6476 P Pedestrian, Crowd and Evacuation Dynamics

Pedestrian, Crowdand Evacuation DynamicsDIRK HELBING1,2, ANDERS JOHANSSON11 ETH Zurich, Zurich, Switzerland2 Institute for Advanced Study, Collegium Budapest,Budapest, Hungary

Article Outline

GlossaryDenition of the SubjectIntroductionPedestrian DynamicsCrowd DynamicsEvacuation DynamicsFuture DirectionsAcknowledgmentsBibliography

Glossary

Collective intelligence Emergent functional behavior ofa large number of people that results from interactionsof individuals rather than from individual reasoning orglobal optimization.

Crowd Agglomeration of many people in the same area atthe same time. The density of the crowd is assumed tobe high enough to cause continuous interactions withor reactions to other individuals.

Crowd turbulence Unanticipated and unintended irreg-ular motion of individuals into dierent directions dueto strong and rapidly changing forces in crowds of ex-treme density.

Emergence Spontaneous establishment of a qualitativelynew behavior through non-linear interactions of manyobjects or subjects.

Evolutionary optimization Gradual optimization basedon the eect of frequently repeated randommutationsand selection processes based on some success func-tion (tness).

Faster-is-slower eect This term reects the observationthat certain processes (in evacuation situations, pro-duction, trac dynamics, or logistics) take more timeif performed at high speed. In other words, waiting can

often help to coordinate the activities of several com-peting units and to speed up the average progress.

Freezing-by-heating eect Noise-induced blockage ef-fect caused by the breakdown of direction-segregatedwalking patterns (typically two or more lanes char-acterized by a uniform direction of motion). Noisemeans frequent variations of the walking direction dueto nervousness or impatience in the crowd, e. g. alsofrequent overtaking maneuvers in dense, slowly mov-ing crowds.

Panic Breakdown of ordered, cooperative behavior of in-dividuals due to anxious reactions to a certain event.Often, panic is characterized by attempted escape ofmany individuals from a real or perceived threat insituations of a perceived struggle for survival, whichmay end up in trampling or crushing of people ina crowd.

Self-organization Spontaneous organization (i. e. forma-tion of ordered patterns) not induced by initial orboundary conditions, by regulations or constraints.Self-organization is a result of non-linear interactionsbetween many objects or subjects, and it often causesdierent kinds of spatio-temporal patterns of motion.

Social force Vector describing acceleration or decelera-tion eects that are caused by social interactions ratherthan by physical interactions or elds.

Definition of the Subject

The modeling of pedestrian motion is of great theoret-ical and practical interest. Recent experimental eortshave revealed quantitative details of pedestrian interac-tions, which have been successfully cast into mathemati-cal equations. Furthermore, corresponding computer sim-ulations of large numbers of pedestrians have been com-pared with the empirically observed dynamics of crowds.Such studies have led to a deeper understanding of howcollective behavior on a macroscopic scale emerges fromindividual human interactions. Interestingly enough, thenon-linear interactions of pedestrians lead to various com-plex, spatio-temporal pattern-formation phenomena. Thisincludes the emergence of lanes of uniform walking di-rection, oscillations of the pedestrian ow at bottlenecks,and the formation of stripes in two intersecting ows.Such self-organized patterns of motion demonstrate thatecient, intelligent collective dynamics can be basedon simple, local interactions. Under extreme conditions,however, coordinationmay break down, giving rise to crit-ical crowd conditions. Examples are freezing-by-heatingand faster-is-slower eects, but also the transition toturbulent crowd dynamics. These observations have im-

Pedestrian, Crowd and Evacuation Dynamics P 6477portant implications for the optimization of pedestrian fa-cilities, in particular for evacuation situations.

Introduction

The emergence of new, functional or complex collectivebehaviors in social systems has fascinated many scientists.One of the primary questions in this eld is how cooper-ation or coordination patterns originate based on elemen-tary individual interactions. While one could think thatthese are a result of intelligent human actions, it turns outthat much simpler models assuming automatic responsescan reproduce the observations very well. This suggeststhat humans are using their intelligence primarily formorecomplicated tasks, but also that simple interactions canlead to intelligent patterns of motion. Of course, it is rea-sonable to assume that these interactions are the result ofa previous learning process that has optimized the auto-matic response in terms of minimizing collisions and de-lays. This, however, seems to be sucient to explain mostobservations.

In this contribution, we will start with a short his-tory of pedestrian modeling and, then, introduce a sim-plied model of pedestrian interactions, the social forcemodel. Furthermore, we will discuss its calibration us-ing video tracking data. Next, we will turn to the subjectof crowd dynamics, as one typically nds the formationof large-scale spatio-temporal patterns of motion, whenmany pedestrians interact with each other. These patternswill be discussed in some detail before we will turn to evac-uation situations and cases of extreme densities, whereone can sometimes observe the breakdown of coordina-tion. Finally, we will address possibilities to design im-proved pedestrian facilities, using special evolutionary al-gorithms.

Pedestrian Dynamics

Short History of Pedestrian Modeling

Pedestrians have been empirically studied for more thanfour decades [1,2,3]. The evaluation methods initially ap-plied were based on direct observation, photographs, andtime-lapse lms. For a long time, the main goal of thesestudies was to develop a level-of-service concept [4], de-sign elements of pedestrian facilities [5,6,7,8], or planningguidelines [9,10]. The latter have usually the form of re-gression relations, which are, however, not very well suitedfor the prediction of pedestrian ows in pedestrian zonesand buildings with an exceptional architecture, or in chal-lenging evacuation situations. Therefore, a number of sim-ulation models have been proposed, e. g. queuing mod-

els [11], transition matrix models [12], and stochastic mod-els [13], which are partly related to each other. In addition,there are models for the route choice behavior of pedestri-ans [14,15].

None of these concepts adequately takes into ac-count the self-organization eects occurring in pedestriancrowds. These are the subject of recent experimental stud-ies [8,16,17,18,19,20]. Most pedestrian models, however,were formulated before. A rst modeling approach thatappears to be suited to reproduce spatio-temporal patternsof motion was proposed by Henderson [21], who conjec-tured that pedestrian crowds behave similar to gases or u-ids (see also [22]). This could be partially conrmed, buta realistic gas-kinetic or uid-dynamic theory for pedes-trians must contain corrections due to their particularinteractions (i. e. avoidance and deceleration maneuvers)which, of course, do not obeymomentum and energy con-servation. Although such a theory can be actually formu-lated [23,24], for practical applications a direct simula-tion of individual pedestrian motion is favorable, sincethis is more exible. As a consequence, pedestrian re-search mainly focuses on agent-based models of pedes-trian crowds, which also allow one to consider local co-ordination problems. The social force model [25,26] ismaybe the most well-known of these models, but we alsolike to mention cellular automata of pedestrian dynam-ics [27,28,29,30,31,32,33] and AI-based models [34,35].

