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The Inflection Point of the Speed-Density Relation and the Social Force Model

Tobias Kretz, Jochen Lohmiller, Johannes Schlaich

Abstract


It has been argued that the speed-density diagram of pedestrian movement has an inflection point. This inflection point was found empirically in investigations of closed-loop single-file pedestrian movement.

The reduced complexity of single-file movement does not only allow a higher precision for the evaluation of empirical data, but it also significantly simplifies analytical considerations. This is especially true if one assumes homogeneous conditions, i.e. neglects temporal variations (consider time averages, neglect stop-and-go waves), individual differences of pedestrians (all simulated pedestrians have identical parameters) and investigates only steady-state (not the initial phase). As will be shown in this contribution one then can make a transition from the microscopic to a continuous and macroscopic perspective.

Building on that it will be shown that certain (common) variants of the Social Force Model (SFM) do not produce an inflection point in the speed-density diagram if – assuming periodic boundary conditions – infinitely many pedestrians contribute to the force computed for one pedestrian. It will furthermore be shown that if – in said 1d movement situation – one only considers nearest neighbors for the computation of the inter-pedestrian forces the Social Force Model in the continuous description results in the so called Kladek formula for the speed-density relation. Since the Kladek formula exhibits the desired inflection point this observation is used as a motivation for an extension of the Social Force Model which allows to transform the continuous description of the SFM continuously to the Kladek formula and which also exhibits the inflection point in the speed density relation. It will be shown then, that this extended SFM yields astonishingly similar speed density relations as the original SFM when only a fixed limited number of (nearest) pedestrians are considered in the computation of the inter-pedestrian force.

Finally it will be discussed, if also the description of the speed-density diagram for (motorized, four-wheel) vehicular and/or bicycle traffic could benefit from these measures.


Keywords


pedestrians; speed density relation; Social Force Model; Kladek Formula

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References


Seyfried, A., Steffen, B., Klingsch, W., Boltes, M.: The fundamental diagram of pedestrian movement revisited. Journal of Statistical Mechanics: Theory and Experiment 2005(10), P10002 (2005). doi:10.1088/1742-5468/2005/10/P10002

Chattaraj, U., Seyfried, A., Chakroborty, P.: Comparison of pedestrian fundamental diagram across cultures. Advances in Complex Systems 12(03), 393-405 (2009). doi:10.1142/S0219525909002209

Seyfried, A., Portz, A., Schadschneider, A.: Phase coexistence in congested states of pedestrian dynamics. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds.) Cellular Automata, Lecture Notes in Computer Science, vol. 6350, pp. 496-505. Springer Berlin Heidelberg (2010)

Seyfried, A., Boltes, M., Kähler, J., Klingsch, W., Portz, A., Rupprecht, T., Schadschneider, A., Steffen, B., Winkens, A.: Enhanced empirical data for the fundamental diagram and the flow through bottlenecks. In: Pedestrian and Evacuation Dynamics 2008, pp. 145-156. Springer (2010)

Portz, A., Seyfried, A.: Analyzing stop-and-go waves by experiment and modeling. In: Peacock, R.D., Kuligowski, E.D., Averill, J.D. (eds.) Pedestrian and Evacuation Dynamics, pp. 577-586. Springer US (2011)

Weidmann, U.: Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturauswertung). ETH, IVT (1993)

Buchmueller, S., Weidmann, U.: Parameters of pedestrians, pedestrian traffic and walking facilities. Institute for Transport Planning and Systems (IVT), Chair of Transport Systems, ETH Zurich (2006)

Kladek, H.: Über die Geschwindigkeitscharakteristik auf Stadtstraßenabschnitten. Ph.D. thesis, Hochschule für Verkehrswesen ,,Friedrich List" Dresden (1966)

Newell, G.: Nonlinear effects in the dynamics of car following. Operations research 9(2), 209-229 (1961). doi:10.1287/opre.9.2.209

Lohse, D., Schnabel, W.: Grundlagen der Straßenverkehrstechnik und der Verkehrsplanung: Band 1-Verkehrstechnik, 3 edn. Beuth Verlag (2011)

Wu, N.: Verkehr auf Schnellstraßen im Fundamentaldiagramm. Straßenverkehrstechnik 8, 378-388 (2000)

Kühne, R.: Das Fundamentaldiagramm-Grundlagen und Anwendungen. FGSV Merkblatt (Entwurf) p. 59 (2004)

Daamen, W.: Modelling passenger flows in public transport facilities. Ph.D. thesis, Delft University of Technology (2004)

Bellomo, N., Dogbe, C.: On the modelling crowd dynamics from scaling to hyperbolic macroscopic models. Mathematical Models and Methods in Applied Sciences 18(supp01), 1317-1345 (2008). doi:10.1142/S0218202508003054

Lämmel, G., Rieser, M., Nagel, K.: Bottlenecks and congestion in evacuation scenarios: A microscopic evacuation simulation for large-scale disasters. In: Proc. of 7th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2008), Estoril, Portugal. Citeseer (2008)

