

However, in the flow-density plane of FIGURE 4 the observations on traffic congestion occurring upstream of the bottleneck do not appear to be concentrated around a single point in the phase space. FIGURE 6b shows the flow-density diagram for pedestrian traffic in the narrow bottleneck experiment. Again, flows and densities have been determined at two cross-sections. The solid fundamental diagram is valid for the total width upstream of the bottleneck (w = 4 m), while the grey stripe-dotted fundamental diagram applies inside of the bottleneck (w = 1 m). However, when pedestrians do not occupy the complete width of the area upstream of the bottleneck, another fundamental diagram applies. Therefore, for each width between 1 m (bottleneck width) and 4 m (complete width of the area upstream of the bottleneck) a different fundamental diagram seems to apply. FIGURE 6b shows several flow-density relations, all having a similar form, but applicable for different widths.

The flow-density diagrams in FIGURE 6b are valid for the total area width. However, also a fundamental diagram may be derived per meter width. FIGURE 7 shows these two types of fundamental diagrams and indicates the locations of corresponding phases in the diagrams.


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Density (P/m)

Flow

(P/s

)

1m wide

2m wide

3m wide

4m wide

Density (P/m )2

Flow

(P/m

s)

1

1

2

2

3

3

4

4

55

6 67 7

Observations in congestion for different widths

Walking speeds

Bottleneck capacity

FIGURE 7 Comparison of the flow-density diagram for the total bottleneck width (on the left) and the flow-density diagram per meter width (on the right)

The width used by pedestrians upstream of the bottleneck is thus very important to be able to derive a proper fundamental diagram. When pedestrians use the complete width upstream of the bottleneck homogeneously, the fundamental diagram for a walkway of 4 m may be applied. This is similar to car traffic flow theory. However, the question is whether pedestrians do occupy the complete width of the area upstream of the bottleneck, and if they do, whether the distribution over the width is homogeneous.

To answer this question, FIGURE 8 gives an overview of various pedestrian trajectories during congestion. This figure shows that pedestrians form a funnel-shaped group while waiting to enter the bottleneck. Only part of the cross-section is thus occupied.

walking direction

Pedestrian trajectory

Lat

eral

wid

th

Cross-section at x = 7 m

FIGURE 8 Trajectories in space projection for pedestrians in congestion

However, FIGURE 8 only shows trajectories during a very short period of time, whereas FIGURE 9 shows the average spatial form that waiting pedestrians adopt during the total congestion period. FIGURE 9 only shows the area upstream of the bottleneck, which is located on the left side of the figure between the lateral positions y = 1.5 m and y = 2.5 m. According to the figure, all pedestrians pass the area just upstream of the bottleneck (between x = 5 m and x = 5.5 m). The further upstream of the bottleneck, the larger width is occupied by pedestrians. However, the outsides of the funnel are only used by about 10% of the pedestrians (in terms of density), while most pedestrians use the centre of a cross-section. The scale values for the density are indicated as percentages of the maximum observed density on the right hand side of FIGURE 9, where 1 indicates that all pedestrians have passed this location, whereas 0 indicates that none of the pedestrians have passed.


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5.55 6 6.5 7 7.5 8 8.5 9 9.5 10

0.5

1

0

1.5

2

2.5

3

3.5

4

x (m)

y (m

)

0

0.17

0.33

0.5

0.67

0.83

1

FIGURE 9 Average density of pedestrians over 2-D space during congestion upstream of the bottleneck

FIGURE 10 shows another macroscopic characteristic of pedestrian flows, namely speed. Flow and speed are namely directly observed on the cross-section, whereas density is derived from these characteristics. The speeds are shown over time in relation to the lateral positions of pedestrians passing cross-section x = 7 m, situated upstream of the bottleneck. Each dot in the figure is an observation of a single pedestrian, labeled in a speed category (see legend of FIGURE 10). The horizontal axis shows the time during the experiment. We can see that at low flows (t < 300 s and t > 700 s) the pedestrian flow is concentrated at the same lateral positions as the bottleneck (lateral position between 1.5 m and 2.5 m). During congestion however, the lateral positions of the pedestrians on the cross section vary significantly. Although FIGURE 9 shows a regular distribution of pedestrians over the width, FIGURE 10 indicates that this distribution varies significantly over time. Looking at the different speed categories, it is clearly visible that walking speeds are high at low flows. This walking speed decreases in congestion. However, at the outsides of the pedestrian flow, pedestrians still encounter (nearly) free flow conditions (even during congestion) and are thus able to maintain a higher speed.