The Social Force Concept

In the following, we shall shortly introduce the social forceconcept, which reproduces most empirical observations ina simple and natural way. Human behavior often seems tobe chaotic, irregular, and unpredictable. So, why and un-der what conditions can we model it by means of forces?First of all, we need to be confronted with a phenomenonof motion in some (quasi-)continuous space, which maybe also an abstract behavioral space such as an opinionscale [36]. Moreover, it is favorable to have a system wherethe uctuations due to unknown inuences are not largecompared to the systematic, deterministic part of motion.This is usually the case in pedestrian trac, where peopleare confronted with standard situations and react auto-matically rather than taking complicated decisions, e. g. ifthey have to evade others.

This automatic behavior can be interpreted as theresult of a learning process based on trial and error [37],which can be simulated with evolutionary algorithms [38].For example, pedestrians have a preferred side of walk-ing, since an asymmetrical avoidance behavior turns outto be protable [25,37]. The related formation of a behav-

6478 P Pedestrian, Crowd and Evacuation Dynamicsioral convention can be described bymeans of evolutionarygame theory [25,39].

Another requirement is the vectorial additivity of theseparate force terms reecting dierent environmental in-uences. This is probably an approximation, but there issome experimental evidence for it. Based on quantitativemeasurements for animals and test persons subject to sep-arately or simultaneously applied stimuli of dierent na-ture and strength, one could show that the behavior inconict situations can be described by a superposition offorces [40,41]. This ts well into a concept by Lewin [42],according to which behavioral changes are guided by so-called social elds or social forces, which has later on beenput into mathematical terms [25,43]. In some cases, socialforces, which determine the amount and direction of sys-tematic behavioral changes, can be expressed as gradientsof dynamically varying potentials, which reect the socialor behavioral elds resulting from the interactions of indi-viduals. Such a social force concept was applied to opinionformation and migration [43], and it was particularly suc-cessful in the description of collective pedestrian behav-ior [8,25,26,37].

For reliable simulations of pedestrian crowds, we donot need to know whether a certain pedestrian, say, turnsto the right at the next intersection. It is sucient to havea good estimate what percentage of pedestrians turns tothe right. This can be either empirically measured or es-timated by means of route choice models [14]. In somesense, the uncertainty about the individual behaviors is av-eraged out at the macroscopic level of description. Never-theless, we will use the more exible microscopic simula-tion approach based on the social force concept. Accord-ing to this, the temporal change of the location r(t) ofpedestrian obeys the equation of motion

dr(t)dt

D v(t) : (1)

Moreover, if f (t) denotes the sum of social forces inu-encing pedestrian and if (t) are individual uctua-tions reecting unsystematic behavioral variations, the ve-locity changes are given by the acceleration equation

dvdtD f (t)C (t) : (2)

A particular advantage of this approach is that we can takeinto account the exible usage of space by pedestrians, re-quiring a continuous treatment of motion. It turns out thatthis point is essential to reproduce the empirical observa-tions in a natural and robust way, i. e. without having toadjust the model to each single situation andmeasurement

site. Furthermore, it is interesting to note that, if the uc-tuation term is neglected, the social force model can be in-terpreted as a particular dierential game, i. e. its dynamicscan be derived from the minimization of a special utilityfunction [44].

Specication of the Social Force Model

The social force model for pedestrians assumes that eachindividual is trying to move in a desired direction e0with a desired speed v0 , and that it adapts the actual ve-locity v to the desired one, v0 D v0 e0 , within a certainrelaxation time . The systematic part f (t) of the accel-eration force of pedestrian is then given by

f (t) D1

(v0 e0v)C

X

()f (t)C

X

i

f i(t); (3)

where the terms f (t) and f i (t) denote the repulsiveforces describing attempts to keep a certain safety distanceto other pedestrians and obstacles i. In very crowded sit-uations, additional physical contact forces come into play(see Subsect. Force Model for Panicking Pedestrians).Further forces may be added to reect attraction eects be-tween members of a group or other inuences. For detailssee [37].

First, we will assume a simplied interaction force ofthe form

f (t) D fd (t)

; (4)

where d D r r is the distance vector pointing frompedestrian to . Angular-dependent shielding eectsmay be furthermore taken into account by a prefactordescribing the anisotropic reaction to situations in frontof as compared to behind a pedestrian [26,45], see Sub-sect. Angular Dependence. However, we will start witha circular specication of the distance-dependent inter-action force,

f (d ) D Aed /Bdkdk

; (5)

where d D kdk is the distance. The parameter A re-ects the interaction strength, and B corresponds to theinteraction range. While the dependence on explicitlyallows for a dependence of these parameters on the singleindividual, we will assume a homogeneous population, i. e.A D A and B D B in the following. Otherwise, it wouldbe hard to collect enough data for parameter calibration.

Elliptical Specication Note that it is possible to expressEq. (5) as gradient of an exponentially decaying potential

Pedestrian, Crowd and Evacuation Dynamics P 6479V . This circumstance can be used to formulate a gener-alized, elliptical interaction force via the potential

V (b ) D ABeb /B ; (6)where the variable b denotes the semi-minor axis bof the elliptical equipotential lines. This has been speciedaccording to

2b Ds

(kdk C kd (v v)tk)2k(v v)tk2 ; (7)

so that both pedestrians and are treated symmetrically.The repulsive force is related to the above potential via

f (d ) D r dV (b )

D dV (b )db

r db (d ) ; (8)

where r d represents the gradient with respect to d .Considering the chain rule, kzk D pz2, and r zkzk Dz/pz2 D z/kzk, this leads to the explicit formula

f (d ) D Aeb /B kdk C kd yk

2b

12

dkdk

C d ykd yk

!

(9)

with y D (v v)t. We usedt D 0:5 s. Fort D0, we regain the expression of Eq. (5).

The elliptical specication has two major advantagescompared to the circular one: First, the interactions de-pend not only on the distance, but also on the relative ve-locity. Second, the repulsive force is not strictly directedfrom pedestrian to pedestrian , but has a lateral com-ponent. As a consequence, this leads to less confrontative,

Pedestrian, Crowd and Evacuation Dynamics, Figure 1Video tracking used to extract the trajectories of pedestrians from video recordings close to two escalators (after [45]). a Illustrationof the tracking of pedestrian heads. b Resulting trajectories after being transformed onto the two-dimensional plane

smoother (sliding) evading maneuvers. Note that fur-ther velocity-dependent specications of pedestrian inter-action forces have been proposed [7,26], but we will re-strict to the above specications, as these are sucient todemonstrate the method of evolutionary model calibra-tion.

Evolutionary Calibration with Video Tracking Data

For parameter calibration, several video recordings ofpedestrian crowds in dierent natural environments havebeen used. The dimensions of the recorded areas wereknown, and the oor tiling or environment providedsomething like a coordinate system. The heads were au-tomatically determined by searching for round movingstructures, and the accuracy of tracking was improved bycomparing actual with linearly extrapolated positions (soit would not happen so easily that the algorithm inter-changed or lost close by pedestrians). The trajectories ofthe heads were then projected on two-dimensional spacein a way correcting for distortion by the camera perspec-tive. A representative plot of the resulting trajectories isshown in Fig. 1. Note that trajectory data have been ob-tained with infra-red sensors [47] or video cameras [48,49]for several years now, but algorithms that can simultane-ously handlemore than one thousand pedestrians have be-come available only recently [87].

