Constant net-time headway as key mechanism behind pedestrian flow dynamics
CConstant-net-time headway as key mechanismbehind pedestrian (cid:29)ow dynamics
Anders Johansson ∗ ETH Zurich,UNO C 11 Universit(cid:228)tstrasse 41,8092 Zurich, Switzerland(Dated: November 5, 2018)We show that keeping a constant lower limit on the net-time headway is the key mechanismbehind the dynamics of pedestrian streams. There is a large variety in (cid:29)ow and speed as functionsof density for empirical data of pedestrian streams, obtained from studies in di(cid:27)erent countries. Thenet-time headway however, stays approximately constant over all these di(cid:27)erent data sets.By using this fact, we demonstrate how the underlying dynamics of pedestrian crowds, naturallyfollows from local interactions. This means that there is no need to come up with an arbitrary (cid:28)tfunction (with arbitrary (cid:28)t parameters) as has traditionally been done.Further, by using not only the average density values, but the variance as well, we show how therecently reported stop-and-go waves [Helbing et al., Physical Review E, 75, 046109] emerge whenlocal density variations take values exceeding a certain maximum global (average) density, whichmakes pedestrians stop.
PACS numbers: 89.40.-a, 87.23.Ge, 12.00.00
I. INTRODUCTION
With an increasing population and with more cost ef-fective transportation, mass gatherings become more fre-quent. The total size of such gatherings are often as largeas millions of people, for example during the inaugurationceremony of president Obama [1] and the Hajj pilgrimageto Mecca [2].To guarantee the safety of the participants during suchlarge mass gatherings, careful planning needs to be car-ried out by the organizer. During the last decades, nu-merous empirical studies [2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13] have been performed on pedestrian crowds in dif-ferent countries, in order to understand the dynamics ofthese crowds. Even though an understanding of crowddynamics is a prerequisite for being able to plan a massgathering, there is still no concensus on some of the mostbasic relations, such as how the (cid:29)ow of people (people permeter per second) depends on the crowd density (peopleper m ). Misconceptions of these basic relations mayresult in serious safety risks during mass gatherings [13].Let us now start from the bottom up, and show howlocal interactions lead to certain (cid:29)ow-density relation-ships for the stream of pedestrians. Since movement andavoidance patterns of pedestrians tend to be rather com-plex, the traditional way to reduce complexity is to (cid:28)nda relation of the (cid:29)ow Q ( m − s − ) as a function of theaverage density (cid:37) ( m − ). Using this relation, called thefundamental diagram, has been successful to some extent,but unfortunately there are large variations of these re-lations, among empirical studies carried out in di(cid:27)erent ∗ Electronic address: [email protected] countries. All these studies agree on that walking speedof pedestrians is a decreasing function of density, butthey disagree on how this function looks like. We willnow demonstrate how the net-time headway, as a resultof (cid:28)nite reaction times, is the key mechanism which canexplain the discrepancies between data sets from di(cid:27)erentstudies.