11

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

Time (seconds)

Loc

atio

n (m

etre

s)

v [0.0;0.4]v [0.4;0.8]v [0.8;1.2]v [1.2;1.6]v [1.6;2.0]

FIGURE 10 Speeds according to the lateral location where pedestrians pass the cross-section at x=7m for the total simulation period

FIGURE 11 shows the minimum, average and maximum walking speeds over the lateral position where a pedestrian passes the cross-section at x = 7 m. As well as in FIGURE 10, it is clear that speeds are higher at the outsides of the flow than in front of the bottleneck. The three lines come together on the right side of the figure, as only during a single time period pedestrians have passed at this position.


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0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

Lateral position in front of bottleneck (m)

Spee

d (m

/s)

Average

Minimum

Maximum

FIGURE 11 Speeds as function of the lateral position in a cross-section for different time periods in congestion

The previous paragraphs have indicated that it does not make very much sense to derive a single flow-density relation for the complete width of an area upstream of the bottleneck. In fact, a number of points may be distinguished, forming together one observation in the flow-density phase-space in FIGURE 4. This is made clear in FIGURE 12. As an example, we will distinguish three equilibrium regions, having similar speed and flow. The two outer regions (with densities k1 and k3) are more or less free flow, whereas congestion occurs upstream of the bottleneck (with density k2). The three observations are indicated in the flow-density diagram by grey dots. However, the observation in the flow-density plane in FIGURE 4 is based on a combination of these three equilibrium points. The result is that the aggregate observation (indicated by the black dot in FIGURE 12) may be located anywhere on the horizontal dotted line and does not belong to a specific fundamental diagram. The flow of the aggregate observations only varies slightly when the flow through the bottleneck approaches bottleneck capacity, which is constant. The congestion part of the fundamental diagram therefore cannot be estimated using aggregate observations for the complete width of a cross-section upstream of the bottleneck.


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Density (P/m)

Flow

(P/s

)

1m wide

2m wide

3m wide

4m wide

1.5m

1.5m

1.0mk1

qk1

k2

qk2

qktot

k3

qk3

Measurement for part of the cross-section

Measurement for complete cross-section

FIGURE 12 Composition of a measurement point in the flow-density diagram for the complete cross-section

CONCLUSIONS

In this contribution, we have discussed results of dedicated experiments conducted to gain more insights into walking behavior. This paper focuses on the applicability of fundamental diagrams describing pedestrian flow operations in congestion.

One of the experiments concerned a narrow bottleneck. Pedestrian demand was so large that congestion occurred. Based on observations at two cross-sections (one inside of the bottleneck and one upstream), fundamental diagrams are shown describing pedestrian traffic.

It turns out that during congestion, pedestrians form a funnel-shaped group upstream of the bottleneck. However, the width of this group at the cross-section varies over time. Also, pedestrians meet different conditions at the cross-section (at the same time), mainly depending on their lateral position. In the lateral centre, high densities and low speeds are observed. However, on the boundary of the congested region, pedestrians may walk in nearly free flow conditions and literally walk around the congestion. An observation in the flow-density diagram thus consists of several observations, each describing a smaller range of the cross-section, in which similar, homogeneous conditions occur.

We can therefore state that the congestion part of the fundamental diagram cannot be estimated using aggregate observations for the complete width of a cross-section upstream of the bottleneck. Instead, homogeneous parts should be distinguished in a cross-section. Since we know the widths of these homogeneous parts, we also know to which fundamental diagram a specific (aggregate) observation belongs.

Using experimental data, some remarks have to be made with respect to the generalization of the results. Real pedestrians in true high demand situations may exhibit behavioral characteristics that are different from what was observed in the experiments. Real pedestrians may have specific objectives of where they need to be and when they need to go there. As such, their behavior, in interaction with large masses of other similarly goal-oriented pedestrians may generate interactions of a kind that was not present in the experiments. Weidmann [2] already found different free speeds according to the trip motive of pedestrians. Furthermore, the experiments have been performed in the Netherlands and will be representative for the Dutch situation. Especially in Asian countries, densities and speeds differ significantly from those in the western countries [11]. Finally, pedestrian behavior might depend on queue length and waiting time, that is significantly longer queues or waiting times may result in changes in pedestrian behavior. Although we assume that the above-mentioned remarks do not have much influence on the results presented in this paper, a subsequent paper will contain similar calculations on observations from reality.

ACKNOWLEDGEMENTS This research has been funded by the Social Science Research Council (MaGW) of the Netherlands Organization


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for Scientific Research (NWO). We are grateful to the constructive comments of a number of anonymous reviewers. REFERENCES

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Daamen, Hoogendoorn e Bovy, 2005

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