For model calibration, it is recommended to use a hy-brid method fusing empirical trajectory data and micro-scopic simulation data of pedestrian movement in space.In corresponding algorithms, a virtual pedestrian is as-signed to each tracked pedestrian in the simulation do-main. One then starts a simulation for a time period T(e. g. 1.5 s), in which one pedestrian is moved accord-ing to a simulation of the social force model, while theothers are moved exactly according to the trajectories ex-

6480 P Pedestrian, Crowd and Evacuation Dynamicstracted from the videos. This procedure is performed forall pedestrians and for several dierent starting times t,using a xed parameter set for the social force model.

Each simulation run is performed according to the fol-lowing scheme:

1. Dene a starting point and calculate the state (positionr , velocity v , and acceleration a D dv/dt) for eachpedestrian .

2. Assign a desired speed v0 to each pedestrian, e. g. themaximum speed during the pedestrian tracking time.This is suciently accurate, if the overall pedestriandensity is not too high and the desired speed is constantin time.

3. Assign a desired goal point for each pedestrian, e. g. theend point of the trajectory.

4. Given the tracked motion of the surrounding pedes-trians , simulate the trajectory of pedestrian overa time period T based on the social force model, start-ing at the actual location r(t).

After each simulation run, one determines the relative dis-tance error

krsimulated (t C T) rtracked (t C T)kkrtracked (t C T) rtracked (t)k

: (10)

After averaging the relative distance errors over the pedes-trians and starting times t, 1 minus the result can betaken asmeasure of the goodness of t (the tness) of theparameter set used in the pedestrian simulation. Hence,the best possible value of the tness is 1, but any devia-tion from the real pedestrian trajectories implies lower val-ues.

One result of such a parameter optimization is that, foreach video, there is a broad range of parameter combina-tions of A and B which perform almost equally well [45].This allows one to apply additional goal functions in theparameter optimization, e. g. to determine among the bestperforming parameter values such parameter combina-tions, which perform well for several video recordings, us-ing a tness function which equally weights the tnessreached in each single video. This is how the parametervalues listed in Table 1 were determined. It turns out that,in order to reach a good model performance, the pedes-trian interaction force must be specied velocity depen-dent, as in the elliptical model.

Note that our evolutionary tting method can be alsoused to determine interaction laws without prespeciedinteraction functions. For example, one can obtain the dis-tance dependence of pedestrian interactions without a pre-specied function. For this, one adjusts the values of the

Pedestrian, Crowd and Evacuation Dynamics, Table 1Interaction strength A and interaction rangeB resulting fromourevolutionary parameter calibration for the circular and ellipticalspecification of the interaction forces between pedestrians (seemain text). The calibration was based on three different videorecordings, one for low crowd density, one for medium, and onefor high density. The parameter values are specified as meanvalue standarddeviation. Thebest fitness value obtainedwiththe elliptical specification for the video with the lowest crowddensity was as high as 0.9

Model A B FitnessExtrapolation 0 0.34Circular 0:11 0:06 0:84 0:63 0.35Elliptical 4:30 3:91 1:07 1:35 0.53

force at given distances dk D kd1 (with k 2 f1; 2; 3; : : :g)in an evolutionary way. To get some smoothness, linear in-terpolation is applied. The resulting t curve is presentedin Fig. 2 (left). It turns out that the empirical dependenceof the force with distance can be well tted by an exponen-tial decay.

Angular Dependence

A closer study of pedestrian interactions reveals that theseare not isotropic, but dependent on the angle ' of theencounter, which is given by the formula

cos(' ) D vkvk dkdk : (11)

Generally, pedestrians show little response to pedestriansbehind them. This can be reected by an angular-depen-dent prefactor w(' ) of the interaction force [45]. Em-pirical results are represented in Fig. 2 (right). Reasonableresults are obtained for the following specication of theprefactor:

w' (t)

D

C (1 )

1C cos(' )2

; (12)

where with 0 1 is a parameter which growswith the strength of interactions from behind. An evolu-tionary parameter optimization gives values 0:1 [45],i. e. a strong anisotropy. With such an angle-dependentprefactor, the tness of the elliptical force increases from0.53 to 0.61, when calibrated to the same set of videos.Other angular-dependent specications split up the inter-action force between pedestrians into a component againstthe direction of motion and another one perpendicular toit. Such a description allows for even smoother avoidancemaneuvers.

Pedestrian, Crowd and Evacuation Dynamics P 6481

Pedestrian, Crowd and Evacuation Dynamics, Figure 2Results of an evolutionary fitting of pedestrian interactions. a Empirically determined distance dependence of the interaction forcebetween pedestrians (after [45]). An exponential decay fits the empirical data quite well. The dashed fit curve corresponds to Eq. (5)with the parameters A D 0:53 and B D 1:0. b Angular dependence of the influence of other pedestrians. The direction along thepositive x-axis corresponds to the walking direction of pedestrians, y to the perpendicular direction

CrowdDynamics

Analogies with Gases, Fluids, and Granular Media

When the density is low, pedestrians can move freely, andthe observed crowd dynamics can be partially comparedwith the behavior of gases. At medium and high densities,however, the motion of pedestrian crowds shows somestriking analogies with the motion of uids:

1. Footprints of pedestrians in snow look similar tostreamlines of uids [15].

2. At borderlines between opposite directions of walkingone can observe viscous ngering [50,51].

3. The emergence of pedestrian streams through standingcrowds [7,37,52] appears analogous to the formation ofriver beds [53,54].

At high densities, however, the observations have ratheranalogies with driven granular ows. This will be elabo-rated in more detail in Sects. Force Model for Panick-ing Pedestrians and Collective Phenomena in Panic Sit-uations. In summary, one could say that uid-dynamicanalogies work reasonably well in normal situations, whilegranular aspects dominate at extreme densities. Neverthe-less, the analogy is limited, since the self-driven motionand the violation of momentum conservation imply spe-cial properties of pedestrian ows. For example, one usu-ally does not observe eddies, which typically occur in reg-ular uids at high enough Reynolds numbers.

Self-Organization of Pedestrian Crowds

Despite its simplications, the social force model of pedes-trian dynamics describes a lot of observed phenomena

quite realistically. Especially, it allows one to explain var-ious self-organized spatio-temporal patterns that are notexternally planned, prescribed, or organized, e. g. by traf-c signs, laws, or behavioral conventions. Instead, the spa-tio-temporal patterns discussed below emerge due to thenon-linear interactions of pedestrians even without as-suming strategical considerations, communication, or im-itative behavior of pedestrians. Despite this, we may stillinterpret the forming cooperation patterns as phenomenathat establish social order on short time scales. It is actu-ally surprising that strangers coordinate with each otherwithin seconds, if they have grown up in a similar environ-ment. People from dierent countries, however, are some-times irritated about local walking habits, which indicatesthat learning eects and cultural backgrounds still playa role in social interactions as simple as random pedestrianencounters. Rather than on particular features, however,in the following we will focus on the common, interna-tionally reproducible observations.

