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December  2013, 8(4): 985-1007. doi: 10.3934/nhm.2013.8.985

Structured first order conservation models for pedestrian dynamics

1. 

Siemens AG, Corporate Technology, 80200 Munich, Germany, Germany

Received  August 2012 Revised  April 2013 Published  November 2013

In this contribution, we revisit multiple first order macroscopic modelling approaches to pedestrian flows and computationally compare the results with a microscopic approach to pedestrian dynamics. We find that widely used conservation schemes show significantly different results than microscopic models. Thus, we propose to adopt on a macroscopic level a structured continuum model. The approach basically relies on fundamental diagrams - the relationship between fluxes and local densities - as well as the explicit consideration of individual velocities, thus showing similarities to generalised kinetic models. The macroscopic model is outlined in detail and shows a significantly better agreement with microscopic pedestrian models. The increased realism, important for safety relevant real life applications, is underlined considering several scenarios.
Citation: Dirk Hartmann, Isabella von Sivers. Structured first order conservation models for pedestrian dynamics. Networks and Heterogeneous Media, 2013, 8 (4) : 985-1007. doi: 10.3934/nhm.2013.8.985
References:
[1]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[2]

N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Netw. Heterog. Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.

[3]

N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow, CR Mecanique, 333 (2005), 843-851. doi: 10.1016/j.crme.2005.09.004.

[4]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: a survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.

[5]

N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling, Math. Mod. Meth. Appl. S., 12 (2002), 1801-1843. doi: 10.1142/S0218202502002343.

[6]

V. Blue, M. Embrechts and J. Adler, Cellular automata modeling of pedestrian movements, IEEE Int. Conf. on Syst., Man and Cybern., (1997), 2320. doi: 10.1109/ICSMC.1997.635272.

[7]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525, arXiv:cond-mat/0102397. doi: 10.1016/S0378-4371(01)00141-8.

[8]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics, Math. Mod. Meth. Appl. S., 18 (2008), 1217-1247. doi: 10.1142/S0218202508003017.

[9]

C. Canuto, F. Fagnani and P. Tilli, A eulerian approach to the analysis of rendez-vous algorithms, Proceedings of the 17th IFAC World Congress, (2008), 9039-9044.

[10]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Appl. Math., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[11]

M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics, Netw. Heterog. Media, 6 (2011), 425-442. doi: 10.3934/nhm.2011.6.425.

[12]

E. Christiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with applications to crowd dynamics, Multiscale Model. Sim., 9 (2011), 155-182. doi: 10.1137/100797515.

[13]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks, Mathematical Methods in the Applied Sciences, 28 (2005), 1553-1567. doi: 10.1002/mma.624.

[14]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics, Phys. Rev. E, 82 (2010), 046111. doi: 10.1103/PhysRevE.82.046111.

[15]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon, Oxford, 1997.

[16]

M. Fukui and Y. Ishibashi, Self-organized phase transitions in cellular automaton models for pedestrians, J. Phys. Soc. Jap., 68 (1999), 2861. doi: 10.1143/JPSJ.68.2861.

[17]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559. doi: 10.4310/CMS.2005.v3.n4.a5.

[18]

S. Göttlich, S. Kühn, J. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Netw. Heterog. Media, 6 (2011), 443-464. doi: 10.3934/nhm.2011.6.443.

[19]

D. Hartmann, Adaptive pedestrian dynamics based on geodesics, New J. Phys., 12 (2010), 043032. doi: 10.1088/1367-2630/12/4/043032.

[20]

D. Helbing, A fluid-dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415.

[21]

L. F. Henderson, On the fluid mechanics of human crowd motion, Transport. Res., 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.

[22]

H.-J. Huang and R.-Y. Guo, Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits, Phys. Rev. E., 78 (2008), 021131. doi: 10.1103/PhysRevE.78.021131.

[23]

D. Helbing and A. Johansson, Pedestrian, crowd and evacuation dynamics, Encyclopedia of Complexity and Systems Science, 16 (2010), 6476-6495.

[24]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.

[25]

D. Helbing and P. Molnár, Social Force Model for Pedestrian Dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[26]

H. Holden and N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.

[27]

H. W. Hamacher and S. A. Tjandra, Mathematical modelling of evacuation problems: A state of the art, in Pedestrian and Evacuation Dynamics (eds. M. Schreckenberg and S. D. Sharma), Springer, 2002, 227-266.

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transport. Res. B - Meth., 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.

[29]

R. L. Hughes, The flow of human crowds, Annu. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.

[30]

L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transport. Res. B - Meth., 43 (2009), 127-141. doi: 10.1016/j.trb.2008.06.003.

