Self-organization in walker experiments
Serge Hoogendoorn1 and Winnie Daamen1.
1
Delft University of Technology
Faculty of Civil Engineering and Geosciences
Stevinweg 1, email: [email protected]
Abstract. Microscopic simulation models predict different forms of selforganization in pedestrian flows, such as the dynamic formation of lanes in
bi-directional pedestrian flows. This paper proposes a theory predicting selforganization, as well as results from experimental research that provides
more insight into these dynamic phenomena. Groups of pedestrians that are
homogeneous in terms of desired walking speeds and direction appear to
form structures consisting of overlapping layers. This basic pattern forms the
basis of other more complex patterns emerging in multi-directional pedestrian flow: in a bi-directional pedestrian flow, dynamic lanes are formed
which can be described by the layer structure. Diagonal patterns can be identified in crossing pedestrian flows. This paper both describes these structures,
as well as the implications for theory and modeling of pedestrian flows.
Keywords: pedestrian flow, self-organization, experimental research
1 Introduction
In the early seventies, a substantial body of research was established on pedestrian behavior and the interaction of pedestrians with their environment
and other pedestrians. These studies focused primarily on the social and psychological aspects of walking, such as the way pedestrians observe the walking infrastructure via subliminal scanning [1], interact with other pedestrians
via subconscious communication and cooperation [2-5]. Also, attention was
paid to the way social and cultural differences affect these different processes
[2,3,6].
For the design of walking infrastructure, a working knowledge of the characteristics of pedestrian flows is required in order to design the infrastructure as
well as to assess its efficiency and safety. In particular, a good understanding
of the emergent patterns is required to predict how the flow will behave under different circumstances. This knowledge is generally based on results of
microscopic simulation studies [7-9] rather than detailed empirical or experimental research on how individual pedestrians act under a variety of traffic flow or environmental conditions. These models predict the formation of a
variety of structures in pedestrian flows, such as dynamic lanes in bidirectional flows, or strips in crossing flows. Empirical evidence indeed
points towards the existence of these self-organizing phenomena [10,11], but
so far no experimental or empirical studies have provided definite results
regarding quantitative characteristics of these self-organized structures.
2
Behavioral theory of self-organization
Before presenting the results of the walking experiments and the observed
self-organization therein, we propose a theory of self-organization in pedestrian flow. The theory is based on the assumption that each pedestrian aims to
minimize his or hers predicted disutility of walking (i.e. the pedestrian
economicus [9]): for all available options (e.g. accelerating, decelerating,
changing direction, do nothing), a pedestrian tries to choose the option that
will yield the smallest predicted disutility. In psychology, this behavioral
assumption is referred to as the principle of least effort, i.e. an individual will
try to adapt to his or her environment (or will try to change the environment
to suit its needs, whichever is easier); see [15]. In predicting the disutility of
walking, a pedestrian values the different attributes characterizing the available options (e.g. risk to collide with another pedestrian, straying from the
intended walking path, physical contact with other pedestrians, etc.) differently.
Under specific conditions, we may expect that given this assumption, the
pedestrian flow will evolve to an user-equilibrium state (which is either stable or meta-stable) in which no pedestrian can improve his or hers condition
by unilaterally undertaking an action. The hypothesis is that the selfformation of homogeneous patterns (e.g. dynamic lane formation, or the formation of diagonal strips) is equivalent to the evolution of the aforementioned user-equilibrium state.
In illustration, consider a situation in which all pedestrians have the same
physical characteristics (e.g. desired walking speed, size, etc.). Clearly, in
case of a unidirectional flow, the theory predicts a homogeneous distribution
of pedestrians over the walking: each pedestrian has the same speed and there
is no use of overtaking other pedestrians. This results a homogeneous pattern
in which pedestrians walk behind each other in layers that overlap partially.
Note that this staggered pattern is optimal from the viewpoint of efficient use
of available walking space.
When the flow is heterogeneous (e.g. regarding the walking speeds), changing speed or direction may improve the conditions of an individual, since
slower pedestrians with hold back faster ones. Faster pedestrians will overtake only if overtaking will result in a substantially improved situation. When
such benefits are not sufficient, that is, when the switching costs are higher
than the disutility of staying in the current layer, the pedestrian aiming to
walk fast will stay behind the slow pedestrian: the disutility of not being able
to walk at the desired velocity is in this case compensated by the reduced
probability of colliding with other pedestrians and the switching cost from
the current layer to another (caused by e.g. needing to cross through another
layer). Depending on the pedestrian population, for inhomogeneous unidirectional flows, the theory thus predicts the formation of homogeneous structures (pedestrian have equals speeds) with in between individual pedestrians
(or small groups) walking at a higher speed.