Lane Formation In pedestrian ows one can often ob-serve that oppositely moving pedestrians are forminglanes of uniform walking direction (see Fig. 3). This phe-nomenon even occurs when there is not a large distance toseparate each other, e. g. on zebra crossings. However, thewidth of lanes increases (and their number decreases), ifthe interaction continues over longer distances (and if per-turbations, e. g. by ows entering or leaving on the sides,are low; otherwise the phenomenon of lane formationmaybreak down [55]).

Lane formation may be viewed as segregation phe-nomenon [56,57]. Although there is a weak preference forone side (with the corresponding behavioral convention

6482 P Pedestrian, Crowd and Evacuation Dynamics

Pedestrian, Crowd and Evacuation Dynamics, Figure 3Self-organization of pedestrian crowds. a Photograph of lanes formed in a shopping center. Computer simulations reproduce theself-organization of such lanes very well. b Evaluation of the cumulative number of pedestrians passing a bottleneck from differentsides.One can clearly see that the narrowing is often passedbygroups of people in an oscillatoryway rather than one byone. cMulti-agent simulation of two crossing pedestrian streams, showing the phenomenon of stripe formation. This self-organized patternallows pedestrians to pass the other stream without having to stop, namely by moving sidewards in a forwardly moving stripe.(After [8])

depending on the country), the observations can only bewell reproduced when repulsive pedestrian interactionsare taken into account. The most relevant factor for thelane formation phenomenon is the higher relative velocityof pedestrians walking in opposite directions. Comparedto people following each other, oppositely moving pedes-trians havemore frequent interactions until they have seg-regated into separate lanes by stepping aside whenever an-other pedestrian is encountered. The most long-lived pat-terns of motion are the ones which change the least. Itis obvious that such patterns correspond to lanes, as theyminimize the frequency and strength of avoidancemaneu-vers. Interestingly enough, as computer simulations show,lane formation occurs also when there is no preference forany side.

Lanes minimize frictional eects, accelerations, energyconsumption, and delays in oppositely moving crowds.Therefore, one could say that they are a pattern reect-ing collective intelligence. In fact, it is not possible fora single pedestrian to reach such a collective pattern ofmotion. Lane formation is a self-organized collaborative

pattern of motion originating from simple pedestrian in-teractions. Particularly in cases of no side preference, thesystem behavior cannot be understood by adding up thebehavior of the single individuals. This is a typical featureof complex, self-organizing systems and, in fact, a wide-spread characteristics of social systems. It is worth noting,however, that it does not require a conscious behavior toreach forms of social organization like the segregation ofoppositely moving pedestrians into lanes. This organiza-tion occurs automatically, although most people are noteven aware of the existence of this phenomenon.

Oscillatory Flows at Bottlenecks At bottlenecks, bidi-rectional ows of moderate density are often characterizedby oscillatory changes in the ow direction (see Fig. 3). Forexample, one can sometimes observe this at entrances ofmuseums during crowded art exhibitions or at entrancesof sta canteens during lunch time.While these oscillatoryows may be interpreted as an eect of friendly behavior(you go rst, please), computer simulations of the socialforce model indicate that the collective behaviormay again

Pedestrian, Crowd and Evacuation Dynamics P 6483be understood by simple pedestrian interactions. That is,oscillatory ows occur even in the absence of communica-tion. Therefore, they may be viewed as another self-orga-nization phenomenon, which again reduces frictional ef-fects and delays. That is, oscillatory ows have features ofcollective intelligence.

While this may be interpreted as result of a learningeect in a large number of similar situations (a repeatedgame), our simulations suggest an even simpler, many-particle interpretation: Once a pedestrian is able to passthe narrowing, pedestrians with the same walking direc-tion can easily follow. Hence, the number and pressureof waiting, pushy pedestrians on one side of the bottle-neck becomes less than on the other side. This eventuallyincreases their chance to occupy the passage. Finally, thepressure dierence is large enough to stop the ow andturn the passing direction at the bottleneck. This reversesthe situation, and eventually the ow direction changesagain, giving rise to oscillatory ows.

Stripe Formation in Intersecting Flows In intersectionareas, the ow of people often appears to be irregular orchaotic. In fact, it can be shown that there are severalpossible collective patterns of motion, among them ro-tary and oscillating ows. However, these patterns contin-uously compete with each other, and a temporarily dom-inating pattern is destroyed by another one after a shorttime. Obviously, there has not evolved any social conven-tion that would establish and stabilize an ordered and e-cient ow at intersections.

Self-organized patterns of motion, however, are foundin situations where pedestrian ows cross each other onlyin two directions. In such situations, the phenomenon ofstripe formation is observed [58]. Stripe formation allowstwo ows to penetrate each other without requiring thepedestrians to stop. For an illustration see Fig. 3. Likelanes, stripes are a segregation phenomenon, but not a sta-tionary one. Instead, the stripes are density waves mov-ing into the direction of the sum of the directional vec-tors of both intersecting ows. Naturally, the stripes ex-tend sidewards into the direction which is perpendicularto their direction of motion. Therefore, the pedestriansmove forward with the stripes and sidewards within thestripes. Lane formation corresponds to the particular caseof stripe formation where both directions are exactly op-posite. In this case, no intersection takes place, and thestripes do not move systematically. As in lane formation,stripe formation allows to minimize obstructing interac-tions and to maximize the average pedestrian speeds, i. e.simple, repulsive pedestrian interactions again lead to anintelligent collective behavior.

EvacuationDynamics

While the previous section has focused on the dynamics ofpedestrian crowds in normal situations, we will now turnto the description of situations in which extreme crowddensities occur. Such situations may arise at mass events,particularly in cases of urgent egress. While most evacu-ations run relatively smoothly and orderly, the situationmay also get out of control and end up in terrible crowddisasters (see Table 2). In such situations, one often speaksof panic, although, from a scientic standpoint, the useof this term is rather controversial. Here, however, we willnot be interested in the question whether panic actuallyoccurs or not. We will rather focus on the issue of crowddynamics at high densities and under psychological stress.

Evacuation and Panic Research

Computer models have been also developed for emer-gency and evacuation situations [32,60,61,62,63,64,65,66,67,68]. Most research into panic, however, has been of em-pirical nature (see, e. g. [69,70,71,72]), carried out by socialpsychologists and others.

With some exceptions, panic is observed in cases ofscarce or dwindling resources [69,73], which are eitherrequired for survival or anxiously desired. They are usu-ally distinguished into escape panic (stampedes, bank orstockmarket panic) and acquisitive panic (crazes, specu-lative manias) [74,75], but in some cases this classicationis questionable [76].

It is often stated that panicking people are obsessedby short-term personal interests uncontrolled by socialand cultural constraints [69,74]. This is possibly a re-sult of the reduced attention in situations of fear [69],which also causes that options like side exits are mostly ig-nored [70]. It is, however, mostly attributed to social con-tagion [69,71,73,74,75,76,77,78,79,80,81], i. e., a transitionfrom individual to mass psychology, in which individualstransfer control over their actions to others [75], leading toconformity [82]. This herding behavior is in some senseirrational, as it often leads to bad overall results like dan-gerous overcrowding and slower escape [70,75,76]. In thisway, herding behavior can increase the fatalities or, moregenerally, the damage in the crisis faced.