Lämmel, G., Klüpfel, H., Nagel, K.: The MATSim network flow model for traffic simulation adapted to large-scale emergency egress and an application to the evacuation of the Indonesian city of Padang in case of a tsunami warning, chap. 11, pp. 245-265. Emerald Insight (2009). doi:10.1108/9781848557512-011

Venuti, F., Bruno, L.: Crowd-structure interaction in lively footbridges under synchronous lateral excitation: A literature review. Physics of Life Reviews 6(3), 176-206 (2009). doi:10.1016/j.plrev.2009.07.001

Beltaief, O., El Hadouaj, S., Ghedira, K.: Multi-agent simulation model of pedestrians crowd based on psychological theories. In: Logistics (LOGISTIQUA), 2011 4th International Conference on, pp. 150-156. IEEE (2011). doi:10.1109/LOGISTIQUA.2011.5939418

Galiza, R., Ferreira, L., Charles, P.: Estimating the effects of older people in pedestrian flow: A micro-simulation approach. In: Transport Research Board TRB 90th Annual Meeting, pp. 1-13. Transportation Research Board (TRB) (2011)

Papadrakakis, M., Fragiadakis, M., Plevris, V.: Footbridge lateral vibrations induced by synchronised pedestrians: An overview on modelling strategies. In: 3rd ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (2011)

Bruno, L., Tosin, A., Tricerri, P., Venuti, F.: Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications. Applied Mathematical Modelling 35(1), 426-445 (2011). doi:10.1016/j.apm.2010.07.007

Bruckner, M., Tauböck, S., Popper, N., Emrich, Š.., Rozsenich, B., Alkilani, S.: A combined cellular automata and devs simulation. In: MathMod 2012 (2012)

Huth, F., Bärwolff, G., Schwandt, H.: Fundamental diagrams and multiple pedestrian streams. Preprint 2012/17 p. eprint (2012)

Lüßmann, J., Sánta, C.: Network-wide evaluation of cooperative traffic systems using microscopic traffic flow simulation. Simulation Notes Europe p. 89 (2012). doi:10.11128/sne.22.tn.10129

Zhang, J.: Pedestrian fundamental diagrams: Comparative analysis of experiments in different geometries. Ph.D. thesis, Forschungszentrum Jülich (2012)

Nikolić, M., Bierlaire, M., Farooq, B.: Probabilistic speed-density relationship for pedestrians based on data driven space and time representation. In: Proceedings of the 14th Swiss Transport Research Conference (2014)

Martinez G., F.A.: Reinforcement learning in a multi-agent framework for pedestrian simulation. Ph.D. thesis, University of Valencia (2014)

Chen, L.: Towards a multi-agent based software framework for fine-scale pedestrian movement modelling. International Journal of Modelling and Simulation 34(2), 5945 (2014). doi:10.2316/Journal.205.2014.2.205-5945

Daniel, J., Truong, C., Lauffenburger, J., Basset, M.: Real-time trajectory generation for advanced driver assistance systems applications. In: International Forum On Strategic Technologies (IFOST09), vol. 26 (2009)

Helbing, D., Farkas, I., Vicsek, T.: Simulating dynamical features of escape panic. Nature 407, 487-490 (2000). doi:10.1038/35035023

Kretz, T.: On oscillations in the social force model. Physica A: Statistical Mechanics and its Applications 438, 272-285 (2015). doi:10.1016/j.physa.2015.07.002

Johansson, A., Helbing, D., Shukla, P.: Specification of the social force pedestrian model by evolutionary adjustment to video tracking data. Advances in Complex Systems 10(supp02), 271-288 (2007). doi:10.1142/S0219525907001355

Seyfried, A., Steffen, B., Lippert, T.: Basics of modelling the pedestrian flow. Physica A: Statistical Mechanics and its Applications 368(1), 232-238 (2006). doi:10.1016/j.physa.2005.11.052

Helbing, D., Molnar, P.: Social force model for pedestrian dynamics. Physical review E 51(5), 4282 (1995). doi:10.1103/PhysRevE.51.4282

Wolfram|Alpha: (2015). Publisher: Wolfram Alpha LLC. Retrieved March 24th 2015. https://www.wolframalpha.com/input/?i=solve+%282x-1%29*exp%281%2Fx%29+-+0.9*%282x%2B1%29+for+x

Yu, W., Chen, R., Dong, L., Dai, S.: Centrifugal force model for pedestrian dynamics. Physical Review E 72(2), 026112 (2005). doi:10.1103/PhysRevE.72.026112

Yu, W., Johansson, A.: Modeling crowd turbulence by many-particle simulations. Physical Review E 76(4), 046105 (2007). doi:10.1103/PhysRevE.76.046105