II. REACTION TIME
It is known from tra(cid:30)c science that (cid:28)nite reactiontimes are needed to explain instabilities in tra(cid:30)c (cid:29)ows[14]. For pedestrian-(cid:29)ow dynamics, the role of (cid:28)nite re-action times has not been investigated in detail. By doingso, it turns out that the (cid:28)nite reaction time gives rise toa certain net-time headway, which is needed as a safetyheadway, to prevent accidental physical encounters withsurrounding pedestrians.Many-particle simulations [15, 16] coupled with empiri-cal pedestrian-trajectory data [17] reveal the probability-density function of delay times T d from a walking exper-iment [18] where two pedestrian streams are intersectingat a ◦ angle. The resulting distribution of delay timesare shown in Fig. 1.Interestingly, the probability-density function of thedelay curve is bi-modal. The (cid:28)rst peak occurs at lowertimes than the typical response times, to visual or acous-tic cues [19, 20]. Therefore, this peak must correspondto anticipated movements of the surrounding pedestrians.The second peak at around 0.45 s occurs at times whichare signi(cid:28)cantly larger than the previously mentioned re-sponse times, but also lower than response times involv-ing conscious reactions [14, 21]. Therefore, we concludethat this second peak corresponds to an unconscious re- a r X i v : . [ phy s i c s . s o c - ph ] A ug P r obab ili y den s i t y Figure 1: Probability-density function of the delay T d . Theerror bars correspond to one standard deviation. Interest-ingly, the probability-density function is bi-modal. When thesurrounding pedestrians are acting in a way that is easy topredict, extrapolation allows to anticipate their behaviors,while a delayed reaction results in cases of unexpected be-haviors. sponse, which is more complex than a simple reaction.In fact, it has been shown that reactions where there aremore than one possible response (choice reaction time)as well as reactions to more complex cues (recognitionreaction time) take signi(cid:28)cantly longer time [22].We interpret the bi-modality as follows: When the sur-rounding pedestrians act in a way that is easy to predict,extrapolation allows to anticipate their behaviors, while adelayed reaction results in cases of unexpected behaviors. III. MODEL
There has been a rich amount of microscopic models ofpedestrian dynamics, for example the social-force model[15, 16] and cellular-automata models [23, 24, 25]. Thesemodels are able to reproduce various self-organizationphenomena, such as lane and stripe formation [26], freez-ing by heating [27], Mexican waves in excitable media[28], intermittent out(cid:29)ows [29], stop-and-go waves andcrowd turbulence [15].When measuring empirical pedestrian (cid:29)ows and densi-ties and then (cid:28)tting a suitable curve to the data, one ob-tains a function which is useful for engineering involvedin planning of pedestrian facilities. This pragmatic (cid:28)tcurve, however, does not provide any insight into themechanisms and dynamics behind the pedestrian inter-actions and behaviours, leading to the aggregated data.However, when plotting the fundamental diagrams ob-tained in various empirical studies (Fig. 3 (top)), one cansee that each of the curves has a similar parabola-like −2 ) M i n i m u m d i s t an c e ( m ) Figure 2: The distance between an arbitrary pedestrian α andthe closest surrounding pedestrian β , as a function of global(average) crowd density (cid:37) . The solid line shows the averagevalue ± one standard deviation as error bars, and the dashedline shows the (cid:28)t curve / p ( (cid:37) ) . The data is from Ref. [18] shape. Nevertheless, the curves are quite di(cid:27)erent fromone measurement site to another. One question remainsto be answered: What, if any, is the common underlyingprinciple of these curves?In an attempt to bridge this knowledge gap, let us comeback to the issue of reaction times, mentioned before.Since pedestrians have a typical