The various socio-psychological theories for this con-tagion assume hypnotic eects, rapport, mutual excitationof a primordial instinct, circular reactions, social facili-tation (see the summary by Brown [80]), or the emer-gence of normative support for selsh behavior [81].Brown [80] and Coleman [75] add another explanation re-lated to the prisoners dilemma [83,84] or common goodsdilemma [85], showing that it is reasonable to make ones

6484 P Pedestrian, Crowd and Evacuation DynamicsPedestrian, Crowd and Evacuation Dynamics, Table 2Incomplete list of major crowd disasters since 1970 after J. F. Dickie in [59], http://www.crowddynamics.com/Main/Crowddisasters.html, http://SportsIllustrated.CNN.com/soccer/world/news/2000/07/09/stadium_disasters_ap/, and other internet sources, exclud-ing fires, bomb attacks, and train or plane accidents. The number of injured people was usually a multiple of the fatalities

Date Place Venue Deaths Reason1971 Ibrox, UK Stadium 66 Collapse of barriers1974 Cairo, Egypt Stadium 48 Crowds break barriers1982 Moscow, USSR Stadium 340 Re-entering fans after last minute goal1988 Katmandu, Nepal Stadium 93 Stampede due to hailstorm1989 Hillsborough, Sheffield, UK Stadium 96 Fans trying to force their way into the stadium1990 New York City Bronx 87 Illegal happy land social club1990 Mena, Saudi Arabia Pedestrian Tunnel 1426 Overcrowding1994 Mena, Saudi Arabia Jamarat Bridge 266 Overcrowding1996 Guatemala City, Guatemala Stadium 83 Fans trying to force their way into the stadium1998 Mena, Saudi Arabia 118 Overcrowding1999 Kerala, India Hindu Shrine 51 Collapse of parts of the shrine1999 Minsk, Belarus Subway Station 53 Heavy rain at rock concert2001 Ghana, West Africa Stadium > 100 Panic triggered by tear gas2004 Mena, Saudi Arabia Jamarat Bridge 251 Overcrowding2005 Wai, India Religious Procession 150 Overcrowding (and fire)2005 Bagdad, Iraque Religious Procession > 640 Rumors regarding suicide bomber2005 Chennai, India Disaster Area 42 Rush for flood relief supplies2006 Mena, Saudi Arabia Jamarat Bridge 363 Overcrowding2006 Pilippines Stadium 79 Rush for game show tickets2006 Ibb, Yemen Stadium 51 Rally for Yemeni president

subsequent actions contingent upon those of others. How-ever, the socially favorable behavior of walking orderly isunstable, which normally gives rise to rushing by every-one. These thoughtful considerations are well compatiblewith many aspects discussed above and with the classicalexperiments byMintz [73], which showed that jamming inescape situations depends on the reward structure (payomatrix).

Nevertheless and despite of the frequent reports in themedia and many published investigations of crowd dis-asters (see Table 2), a quantitative understanding of theobserved phenomena in panic stampedes was lacking fora long time. In the following, we will close this gap.

Situations of Panic

Panic stampede is one of the most tragic collective behav-iors [71,72,73,74,75,77,78,79,80,81], as it often leads to thedeath of people who are either crushed or trampled downby others. While this behavior may be comprehensiblein life-threatening situations like res in crowded build-ings [69,70], it is hard to understand in cases of a rushfor good seats at a pop concert [76] or without any ob-vious reasons. Unfortunately, the frequency of such disas-ters is increasing (see Table 2), as growing population den-

sities combined with easier transportation lead to greatermass events like pop concerts, sport events, and demon-strations. Nevertheless, systematic empirical studies ofpanic [73,86] are rare [69,74,76], and there is a scarcity ofquantitative theories capable of predicting crowd dynam-ics at extreme densities [32,60,61,64,65,68]. The followingfeatures appear to be typical [46,55]:

1. In situations of escape panic, individuals are gettingnervous, i. e. they tend to develop blind actionism.

2. People try to move considerably faster than nor-mal [9].

3. Individuals start pushing, and interactions amongpeople become physical in nature.

4. Moving and, in particular, passing of a bottleneck fre-quently becomes incoordinated [73].

5. At exits, jams are building up [73]. Sometimes, inter-mittent ows or arching and clogging are observed [9],see Fig. 4.

6. The physical interactions in jammed crowds add upand can cause dangerous pressures up to 4,500 New-tons per meter [59,70], which can bend steel barriersor tear down brick walls.

7. The strength and direction of the forces acting inlarge crowds can suddenly change [87], pushing peo-

Pedestrian, Crowd and Evacuation Dynamics P 6485

Pedestrian, Crowd and Evacuation Dynamics, Figure 4Panicking football fans trying to escape the football stadium inSheffield. Because of a clogging effect, it is difficult to pass theopen door

ple around in an uncontrollable way. This may causepeople to fall.

8. Escape is slowed down by fallen or injured peopleturning into obstacles.

9. People tend to show herding behavior, i. e., to do whatother people do [69,78].

10. Alternative exits are often overlooked or not ecientlyused in escape situations [69,70].

The following quotations give a more personal impressionof the conditions during crowd panic:

1. They just kept pushin forward and they would justwalk right on top of you, just trample over ya like youwere a piece of the ground. (After the panic at TheWho Concert Stampede in Cincinnati.)

2. People were climbin over people ta get in . . . an at onepoint I almost started hittin em, because I could notbelieve the animal, animalistic ways of the people, youknow, nobody cared. (After the panic at The WhoConcert Stampede.)

3. Smaller people began passing out. I attempted to liftone girl up and above to be passed back . . . After sev-eral tries I was unsuccessful and near exhaustion. (Af-ter the panic at The Who Concert Stampede.)

4. I couldnt see the oor because of the thickness of thesmoke. (After the Hilton Hotel Fire in Las Vegas.)

5. The club had two exits, but the young people had ac-cess to only one, said Narend Singh, provincial min-ister for agriculture and environmental aairs. How-ever, the clubs owner, Rajan Naidoo, said the club hadfour exits, and that all were open. I think the childrenpanicked and headed for the main entrance where they

initially came in, he said. (After the Durban DiscoStampede.)

6. At occupancies of about 7 persons per square meterthe crowd becomes almost a uid mass. Shock wavescan be propagated through the mass, sucient to . . .propel them distances of 3 meters or more. . . . Peoplemay be literally lifted out of their shoes, and have cloth-ing torn o. Intense crowd pressures, exacerbated byanxiety, make it dicult to breathe, which may nallycause compressive asphyxia. The heat and the ther-mal insulation of surrounding bodies cause some to beweakened and faint. Access to those who fall is impos-sible. Removal of those in distress can only be accom-plished by lifting them up and passing them overheadto the exterior of the crowd. (J. Fruin in [88].)

7. It was like a huge wave of sea gushing down on thepilgrims (P. K. Abdul Ghafour, Arab News, after thesad crowd disaster in Mena on January 12, 2006).