[31]

P. Kachroo, S. J. Al-Nasur, S. A. Wadoo and A. Shende, Pedestrian Dynamics - Feedback Control of Crowd Evacuation, Springer, New York, 2008.

[32]

B. S. Kerner, The Physics of Traffic, Springer, New York, 2004. doi: 10.1007/978-3-540-40986-1.

[33]

G. Köster, D. Hartmann, and W. Klein, Microscopic pedestrian simulations: From passenger exchange times to regional evacuation, in Operations Research Proceedings (eds. B. Hum, K. Morasch, S. Pickl and M. Siegle), Springer, 2010, 571-576.

[34]

H. Klüpfel, A Cellular Automaton Model for Crowd Movement and Egress Simulation, Ph.D. thesis, Universität Duisburg-Essen in Duisburg, 2003.

[35]

A. Kneidl, M. Thiemann, A. Borrmann, S. Ruzika, H. W. Hamacher, G. Köster and E. Rank, Bidirectional Coupling of Macroscopic an Microscopic Approaches for Pedestrian Behavior Prediction, in Pedestrian and Evacuation Dynamics (eds. R. D. Peacock, E. D. Kuligowski and J. D. Averill), Springer, 2011, 459-470. doi: 10.1007/978-1-4419-9725-8_41.

[36]

A. Kneidl, M. Thiemann, D. Hartmann and A. Borrmann, Combining pedestrian simulation with a network flow optimization to support security staff in handling an evacuation of a soccer stadium, in Proceedings of European Conference Forum 2011 (eds. E. Tobin and M. Otreba), University College Cork, Cork, 2011.

[37]

R. Löhner, On the modeling of pedestrian motion, Appl. Math. Model., 34 (2010), 366-382. doi: 10.1016/j.apm.2009.04.017.

[38]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, 2005. doi: 10.1007/978-3-0348-8629-1.

[39]

T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution, Simulation, 81 (2005), 339-352. doi: 10.1177/0037549705052772.

[40]

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics, IEICE Trans. Inf. Syst., E87D (2004), 726-732.

[41]

M. Oppenhäuser, Realisierung und Potenzialanalyse von wissenschaftlichen Konzepten zur Regionalen Evakuierung aus Polizeilicher Sicht am Beispiel des Projektes REPKA, Master's Thesis, German Police University, Münster, 2011, http://opac.pfa-ms.de/onlinedokumente/masterarbeiten/2011/Oppenhaeuser_Markus.pdf.

[42]

N. Pelechano, J. M. Allbeck and N. Badler, Virtual Crowds: Methods, Simulation, and Control, Morgan & Claypool Publishers, San Rafael, Calif., 2008. doi: 10.2200/S00123ED1V01Y200808CGR008.

[43]

D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions, Physica A, 388 (2009), 3600-3608. doi: 10.1016/j.physa.2009.05.027.

[44]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Flow, Elsevier, New York, 1971.

[45]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 2007.

[46]

, REPKA, Regionale Evakuierung: Planung, Kontrolle und Anpassunghttp://www.repka-evakuierung.de/.

[47]

E. V. RiMEA, Richtlinie für Mikroskopische Entfluchtungsanalysenhttp://www.rimea.de/.

[48]

A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications, Encyclopedia of Complexity and System Science, (2009), 3142-3176.

[49]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling, Netw. Heterog. Media, 6 (2011), 545-560. doi: 10.3934/nhm.2011.6.545.

[50]

A. Varas, M. D. Cornejo, D. Mainemer, B. Toledo, J. Rogan, V. Munoz and J. A. Valdivia, Cellular automaton model for evacuation process with obstacles, Physica A, 382 (2007), 631-642. doi: 10.1016/j.physa.2007.04.006.

[51]

U. Weidmann, Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturauswertung) Schriftenreihe des IVT, University of Zurich in Zurich, 1993.

[52]

Y. Xia, S. C. Wong and C. W. Shu, Dynamic continuum pedestrian flow model with memory effect, Phys. Rev. E, 79 (2009), 066113. doi: 10.1103/PhysRevE.79.066113.

[53]

W. J. Yu, R. Chen, L. Y. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics, Phys. Rev. E, 72 (2005), 026112. doi: 10.1103/PhysRevE.72.026112.

[54]

K. Yamamoto, S. Kokubo and K. Nishinari, Simulation for pedestrian dynamics by real-coded cellular automata, Physica A, 379 (2007), 654. doi: 10.1016/j.physa.2007.02.040.

show all references

References:
[1]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[2]

N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Netw. Heterog. Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.

[3]

N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow, CR Mecanique, 333 (2005), 843-851. doi: 10.1016/j.crme.2005.09.004.

[4]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: a survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.

[5]

N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling, Math. Mod. Meth. Appl. S., 12 (2002), 1801-1843. doi: 10.1142/S0218202502002343.