reaction time T d =0.45s to unexpected behaviors of surrounding pedestrians, itwould be natural that they compensate the risk of bump-ing into others, by keeping a certain safety time headwayto the surrounding pedestrians [34].To connect the aggregated density to local interactions,let us approximate the mean distance between the center-of-masses of a pedestrian α and the closest pedestrian β by d = (cid:104) d αβ (cid:105) = 1 / √ (cid:37) , where (cid:37) is the global [35] (average)density. Note that this would hold only if the pedestri-ans were distributed into a square lattice, but for otherdensity distributions, it will serve as a fair approximation(see Fig. 2).The net distance is de(cid:28)ned as ˆ d = d − r , where r =1 / (2 (cid:112) (cid:37) max ) is the e(cid:27)ective radius of a pedestrian, and (cid:37) max is the largest measured density.Assuming that the predecessor β [36] would sud-denly stop [29], it would take ˆ T = ˆ d/v α = (1 / √ (cid:37) − / √ (cid:37) max ) /v α seconds before a physical encounter withpedestrian α occurs, if v α is the speed of pedestrian α . We now show how the net-time headway ˆ T dependson the global density (cid:37) by applying the above schemeto empirical data determined from di(cid:27)erent studies (seeFig. 3(bottom)).Note that ˆ T saturates at a constant value, that is verysimilar to the response time to unexpected behaviors (seeFig. 1). However, in the data of Ref. [2] there is a transi-tion at very high densities, where ˆ T suddenly increases.This can be interpreted in at least two ways: • Hypothesis 1: When the density is very high, pedes-trians start to have fear of crushing or asphyxia [30],and therefore want to increase the space aroundthemselves (leading to higher net-time headways ˆ T ). • Hypothesis 2: If the space in front of a pedestrian istoo small (or the velocity is too low) it will no longerbe possible to take normal steps. Rather, pedes-trians will completely stop until they have gainedenough space to make a step.In previous work [30], Hypothesis 1 has been used. Inthis study, however, we will investigate Hypothesis 2.This interpretation would naturally explain the empir-ically observed stop-and-go waves analyzed in Ref. [2],and would further imply: Above an average density of 5persons per m , the fundamental diagram will no longerdescribe the dynamics of the crowd well, since the (cid:29)owrate is then alternating between movement and standstillrather than continuous.In an attempt to unify all fundamental diagrams inthe same framework, the following scheme is proposed:Each pedestrian α has a free speed v α = v max (whichis an upper speed limit, occurring when (cid:37) → ). Eachpedestrian also has a lower limit v min of the speed. For v < v min , pedestrians can no longer make normal steps,and would rather stop completely. For simplicity, thesevalues are assumed to be the same for all pedestrians.It has been reported in Ref. [31] that the free (uncon-strained) headways are exponentially distributed, wherethe constrained headways on the other hand, are limitedby a desired minimum headway. Therefore, we proposethat the net-time headway ˆ T is the key control parameterfor the fundamental diagram. That is, pedestrians willdecrease their speeds, if necessary, to assure a constantlower limit of the net-time headway ˆ T . The fundamentaldiagram can now be speci(cid:28)ed as: v ( (cid:37) ) = d − r ˆ T = 1 / √ (cid:37) − / √ (cid:37) max ˆ T (1)and bounded by [ v min , v max ] It has been shown in Ref. [32] that each average density (cid:37) corresponds to a distribution of local densities ρ , andwe therefore approximate the density distribution witha Gaussian distribution N ( (cid:37), (cid:112) (cid:37)/ with mean (cid:37) andstandard deviation (cid:112) (cid:37)/ (see Fig. 4).According to Hypothesis 2, de(cid:28)ned above, we get an ex-tra constraint, saying that pedestrians will stop walkingif they are too close to other pedestrians, which happensfor ρ ≥ (cid:37) max (physical interaction). They will then re-sume walking again when they have enough space L fortaking a step. Since one step (for low walking speed) -2 ) F l o w ( m - s - ) Density (m -2 ) N e t - t i m e head w a y T ( s ) Seyfried et al.Mori and TsukaguchiHelbing et al.