Force Model for Panicking Pedestrians

Additional, physical interaction forces f ph

come into playwhen pedestrians get so close to each other that they havephysical contact (i. e. d < r D r C r , where rmeans the radius of pedestrian ). In this case, which ismainly relevant to panic situations, we assume also a bodyforce k(r d )n counteracting body compressionand a sliding friction force (r d )vt t im-peding relative tangential motion. Inspired by the formu-las for granular interactions [89,90], we assume

f ph

(t) D k(rd )nC(rd )vt t ;(13)

where the function (z) is equal to its argument z, ifz 0, otherwise 0. Moreover, t D (n2 ; n1 ) meansthe tangential direction and vt

D (v v) t the

tangential velocity dierence, while k and represent largeconstants. (Strictly speaking, friction eects already set inbefore pedestrians touch each other, because of the psy-chological tendency not to pass other individuals witha high relative velocity, when the distance is small.)

The interactions with the boundaries of walls andother obstacles are treated analogously to pedestrian in-teractions, i. e., if d i (t) means the distance to obstacle orboundary i, n i (t) denotes the direction perpendicular toit, and t i (t) the direction tangential to it, the correspond-ing interaction force with the boundary reads

f i DA exp[(r d i )/B]C k(r d i )

n i (r d i )(v t i)t i : (14)

6486 P Pedestrian, Crowd and Evacuation DynamicsFinally, re fronts are reected by repulsive social

forces similar those describing walls, but they are muchstronger. The physical interactions, however, are qualita-tively dierent, as people reached by the re front becomeinjured and immobile (v D 0).

Collective Phenomena in Panic Situations

In panic situations (e. g. in some cases of emergency evac-uation) the following characteristic features of pedestrianbehavior are often observed:

1. People are getting nervous, resulting in a higher level ofuctuations.

2. They are trying to escape from the source of panic,which can be reected by a signicantly higher desiredvelocity v0 .

3. Individuals in complex situations, who do not knowwhat is the right thing to do, orient at the actions oftheir neighbors, i. e. they tend to do what other peopledo. We will describe this by an additional herding in-teraction.

We will now discuss the fundamental collective eectswhich uctuations, increased desired velocities, and herd-

Pedestrian, Crowd and Evacuation Dynamics, Figure 5aNormal leaving of a room, when the exit is well visible. b Escape from a roomwith no visibility, e. g. due to dense smoke or a powerblackout. (After [18])

ing behavior can have. In contrast to other approaches, wedo not assume or imply that individuals in panic or emer-gency situations would behave relentless and asocial, al-though they sometimes do.

Herding and Ignorance of Available Exits If people arenot sure what is the best thing to do, there is a tendencyto show a herding behavior, i. e. to imitate the behaviorof others. Fashion, hypes and trends are examples for this.The phenomenon is also known from stock markets, andparticularly pronounced when people are anxious. Sucha situation is, for example, given if people need to escapefrom a smoky room. There, the evacuation dynamics isvery dierent from normal leaving (see Fig. 5).

Under normal visibility, everybody easily nds an exitand uses more or less the shortest path. However, whenthe exit cannot be seen, evacuation is much less ecientand may take a long time. Most people tend to walk rela-tively straight into the direction in which they suspect anexit, but in most cases, they end up at a wall. Then, theyusually move along it in one of the two possible directions,until they nally nd an exit [18]. If they encounter oth-ers, there is a tendency to take a decision for one direction

Pedestrian, Crowd and Evacuation Dynamics P 6487andmove collectively. Also in case of acoustic signals, peo-ple may be attracted into the same direction. This can leadto over-crowded exits, while other exits are ignored. Thesame can happen even for normal visibility, when peopleare not well familiar with their environment and are notaware of the directions of the emergency exits.

Computer simulations suggest that neither individual-istic nor herding behavior performs well [46]. Pure indi-vidualistic behavior means that each pedestrian nds anexit only accidentally, while pure herding behavior im-plies that the complete crowd is eventuallymoving into thesame and probably congested direction, so that availableemergency exits are not eciently used. Optimal chancesof survival are expected for a certain mixture of individ-ualistic and herding behavior, where individualism allowssome people to detect the exits and herding guarantees thatsuccessful solutions are imitated by small groups of oth-ers [46].

Freezing by Heating Another eect of getting ner-vous has been investigated in [55]. Let us assume the in-dividual uctuation strength is given by

D (1 n)0 C nmax ; (15)where n with 0 n 1 measures the nervousness ofpedestrian . The parameter 0 means the normal andmax the maximum uctuation strength. It turns out that,at suciently high pedestrian densities, lanes are de-stroyed by increasing the uctuation strength (which isanalogous to the temperature). However, instead of theexpected transition from the uid lane state to a disor-dered, gaseous state, a solid state is formed. It is charac-terized by an at least temporarily blocked, frozen situa-tion so that one calls this paradoxical transition freezingby heating (see Fig. 6). Notably enough, the blocked statehas a higher degree of order, although the internal energyis increased [55].

The preconditions for this unusual freezing-by-heat-ing transition are the driving term v0 e0/ and the dis-sipative friction v/ , while the sliding friction forceis not required. Inhomogeneities in the channel diameter

Pedestrian, Crowd and Evacuation Dynamics, Figure 6Result of the noise-induced formation of a frozen state ina (periodic) corridor used by oppositely moving pedestrians (af-ter [55])

or other impurities which temporarily slow down pedes-trians can further this transition at the respective places.Finally note that a transition from uid to blocked pedes-trian counter ows is also observed, when a critical densityis exceeded [31,55].

Intermittent Flows, Faster-Is-Slower Eect, and Phan-tom Panic If the overall ow towards a bottleneck ishigher than the overall outow from it, a pedestrian queueemerges [91]. In other words, a waiting crowd is formedupstream of the bottleneck. High densities can result, ifpeople keep heading forward, as this eventually leads tohigher and higher compressions. Particularly critical situ-ations may occur if the arrival ow is much higher thanthe departure ow, especially if people are trying to get to-wards a strongly desired goal (acquisitive panic) or awayfrom a perceived source of danger (escape panic) with anincreased driving force v0e0/ . In such situations, the highdensity causes coordination problems, as several peoplecompete for the same few gaps. This typically causes bodyinteractions and frictional eects, which can slow downcrowd motion or evacuation (faster is slower eect).

A possible consequence of these coordination prob-lems are intermittent ows. In such cases, the outowfrom the bottleneck is not constant, but it is typically in-terrupted. While one possible origin of the intermittentows are clogging and arching eects as known from gran-ular ows through funnels or hoppers [89,90], stop-and-go waves have also been observed in more than 10 meterwide streets and in the 44 meters wide entrance area tothe Jamarat Bridge during the pilgrimage in January 12,2006 [87], see Fig. 7. Therefore, it seems to be importantthat people do not move continuously, but have minimumstrides [25]. That is, once a person is stopped, he or she willnot move until some space opens up in front. However, in-creasing impatience will eventually reduce the minimumstride, so that people eventually start moving again, evenif the outow through the bottleneck is stopped. This willlead to a further compression of the crowd.