For bi-directional flows, user-optimal states are states in which lanes of uniform walking directions. In this case, the system optimal situation would be
two regions of pedestrians moving in opposite directions, since this situation
yields the least friction between the regions, and thus the smallest collective
disutility. This ideal state may not occur due to initial conditions, heterogeneous composition of the flows (with respect to the desired walking speed)
combined with overtaking opportunities, etc. This also implies that the number of lanes that are formed will also depend on the density, since the latter
determines overtaking opportunities.
Applying the theory to crossing pedestrian flows will also lead to the formation of homogeneous structures, i.e. to the formation of diagonal strips. The
shape of these diagonal strips yields a situation in which the system has the
least collective disutility when the two flows cross.
Clearly, which self-organized structures will appear depends on a number of
things amongst which are the initial and boundary conditions. The theory
does also not preclude the emergence of non-optimal meta-stable states (e.g.
the formation of multiple lanes in a bi-directional flow), which can only
change in an optimal state when pedestrians ‘invest’ by (temporarily) accepting high disutilities (e.g. by crossing a flow moving in the opposite direction).
3
Walker experiments
Motivated by the lack of experimental knowledge of self-organization in pedestrian flow, Delft University of Technology conducted a large-scale walking experiment to gain a better understanding of emergent structures in pedestrian flow. In total, 10 different walking experiments were performed, 5 of
which are relevant for the results presented in this paper. In each of these
experiments, approximately 60-90 individuals were involved. With a highpositioned digital camera, an area of approximately 14 m by 12 m was observed. Trajectories of all pedestrians were determined from the video data
using dedicated tracking software. The experiments that are relevant within
the scope of this contribution are shown in Tab. 1.
In all considered experiments, participants were asked to walk as they normally would. The composition of the populations was heterogeneous, both
with respect to age, gender, etc. The pedestrians were asked to enter the
walking area, walk across, leave, and walk around to the entry of the walking
areas. Upon entering, care was taken to distribute pedestrians across the entire width of the area. For a detailed description of the experimental set-up,
we refer to [12].
Tab. 1 Overview of relevant walker experiments.
Nr. Experiment Size of walkDescription
type
ing area
Homogeneous walking conditions.
10 x 4
1.
2.
10 x 4
3.
10 x 4
4.
10 x 4
5.
8x8
4
Unidirectional flow through wide bottleneck (2 meter width)
Unidirectional flow through narrow
bottleneck (1 meter width)
50% from East to West; 50% from
West to East
50% from East to West; 50% from
North to South
Observed spatial patterns
This section discusses the different spatial patterns that have been observed
in the pedestrian experiment, and how these observations support our theory
on self-organization in pedestrian flow.
4.1
Standard pattern
The standard patterns occur in case of unidirectional pedestrian flows or sub
flows (i.e. homogeneous clusters in multi-directional flows) when there is
insufficient space to overtake slower pedestrians walking in the same direction (experiments 1-3). The standard pattern consists of a number of partly
overlapping layers. We emphasize that overlapping does not imply physical
overlapping. In [13] it is shown how pedestrians make a swerving motion
while walking, thus requiring more space than just their width of their shoulders. For walking speeds higher than 0.3 m/s, the swerving amplitude a(v) (in
m) equals approximately [13]:
a(v) = 0.068 − 0.017v
(1)
where v is the pedestrian speed in m/s.
The photographs shown in Fig. 1 depict the four types of homogenous basic
patterns that emerge, in this case in the narrow bottleneck experiment. Note
that these patterns are in line with the prediction of the formation of staggered layers based on our theory presented in section 2. The arrows in the
figure all have the same length and are used to show that the distance headways between pedestrians in a layer are approximately constant (except for
pattern 4). Pattern 1 shows the case that the distances between pedestrians in
different layers are also approximately constant, i.e. the lead gap and the lag
gap are almost equal. Patterns 2 and 3 are similar, and reflect cases in which
the lead gap for the pedestrians in one layer is relatively small, while the lag
gap is relatively large (or vice-versa).