Weidmann ^ Figure 3: (Color online) Top: Flow as a function of density,for data from a number of empirical studies. Bottom: Thenet-time headways ˆ T as a function of density (cid:37) . ˆ T is mostoften bounded by a constant lower value of about 0.5 s. Inthe data of Ref. [2], however, there is a transition for highdensities where ˆ T suddenly increases. The data sets are thesame as used in Ref. [2], i.e. the data from (Helbing et al.)correspond to local densities and (cid:29)ows. needs approximately L ≈ . m [31], we get a new net-time headway ˆ T (cid:48) = L/v min ≈ s, whenever ρ ≥ (cid:37) max .The fraction of pedestrians that are physically col-liding with others, can be measured by integrating theprobability-density-function of the Gaussian distribution(see Fig. 6 (top) ). f stop = (cid:90) ∞ (cid:37) max N ( ρ ) dρ (2)with mean µ = (cid:37) and standard deviation σ = (cid:112) (cid:37)/ .Then, the mean net-time headway (see Fig. 6 (bottom))is given by the fraction of stopped pedestrians as (cid:104) ˆ T (cid:105) = (1 − f stop ) ˆ T + f stop Lv min . (3)Figure 5 shows generated fundamental diagrams fromEq. (1) with the parameters ˆ T = 0 . s, (cid:37) max = 5 . m − , −2 ) D en s i t y d i s t r i bu t i on Figure 4: (Color online) Distributions of local densities ρ for three di(cid:27)erent global (average) densities (cid:37) = 1 . , and5 pedestrians/ m . The data comes from Ref. [2]. For eachglobal density (cid:37) a Beta distribution is (cid:28)tted (dashed lines).However, Gaussian distributions (solid lines) also (cid:28)t the datafairly well. The Gaussian distributions are produced with theparameters µ = (cid:37) and σ = p (cid:37)/ . and for di(cid:27)erent values of the free speed v = v max . Since v only gives the upper limit of the velocity, fundamen-tal diagrams with di(cid:27)erent v converge at high enoughcrowd densities, given that all other parameters are (cid:28)xed.The reason is that, for high density, the movement istransformed from individual walking to walking which isconstrained by other pedestrians.We now apply the method outlined above on di(cid:27)erentempirical fundamental diagrams. In all cases we use ˆ T =0 . s and v min = 0 . m/s. Starting with Weidmann’s[3] fundamental diagram, we have the parameters (cid:37) max =5 . m − and v = 1 . m/s, which is displayed in Fig. 7(a, b) together with our curve, obtained by Eqs. (1) to(3).Next, we apply our method to the fundamental dia-grams of Refs. [4], [2] and [11], and obtain the resultspresented in Fig. 7 (c-h). The (cid:28)t functions match allfour di(cid:27)erent empirical data sets well. All parametersare kept constant over the di(cid:27)erent data sets, except themaximum density and the free speed, but these two val-ues are obtained from the data, rather than tuned inorder to (cid:28)t the data. IV. CONCLUSIONS
The constant net-time headway is a natural safetymechanism to compensate for the reaction time to un-expected events. It has been demonstrated that variousdata sets, from di(cid:27)erent countries, all share the same net- time headway ˆ T = 0.5 s. The particular advantage of our -2 ) V e l o c i t y ( m / s ) Weidmannv =0.8 m/sv =1.3 m/sv =1.8 m/s -2 ) F l o w ( m s ) - - Figure 5: (Color online) Fundamental diagrams and velocity-density relations generated by Eq. (1), for di(cid:27)erent free speeds v . Note how they all converge for large densities. As acomparison, the empirical (cid:28)t curve by Weidmann [3] is shown. method is that it follows naturally, without the need ofan arbitrary (cid:28)t function. Further, all the parameters aremeasurables, such as the free speed and the maximumdensity. There is not a single free parameter that mustbe tuned in order to (cid:28)t the di(cid:27)erent data sets. However,it should be mentioned that even though the maximumdensity can be estimated, it can normally not be exactlydetermined from the data. This is addressed in recentwork [33] that may make it possible to obtain culturallydependent parameters, such as the maximum density. V. ACKNOWLEDGMENTS