In the worst case, such behavior can trigger a phantompanic, i. e. a crowd disaster without any serious reasons(e. g., in Moscow, 1982). For example, due to the faster-is-slower eect panic can be triggered by small pedestriancounterows [70], which cause delays to the crowd in-tending to leave. Consequently, stopped pedestrians in theback, who do not see the reason for the temporary slow-down, are getting impatient and pushy. In accordance withobservations [7,25], one may model this by increasing thedesired velocity, for example, by the formula

v0(t) D [1 n(t)]v0(0)C n(t)vmax : (16)

6488 P Pedestrian, Crowd and Evacuation Dynamics

Pedestrian, Crowd and Evacuation Dynamics, Figure 7a Long-term photograph showing stop-and-go waves in a densely packed street. While stopped people appear relatively sharp,people moving from right to left have a fuzzy appearance. Note that gaps propagate from right to left. b Empirically observed stop-and-gowaves in front of the entrance to the Jamarat Bridge on January 12, 2006 (after [87]), where pilgrimsmoved from left to right.Dark areas correspond to phases of motion, light colors to stop phases. c Illustration of the shell model, in particular of situationswhere several pedestrians compete for the same gap, which causes coordination problems. d Stop-and-go waves resulting from thealternation of forward pedestrianmotion and backward gap propagation

Herein, vmax is the maximum desired velocity and v0(0)the initial one, corresponding to the expected velocity ofleaving. The time-dependent parameter

n(t) D 1 v(t)v0(0)(17)

reects the nervousness, where v(t) denotes the aver-age speed into the desired direction of motion. Altogether,long waiting times increase the desired speed v0 or drivingforce v0(t)e0/ , which can produce high densities and in-ecient motion. This further increases the waiting times,and so on, so that this tragic feedback can eventually trig-

Pedestrian, Crowd and Evacuation Dynamics P 6489ger so high pressures that people are crushed or falling andtrampled. It is, therefore, imperative, to have sucientlywide exits and to prevent counterows, when big crowdswant to leave [46].

Transition to Stop-and-Go Waves Recent empiricalstudies of pilgrim ows in the area of Makkah, SaudiArabia, have shown that intermittent ows occur notonly when bottlenecks are obvious. On January 12, 2006,pronounced stop-and-go waves have been even observedupstream of the 44m wide entrance to the JamaratBridge [87].While the pilgrim ows were smooth and con-tinuous (laminar) over many hours, at 11:53 am stop-and-go waves suddenly appeared and propagated over dis-tances of more than 30m (see Fig. 7). The sudden transi-tion was related to a signicant drop of the ow, i. e. withthe onset of congestion [87]. Once the stop-and-go wavesset in, they persisted over more than 20min.

This phenomenon can be reproduced by a recentmodel based on two continuity equations, one for for-ward pedestrian motion and another one for backwardgap propagation [91]. Themodel was derived from a shellmodel (see Fig. 7) and describes very well the observed al-ternation between backward gap propagation and forwardpedestrian motion.

Transition to Crowd Turbulence On the same day,around 12:19, the density reached even higher values andthe video recordings showed a sudden transition fromstop-and-go waves to irregular ows (see Fig. 8). These ir-regular ows were characterized by random, unintended

Pedestrian, Crowd and Evacuation Dynamics, Figure 8Pedestrian dynamics at different densities. a Representative trajectories (space-time plots) of pedestrians during the laminar, stop-and-go, and turbulent flow regime. Each trajectory extends over a range of 8 meters, while the time required for this stretch isnormalized to 1. To indicate the different speeds, symbols are included in the curves every 5 seconds. While the laminar flow (topline) is fast and smooth, motion is temporarily interrupted in stop-and-go flow (medium line), and backward motion can occur inturbulent flows (bottom line). b Example of the temporal evolution of the velocity components vx (t) into the average direction ofmotion and vy (t) perpendicular to it in turbulent flow, which occurs when the crowd density is extreme. One can clearly see theirregular motion into all possible directions characterizing crowd turbulence. For details see [87]

displacements into all possible directions, which pushedpeople around. With a certain likelihood, this caused themto stumble. As the people behind weremoved by the crowdas well and could not stop, fallen individuals were tram-pled, if they did not get back on their feet quickly enough.Tragically, the area of trampled people grew more andmore in the course of time, as the fallen pilgrims becameobstacles for others [87]. The result was one of the biggestcrowd disasters in the history of pilgrimage.

How can we understand this transition to irregularcrowd motion? A closer look at video recordings of thecrowd reveals that, at this time, people were so denselypacked that they were moved involuntarily by the crowd.This is reected by random displacements into all possibledirections. To distinguish these irregular ows from lami-nar and stop-and-go ows and due to their visual appear-ance, we will refer to them as crowd turbulence.

As in certain kinds of uid ows, turbulence incrowds results from a sequence of instabilities in the owpattern. Additionally, one nds a sharply peaked probabil-ity density function of velocity increments

Vx D Vx (r; t C ) Vx (r; t) ; (18)which is typical for turbulence [92], if the time shift issmall enough [87]. One also observes a power-law scalingof the displacements indicating self-similar behavior [87].As large eddies are not detected, however, the similaritywith uid turbulence is limited, but there is still an anal-ogy to turbulence at currency exchange markets [92]. In-stead of vortex cascades like in turbulent uids, one rathernds a hierarchical fragmentation dynamics: At extreme

6490 P Pedestrian, Crowd and Evacuation Dynamics

Pedestrian, Crowd and Evacuation Dynamics, Figure 9Left: Snapshot of the on-line visualization of crowd pressure. Red colors (see the lower ellipses) indicate areas of critical crowd con-ditions. In fact, the sad crowd disaster during theMuslim pilgrimage on January 12, 2006, started in this area. Right: The crowd pres-sure is a quantitative measure of the onset of crowd turbulence. The crowd disaster started when the crowd pressure reachedparticularly high values

densities, individual motion is replaced by mass motion,but there is a stick-slip instability which leads to rupturewhen the stress in the crowd becomes too large. That is, themass splits up into clusters of dierent sizes with strongvelocity correlations inside and distance-dependent corre-lations between the clusters.

Crowd turbulence has further specic features [87].Due to the physical contacts among people in extremelydense crowds, we expect commonalities with granular me-dia. In fact, dense driven granular media may form den-sity waves, while moving forward [93], and can displayturbulent-like states [94,95]. Moreover, under quasi-staticconditions [94], force chains [96] are building up, causingstrong variations in the strengths and directions of localforces. As in earthquakes [97,98] this can lead to events ofsudden, uncontrollable stress release with power-law dis-tributed displacements. Such a power-law has also beendiscovered by video-based crowd analysis [87].

Some Warning Signs of Critical Crowd Conditions

Turbulent waves are experienced in dozens of crowd-in-tensive events each year all over the world [88]. Therefore,it is necessary to understand why, where and when po-tentially critical situations occur. Viewing real-time videorecordings is not very suited to identify critical crowd con-ditions: While the average density rarely exceeds values of6 persons per square meter, the local densities can reachalmost twice as large values [87]. It has been found, how-ever, that even evaluating the local densities is not enoughto identify the critical times and locations precisely, whichalso applies to an analysis of the velocity eld [87]. The de-cisive quantity is rather the crowd pressure, i. e. the den-sity, multiplied with the variance of speeds. It allows oneto identify critical locations and times (see Fig. 9).