Pattern 4 is a special pattern, the properties of which can be described using
analogies with defects in solids. This particular defect is referred to as a vacancy1 in solid-state physics. In a pedestrian flow, this defect is caused by
inefficient merging behavior at the bottleneck entry, e.g. due to overly polite
1
A vacancy is defined by a lattice position that is vacant because an atom is missing.
or aggressive behavior. The latter may be more proficient in case of panic
situations, causing more defects in the pattern and a less efficient usage of the
bottleneck. Despite the fact that quite a number of defects were observed, it is
noted that patterns 1 to 3 are more common. The average distances between
pedestrians in longitudinal and lateral sense depends primarily on the average
speed of the pedestrian flow. For saturation flows (i.e. speeds in the bottleneck after congestion has occurred), this speed is around 1 m/s; the average
longitudinal distance between pedestrians (within a layer) is 1.3 m; the centers of the overlapping layers are on average 45 cm apart.
By estimating a so-called composite headway distribution model, we have
established that the minimal time headway needed for safe and comfortable
following2 (the so-called empty zone3) of pedestrians is on average 1.4 s for
all considered experiments; the standard deviation of the empty zone distribution is between 0.4 s and 0.5 s. This empty zone describes the time headway
that a pedestrians requires to safely and comfortably take a step, and thus
depends on the step size, stepping frequency and walking speed. As is shown
in the following section, this also holds for the self-organized structures that
appear in bi-directional or crossing flows. For details, we refer to [13].
Tab. 2 Overview of mean and standard deviation of empty zone distribution
for different experiments.
Empty zone distribution
ExperiDirecFollowing pedesStandard deviament
tion
trian fraction
Mean (s)
tion (s)
1
28%
1.43
0.49
I
2
61%
1.36
0.45
I
3
I
72%
1.37
0.43
36%
1.42
0.50
I
4
52%
1.42
0.54
J
16%
1.43
0.64
I
5
19%
1.37
0.52
L
4.2
Self-organization of standard patterns
One of the typical phenomena occurring in pedestrian flow is selforganization. Several types of self-organization have been described in literature [7,8,10,11]. This section provides an overview of the self-organized
structures observed in the walking experiments.
4.2.1
Lane formation in bi-directional pedestrian flows
For the bi-directional flow experiment (experiment 4), it was observed that
standard patterns are formed in both walking directions. An example is
shown in Fig. 2, where two of these patterns are formed. Note that due to the
fact that relatively much space is available, overtaking opportunities do exist
and the patterns are relatively short-lived. Nevertheless, analyzing the formed
2
The time headway is defined by the difference between the time instants of consecutive pedestrians in the same layer passing a cross-section x
3
The empty zone is defined by the minimum time headway of a following pedestrian
with respect to the leading pedestrian in the same layer.
structures shows that their characteristics are similar for the bi-directional and
the unidirectional flow experiments. Using the composite headway model
estimation method of [13], it was determined that the mean empty zone of
pedestrians in the same layer is approximately 1.4 s (see Tab. 2).
This type of self-organization in pedestrian flow has been observed by other
researchers and appears to be characteristic of bi-directional pedestrian flows.
In [14], it has been proposed that the number of formed lanes depends on the
density in the walking area. In this paper, we have studies this quantitatively
by application of cluster analysis. A cluster is in this case defined using the
G
G
locations ri (t ) and velocities vi (t ) , according to the following criteria:
G
G
1. The distance || ri (t ) − rj (t ) || between i and j is less than threshold c1, and
G
G
2. The velocity difference || vi (t ) − v j (t ) || is less than some value c2.
Using standard cluster analysis, clusters of pedestrians can be identified.
Please not that the clusters and dynamic lanes are not be definition the same.
In general, a dynamic lane can be made up from several clusters (see Fig. 3).
A cluster can however not belong to more than one lane. This distinction is
due to the distance criterion 1 used to define a cluster.
Fig. 3 shows results from application of cluster analysis for four different
time-slices. The figure depicts different kinds of lane formation. Note that
upon entering the walking area, pedestrians were instructed to spread uniformly over the width of the area.
As predicted by the theory proposed in section 2, it turns out that in the experiment often two lanes are formed that are largely right-hand-sided (e.g.
Fig. 3, t = 229 s). This is typical for countries in which traffic regulations
stipulate walking on the right side of the street. However, quite frequently
different patterns emerge: Fig. 3, t = 300 s shows a fairly common situation
in which three dynamic lanes can be identified, and in which pedestrians obviously do not walk on the right side of the walking area. These situations are
caused by the random conditions at the boundaries of the walking area and
the fact that the transition from this sub-optimal three-lane state to the optimal two-lane situation will inflict a very high disutility (pedestrians having to
cross a lane flowing into the opposite direction). Note that also situations
where four lanes are formed have been recorded (e.g. Fig. 3, t = 201 s).