Chraibi, M., Seyfried, A., Schadschneider, A.: Generalized centrifugal-force model for pedestrian dynamics. Physical Review E 82(4), 046111 (2010). doi:10.1103/PhysRevE.82.046111

Campanella, M., Hoogendoorn, S., Daamen, W.: The nomad model: theory, developments and applications. Transportation Research Procedia 2, 462-467 (2014). doi:10.1016/j.trpro.2014.09.061

Sherif, M.: The psychology of social norms. Harper (1936)

Kretz, T., Große, A., Hengst, S., Kautzsch, L., Pohlmann, A., Vortisch, P.: Quickest Paths in Simulations of Pedestrians. Advances in Complex Systems 14, 733-759 (2011). doi:10.1142/S0219525911003281

Kretz, T., Große, A.: From Unbalanced Initial Occupant Distribution to Balanced Exit Usage in a Simulation Model of Pedestrian Dynamics. In: T.J. Shields et al. (ed.) Human Behaviour in Fire Symposium, pp. 536-540. Interscience Communications (2012)

Kretz, T.: Multi-Directional Flow as Touch-Stone to Assess Models of Pedestrian Dynamics. In: Annual Meeting of the TransportationResearch Board 2013 (2013). 13-1160

Kretz, T.: The Effect of Integrating Travel Time. In: Weidmann, U., Kirsch, U., Schreckenberg, M. (eds.) Pedestrian and Evacuation Dynamics 2012, pp. 1013-1027 (2013)

Ziemer, V., Seyfried, A., Schadschneider, A.: Congestion dynamics in pedestrian single-file motion. In: Knoop, V., Daamen, W. (eds.) Traffic and Granular Flow 2015 (preprint)

Zhang, J., Mehner, W., Andresen, E., Holl, S., Boltes, M., Schadschneider, A., Seyfried, A.: Comparative analysis of pedestrian, bicycle and car traffic moving in circuits. Procedia-Social and Behavioral Sciences 104, 1130-1138 (2013). doi:10.1016/j.sbspro.2013.11.209

Seyfried, A., Zhang, J., Andresen, E., Boltes, M., Mehner, W., Holl, S., Schadschneider, A.: Universalities in fundamental diagrams of cars, bicycles and pedestrians. In: Symposium Celebrating 50 Years of Traffic Flow Theory, FZJ-2014-06019. Jülich Supercomputing Center (2014)

Sugiyama, Y., Fukui, M., Kikuchi, M., Hasebe, K., Nakayama, A., Nishinari, K., Tadaki, S., Yukawa, S.: Traffic jams without bottlenecks - experimental evidence for the physical mechanism of the formation of a jam. New Journal of Physics 10(3), 033001 (2008). doi:10.1088/1367-2630/10/3/033001

Nakayama, A., Fukui, M., Kikuchi, M., Hasebe, K., Nishinari, K., Sugiyama, Y., Tadaki, S., Yukawa, S.: Metastability in the formation of an experimental traffic jam. New Journal of Physics 11(8), 083025 (2009). doi:10.1088/1367-2630/11/8/083025

PTV AG: PTV Vissim 8.0 - User Manual. PTV Group, Haid-und-Neu-Str. 15, D-76131 Karlsruhe, Germany (2015)

Hupfer, C.: Computergestützte Videobildverarbeitung zur Verkehrssicherheitsarbeit - am Beispiel von Fußgängerquerungen an städtischen Hauptverkehrsstraßen. Ph.D. thesis, University of Kaiserslautern (1997)

Zhang, J., Klingsch, W., Schadschneider, A., Seyfried, A.: Transitions in pedestrian fundamental diagrams of straight corridors and t-junctions. Journal of Statistical Mechanics: Theory and Experiment 2011(06), P06004 (2011). doi:10.1088/1742-5468/2011/06/P06004

Lighthill, M., Whitham, G.: On kinematic waves. ii. a theory of traffic flow on long crowded roads. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 229, pp. 317-345. The Royal Society (1955). doi:10.1098/rspa.1955.0089

Heydecker, B., Addison, J.: Analysis and modelling of traffic flow under variable speed limits. Transportation Research Part C: Emerging Technologies 19(2), 206-217 (2011). doi:10.1016/j.trc.2010.05.008

Treiber, M., Kesting, A.: Traffic Flow Dynamics: Data, Models and Simulation. Springer-Verlag Berlin Heidelberg (2013). doi:10.1007/978-3-642-32460-4

Takayasu, M., Takayasu, H.: 1/f noise in a traffic model. Fractals 1(04), 860-866 (1993). doi:10.1142/S0218348X93000885

Schadschneider, A., Schreckenberg, M.: Traffic flow models with ‘slow-to-start’ rules. Annalen der Physik 509(7), 541-551 (1997). doi:10.1002/andp.19975090703




DOI: http://dx.doi.org/10.17815/CD.2016.6

Copyright (c) 2016 Tobias Kretz, Jochen Lohmiller, Johannes Schlaich

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