There are even advance warning signs of critical crowdconditions: The crowd accident on January 12, 2006started about 10 minutes after turbulent crowd motionset in, i. e. after the pressure exceeded a value of 0.02/s2

(see Fig. 9). Moreover, it occurred more than 30min af-ter stop-and-go waves set in, which can be easily detectedin accelerated surveillance videos. Such advance warningsigns of critical crowd conditions can be evaluated on-lineby an automated video analysis system. In many cases,this can help one to gain time for corrective measures likeow control, pressure-relief strategies, or the separationof crowds into blocks to stop the propagation of shock-waves [87]. Such anticipative crowd control could increasethe level of safety during future mass events.

Evolutionary Optimization of Pedestrian Facilities

Having understood some of the main factors causingcrowd disasters, it is interesting to ask how pedestrian fa-cilities can be designed in a way that maximizes the ef-ciency of pedestrian ows and the level of safety. Oneof the major goals during mass events must be to avoidextreme densities. These often result from the onset ofcongestion at bottlenecks, which is a consequence of thebreakdown of free ow and causes an increasing degree ofcompression. When a certain critical density is increased(which depends on the size distribution of people), this po-tentially implies high pressures in the crowd, particularlyif people are impatient due to long delays or panic.

The danger of an onset of congestion can be mini-mized by avoiding bottlenecks. Notice, however, that jam-ming can also occur at widenings of escape routes [46].This surprising fact results from disturbances due topedestrians, who try to overtake each other and expandin the wider area because of their repulsive interactions.

Pedestrian, Crowd and Evacuation Dynamics P 6491

Pedestrian, Crowd and Evacuation Dynamics, Figure 10The evolutionary optimization based on Boolean grids [99] uses a two-stage algorithm. a In the randomization stage, obstacles aredistributed over the grid with some randomness, thereby allowing for the generation of new topologies. b In the agglomerationstage, small nearby obstacles are clustered to form larger objects with smooth boundaries

These squeeze into the main stream again at the end of thewidening, which acts like a bottleneck and leads to jam-ming. The corresponding drop of eciency E is more pro-nounced,

1. if the corridor is narrow,2. if the pedestrians have dierent or high desired veloci-

ties, and3. if the pedestrian density in the corridor is high.

Obviously, the emerging pedestrian ows decisivelydepend on the geometry of the boundaries. They can besimulated on a computer already in the planning phase ofpedestrian facilities. Their conguration and shape can besystematically varied, e. g. by means of evolutionary algo-rithms [28,100] and evaluated on the basis of particularmathematical performance measures [7]. Apart from theeciency

E D 1N

X

v e0v0

(19)

we can, for example, dene the measure of comfortC D (1 D) via the discomfort

D D 1N

X

(v v)2(v)2

D 1N

X

1 v2

(v)2

!

: (20)

The latter is again between 0 and 1 and reects the fre-quency and degree of sudden velocity changes, i. e. thelevel of discontinuity of walking due to necessary avoid-ance maneuvers. Hence, the optimal conguration regard-ing the pedestrian requirements is the one with the highestvalues of eciency and comfort.

During the optimization procedure, some or all of thefollowing can be varied:

1. the location and form of planned buildings,2. the arrangement of walkways, entrances, exits, stair-

cases, elevators, escalators, and corridors,3. the shape of rooms, corridors, entrances, and exits,4. the function and time schedule. (Recreation rooms or

restaurants are often continuously frequented, roomsfor conferences or special events are mainly visited andleft at peak periods, exhibition rooms or rooms forfestivities require additional space for people stand-ing around, and some areas are claimed by queues orthrough trac.)

In contrast to early evolutionary optimization methods,recent approaches allow to change not only the dimen-sions of the dierent elements of pedestrian facilities, butalso to vary their topology. The procedure of such algo-rithms is illustrated in Fig. 10. Highly performing designsare illustrated in Fig. 11. It turns out that, for an emer-gency evacuation route, it is favorable if the crowd doesnot move completely straight towards a bottleneck. Forexample, a zigzag design of the evacuation route can re-duce the pressure on the crowd upstream of a bottleneck(see Fig. 12). The proposed evolutionary optimization pro-cedure can, of course, not only be applied to the designof new pedestrian facilities, but also to a reduction of ex-isting bottlenecks, when suitable modications are imple-mented.

Future Directions

In this contribution, we have presented a multi-agent ap-proach to pedestrian and crowd dynamics. Despite thegreat eort required, pedestrian interactions can be wellquantied by video tracking. Compared to other socialinteractions they turn out to be quite simple. Neverthe-

6492 P Pedestrian, Crowd and Evacuation Dynamics

Pedestrian, Crowd and Evacuation Dynamics, Figure 11Two examples of improved designs for cases with a bottleneck along the escape route of a large crowd, obtained with an evolution-ary algorithmbased on Boolean grids. People were assumed tomove from left to right only. a Funnel-shaped escape route. b Zig-zagdesign

Pedestrian, Crowd and Evacuation Dynamics, Figure 12a Conventional design of a stadium exit in an emergency scenario, where we assume that some pedestrians have fallen at the endof the downwards staircase to the left. The dark color indicates high pressures, since pedestrians are impatient and pushing frombehind. b In the improved design, the increasingdiameter of corridors can reducewaiting times and impatience (evenwith the samenumber of seats), thereby accelerating evacuation. Moreover, the zigzag design of the downwards staircases changes the pushingdirection in the crowd. (After [8])

less, they cause a surprisingly large variety of self-orga-nized patterns and short-lived social phenomena, wherecoordination or cooperation emerges spontaneously. Forthis reason, they are interesting to study, particularly asone can expect new insights into coordination mecha-nisms of social beings beyond the scope of classical gametheory. Examples for observed self-organization phenom-ena in normal situations are lane formation, stripe for-mation, oscillations and intermittent clogging eects atbottlenecks, and the evolution of behavioral conventions(such as the preference of the right-hand side in conti-nental Europe). Under extreme conditions (high densitiesor panic), however, coordination may break down, givingrise to freezing-by-heating or faster-is-slower eects,stop-and-go waves or crowd turbulence.

Similar observations as in pedestrian crowds are madein other social systems and settings. Therefore, we expectthat realistic models of pedestrian dynamics will also pro-mote the understanding of opinion formation and otherkinds of collective behaviors. The hope is that, based onthe discovered elementary mechanisms of emergence andself-organization, one can eventually also obtain a betterunderstanding of the constituting principles of more com-plex social systems. At least the same underlying factorsare found in many social systems: Non-linear interactionsof individuals, time-dependence, heterogeneity, stochas-ticity, competition for scarce resources (here: Space andtime), decision-making, and learning. Future work willcertainly also address issues of perception, anticipation,and communication.

Pedestrian, Crowd and Evacuation Dynamics P 6493Acknowledgments

The authors are grateful for partial nancial supportby the German Research Foundation (research projectsHe 2789/7-1, 8-1) and by the Cooperative Center forCommunication Networks Data Analysis, a NAP projectsponsored by the Hungarian National Oce of Researchand Technology under grant No. KCKHA005.

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