Fig. 4 shows the average number of clusters that are identified during the
experiment each 0.5 s as a function of density. The figure shows how the
number of clusters first increases, reaching its maximum value of 12 near a
density of 0.6 P/m2. Please note that the number of formed clusters depends
the cluster definition. Also note that a cluster may consist of a single pedestrian, and that a lane may be split up in several clusters.
Strip formation in crossing flows
4.2.2
Self-organization is not restricted to bi-directional flows: also in crossing
pedestrian flows self-organization will occur: clusters of pedestrians having
the same direction and speed will form. The homogeneous patterns that
emerge have the form of diagonal strips (see [9-11,14]). The characteristics
of the formed structures are similar to the structures formed in the unidirectional and bi-directional experiments: for instance, Tab. 2 shows that the
mean empty zone is approximately 1.4 s; the standard deviation of the empty
zone is however much larger than for the other experiments. An explanation
for this can be found by examining the way in which the diagonal strips are
formed, as is explained below.
From analyzing the video data, the following observation was made: consider
a situation where pedestrian i meets pedestrian j coming from the side. Suppose that the conflict is resolved in such a way that i will undertake some
action to prevent colliding with pedestrian j. It turns out that rather than
changing direction, i will generally reduce speed until j passes. When j has
passed, i will accelerate and continue along its intentional path. Apparently,
pedestrian i predicts that the disutility of reducing speed is less than the disutility of changing directions and walking around j. As a result, pedestrians
will accelerate and decelerate frequently, explaining the larger variability in
the empty zone distribution.
Fig. 5 shows the results of applying the cluster analysis to the crossing flow
experiment 5. The figure clearly shows the self-formation of diagonal basic
structures, as predicted by the theory proposed.
5
Implications for pedestrian theory and modeling
Self-organization is an intrinsic property of pedestrian flow, which can be
explained by considering a pedestrian as a utility optimizer, minimizing the
predicted disutility of walking. Under this assumption, self-organized structures can be considered as user-optimal equilibrium states, in which none of
the pedestrians can improve his or her predicted utility by unilaterally changing his or her situation.
As a result, predicting traffic flow operations accurately requires theory and
models that somehow capture this property, such as the models described in
[7-9,14]. For one, this is important to correctly capture the highly efficient
flow operations in case of bi-direction or crossing flows. Furthermore, practical applications often require a correct reproduction of the different spatial
structures that will result in order to correctly capture primary and secondary
congestion (spill-back) effects. Failing to capture self-organization phenomena may thus result in the incorrect prediction of these spatial patterns, and
thus yield faulty predictions of pedestrian flow in sport-stadiums, transfer
stations, etc.
On top of this, correct description of the formation of diagonal strips in crossing pedestrian flows requires correct predicted of pedestrians actions in case
of crossing another pedestrian. That is, it is essential that the model captures
the fact that pedestrians will generally not change their directions but will
decelerate and wait until the crossing pedestrian passes. As an example, the
NOMAD model [9], which is based on the concept of utility maximising pedestrians, captures this by penalizing lateral acceleration more severely than
longitudinal acceleration. As a result, the model predicts the formation of
diagonal strips correctly.
6
Summary and conclusions
This paper presents both theory and experimental results of self-organization
in pedestrian flows. We identified a basic homogeneous pattern consisting of
pedestrians having the same speed and direction. The basic pattern consist of
overlapping layers, i.e. pedestrians are aligned in a staggered fashion. Furthermore, it was argued that self-organized structures, for instance in bidirectional or crossing flows consist of these basic patterns.
The self-organized phenomena (i.e. dynamic lane formation in bi-directional
flow and formation of diagonal strips in crossing flows) that are predicted by
the theory of the utility maximizing pedestrian are all identified in the different walking experiments. Furthermore, composite headway model estimation
results show how the characteristics inside the formed patterns are similar for
the different experiments, thereby providing quantitative evidence for the
hypothesis that the formed patterns are indeed the identified basic patterns.
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Figures
Spatial pattern 1
Spatial pattern 2
Spatial pattern 3
Spatial pattern 4
Fig. 1 Emergent standard patterns, in this case for the narrow bottleneck experiment.
Fig. 2 Example of formation of standard pattern in bi-directional flow.
Fig. 3 Example results of cluster analysis for four time slices; numbers indicate pedestrian clusters. The color indicates the average direction at a particular location. Note that a lane can consist of multiple clusters.
Fig. 4 Number of formed clusters as a function of density. Note that the
number of pedestrians per cluster increases with density.
Fig. 5 Formation of diagonal strips in crossing pedestrian flow experiment.
The colors indicate the average direction at a particular location.