The author is grateful for the partial (cid:28)nancial supportby the DFG grant He 2789/7-1.He would also like to thank Dirk Helbing for his valu-able comments, Serge Hoogendoorn for sharing data fromhis experiments, and to Habib Al-Abideen for sharingdata from the Hajj. N e t - t i m e head w a y ( s ) With stoppingWithout stopping
Global density (m -2 ) Local density (m -2 ) P r obab ili t y den s i t y f un c t i on max (cid:63) Area which gives the fraction of stopped pedestrians stop f T ^ Figure 6: Top: The mean net-time headway (cid:104) ˆ T (cid:105) is obtainedvia the fraction of pedestrians who are physically collidingwith others, and is therefore stopping and temporarily in-creasing their net-time headway. This fraction is obtainedby integrating over the probability-density-function of thelocal-density distribution, starting at local densities ρ thatare higher than the maximum global density (cid:37) . Bottom: Themean net-time-headway (cid:104) ˆ T (cid:105) (solid line) as a function of theglobal density (cid:37) . The net-time headway without stopping isdisplayed as a dashed line.[1] P. Taylor, How O(cid:30)cials Will Control the Crowds atObama’s Inauguration, Popular Mechanics, January 20,2009.[2] D. Helbing, A. Johansson, and H. Z, Al-Abideen, TheDynamics of Crowd Disasters: An Empirical Study,Phys. Rev. E 75, 046109 (2007).[3] U. Weidmann, Transporttechnik der Fu(cid:255)g(cid:228)nger, ETH-Z(cid:252)rich, Schriftenreihe IVT-Berichte 90 (1993).[4] M. Mori H. Tsukaguchi, A new method for evaluationof level of service in pedestrian facilities, TransportationResearch A 21(3), pp. 223(cid:21)234 (1987).[5] J. J. Fruin, Designing for pedestrians: A level-of-serviceconcept, Highway Research Record 355, 1-15 (1971).[6] K. Ando, H. Ota, and T. Oki, Forecasting the (cid:29)ow ofpeople, Railway Research Review 45, 8-14 (1988). (inJapanese).[7] A. Polus, J. L. Schofer, and A. Ushpiz, Pedestrian (cid:29)owand level of service, Journal of Transportation Engineer- ing 109, 46-56 (1983).[8] R. A. Smith and J. F. Dickie (eds.), Engineering forCrowd Safety. (Elsevier, Amsterdam) (1993).[9] K. Still, Crowd Dynamics, PhD Thesis (2000).[10] K. Teknomo, Microscopic Pedestrian Flow Characteris-tics: Development of an Image Processing Data Collec-tion and Simulation Model. PhD Thesis, Japan (2002).[11] A. Seyfried, B. Ste(cid:27)en, W. Klingsch, and M. Boltes, Thefundamental diagram of pedestrian movement revisited,J. Stat. Mech. P10002 (2005).[12] T. Kretz, A. Gr(cid:252)nebohm, and M. Schreckenberg, Exper-imental study of pedestrian (cid:29)ow through a bottleneck, J.Stat. Mech P10014 (2006).[13] A. Schadschneider, W. Klingsch, H. Kluepfel, T. Kretz,C. Rogsch, and A. Seyfried, Evacuation Dynamics: Em-pirical Results, Modeling and Applications, Encyclope-dia of Complexity and System Science, B. Meyers (Ed.)(Springer, Berlin, 2008). F l o w ( m - s - ) V e l o c i t y ( m / s ) F l o w ( m - s - ) V e l o c i t y ( m / s ) F l o w ( m - s - ) V e l o c i t y ( m / s ) -2 ) F l o w ( m - s - ) WeidmannMori and TsukaguchiHelbing et al. -2 ) V e l o c i t y ( m / s ) Seyfried et al. a bc de fg h
Figure 7: (Color online) Fundamental diagrams and velocity-density relations generated by Eqs. (1) to (3), assuming aconstant net-time headway ˆ T = 0 . s and a minimum ve-locity v min = 0 . m/s. Red markers represent empiricaldata and solid lines the theoretically expected relationships.a) Flow-density data, and b) velocity-density data by Wei-dmann [3], compared to a fundamental diagram generatedwith the parameters (cid:37) max = 5 . m − and v max = 1 . m/s.The dashed line shows the result when it is assumed thatno pedestrians are stopping, i.e. f stop = 0 . c) Flow-densitydata, and d) velocity-density data by Mori and Tsukaguchi[4], compared to a fundamental diagram generated with theparameters (cid:37) max = 12 m − and v max = 1 . m/s. e) Flow-density data, and f) velocity-density data from Helbing etal. [2], compared to a fundamental diagram generated withthe parameters (cid:37) max = 9 . m − and v max = 0 . m/s. g)Flow-density data, and h) velocity-density data from Seyfriedet al. [11], compared to a fundamental diagram generated withthe parameters (cid:37) max = 5 . m − and v max = 1 . m/s.[14] M. Treiber, A. Kesting, and D. Helbing, Delays, inac-curacies and anticipation in microscopic tra(cid:30)c models,Physica A 360, pp. 71(cid:21)88 (2006).[15] D. Helbing and P. MolnÆr, Social force model for pedes-trian dynamics, Phys, Rev. E 51, pp. 4282-4286 (1995).[16] D. Helbing, I. Farkas, and T. Vicsek, Simulating dynam- ical features of escape panic, Nature 407, pp. 487(cid:21)490(2000).[17] A. Johansson, Data-driven modeling of pedestriancrowds, PhD Thesis, TU Dresden (2009).[18] S. P. Hoogendoorn, W. Daamen, and P. H. L. Bovy,Extracting microscopic pedestrian characteristics fromvideo data, Annual Meeting Transportation Res. BoardPre-print CD-Rom, (Mira Digital Publishing, Washing-ton, D.C) (2003).[19] R. S. Woodworth and H. Schlosberg, Experimental Psy-chology. Henry Holt, New York (1954).[20] A. T. Welford, Choice reaction time: Basic concepts. InA. T. Welford (Ed.), Reaction Times. Academic Press,New York, pp. 73(cid:21)128 (1980).[21] M. Green, How Long Does It Take to Stop? Methodolog-ical analysis of driver perception-brake times, Transport.Hum. Factors 2 pp. 195(cid:21)216, (2000).[22] F. C. Donders, On the speed of mental processes. Trans-lated by W. G. Koster, 1969. Acta Psychologica, 30,pp. 412(cid:21)431 (1868).[23] A. Kirchner, K. Nishinari, and A. Schadschneider, Fric-tion e(cid:27)ects and clogging in a cellular automaton modelfor pedestrian dynamics, Phys. Rev. E 67, 056122 (2003).[24] C. Burstedde, K. Klauck, A. Schadschneider, and J. Zit-tartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295(3-4),pp. 507(cid:21)525 (2001).[25] W. G. Weng, T. Chen, H. Y. Yuan, and W. C. Fan,Cellular automaton simulation of pedestrian counter (cid:29)owwith di(cid:27)erent walk velocities, Phys. Rev. E 74, 036102(2006)[26] D. Helbing, L. Buzna, A. Johansson, T. Werner, Self-organized pedestrian crowd dynamics: Experiments, sim-ulations, and design solutions, Transp. Sci. 39, pp. 1(cid:21)24(2005).[27] D. Helbing, I. Farkas, and T. Vicsek, Freezing by heat-ing in a driven mesoscopic system, Phys. Rev. Lett, 84,pp. 1240(cid:21)1243 (2000).[28] I. Farkas, D. Helbing, and T. Vicsek, Mexican waves inan excitable medium, Nature 419, pp. 131(cid:21)132 (2002).[29] D. Helbing, A. Johansson, J. Mathiesen, H. M. Jensen,and A. Hansen, Analytical approach to continuous andintermittent bottleneck (cid:29)ows, Phys. Rev. Lett. 97,168001 (2006).[30] W. Yu and A. Johansson, Modeling crowd turbulenceby many-particle simulations, Phys. Rev. E 76, 046105(2007).[31] S. P. Hoogendoorn, W. Daamen, Pedestrian behavior atbottlenecks, Transp. Sci. 39(2), pp. 147(cid:21)159 (2005).[32] A. Johansson, D. Helbing, H. Z. A-Abideen, S. Al-Bosta,From crowd dynamics to crowd safety: A video-basedanalysis, Advances in Complex Systems 11(4), pp. 497(cid:21)527 (2008).[33] U. Chattaraj, A. Seyfried, and P. Chakroborty, Compar-ison of pedestrian fundamental diagram across cultures,Advances in Complex Systems 12, 393 (2009).[34] Note that we use the peak at longer delays since this peakcorresponds to unexpected events. Even though there isa peak at lower delay times, there is no guarantee thatthe behaviours of others can always be anticipated, andtherefore the maximum peak must be used as a safetytime.[35] The global density (cid:37) is de(cid:28)ned as the number of peoplewithin a certain area, divided by that area. The local den- sity ρρ