[6]

V. Blue, M. Embrechts and J. Adler, Cellular automata modeling of pedestrian movements, IEEE Int. Conf. on Syst., Man and Cybern., (1997), 2320. doi: 10.1109/ICSMC.1997.635272.

[7]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A, 295 (2001), 507-525, arXiv:cond-mat/0102397. doi: 10.1016/S0378-4371(01)00141-8.

[8]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics, Math. Mod. Meth. Appl. S., 18 (2008), 1217-1247. doi: 10.1142/S0218202508003017.

[9]

C. Canuto, F. Fagnani and P. Tilli, A eulerian approach to the analysis of rendez-vous algorithms, Proceedings of the 17th IFAC World Congress, (2008), 9039-9044.

[10]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Appl. Math., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.

[11]

M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics, Netw. Heterog. Media, 6 (2011), 425-442. doi: 10.3934/nhm.2011.6.425.

[12]

E. Christiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with applications to crowd dynamics, Multiscale Model. Sim., 9 (2011), 155-182. doi: 10.1137/100797515.

[13]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks, Mathematical Methods in the Applied Sciences, 28 (2005), 1553-1567. doi: 10.1002/mma.624.

[14]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics, Phys. Rev. E, 82 (2010), 046111. doi: 10.1103/PhysRevE.82.046111.

[15]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon, Oxford, 1997.

[16]

M. Fukui and Y. Ishibashi, Self-organized phase transitions in cellular automaton models for pedestrians, J. Phys. Soc. Jap., 68 (1999), 2861. doi: 10.1143/JPSJ.68.2861.

[17]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559. doi: 10.4310/CMS.2005.v3.n4.a5.

[18]

S. Göttlich, S. Kühn, J. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Netw. Heterog. Media, 6 (2011), 443-464. doi: 10.3934/nhm.2011.6.443.

[19]

D. Hartmann, Adaptive pedestrian dynamics based on geodesics, New J. Phys., 12 (2010), 043032. doi: 10.1088/1367-2630/12/4/043032.

[20]

D. Helbing, A fluid-dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415.

[21]

L. F. Henderson, On the fluid mechanics of human crowd motion, Transport. Res., 8 (1974), 509-515. doi: 10.1016/0041-1647(74)90027-6.

[22]

H.-J. Huang and R.-Y. Guo, Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits, Phys. Rev. E., 78 (2008), 021131. doi: 10.1103/PhysRevE.78.021131.

[23]

D. Helbing and A. Johansson, Pedestrian, crowd and evacuation dynamics, Encyclopedia of Complexity and Systems Science, 16 (2010), 6476-6495.

[24]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.

[25]

D. Helbing and P. Molnár, Social Force Model for Pedestrian Dynamics, Phys. Rev. E, 51 (1995), 4282-4286. doi: 10.1103/PhysRevE.51.4282.

[26]

H. Holden and N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.

[27]

H. W. Hamacher and S. A. Tjandra, Mathematical modelling of evacuation problems: A state of the art, in Pedestrian and Evacuation Dynamics (eds. M. Schreckenberg and S. D. Sharma), Springer, 2002, 227-266.

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transport. Res. B - Meth., 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.

[29]

R. L. Hughes, The flow of human crowds, Annu. Rev. Fluid Mech., 35 (2003), 169-182. doi: 10.1146/annurev.fluid.35.101101.161136.

[30]

L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transport. Res. B - Meth., 43 (2009), 127-141. doi: 10.1016/j.trb.2008.06.003.

[31]

P. Kachroo, S. J. Al-Nasur, S. A. Wadoo and A. Shende, Pedestrian Dynamics - Feedback Control of Crowd Evacuation, Springer, New York, 2008.

[32]

B. S. Kerner, The Physics of Traffic, Springer, New York, 2004. doi: 10.1007/978-3-540-40986-1.

[33]

G. Köster, D. Hartmann, and W. Klein, Microscopic pedestrian simulations: From passenger exchange times to regional evacuation, in Operations Research Proceedings (eds. B. Hum, K. Morasch, S. Pickl and M. Siegle), Springer, 2010, 571-576.

[34]

H. Klüpfel, A Cellular Automaton Model for Crowd Movement and Egress Simulation, Ph.D. thesis, Universität Duisburg-Essen in Duisburg, 2003.

[35]

A. Kneidl, M. Thiemann, A. Borrmann, S. Ruzika, H. W. Hamacher, G. Köster and E. Rank, Bidirectional Coupling of Macroscopic an Microscopic Approaches for Pedestrian Behavior Prediction, in Pedestrian and Evacuation Dynamics (eds. R. D. Peacock, E. D. Kuligowski and J. D. Averill), Springer, 2011, 459-470. doi: 10.1007/978-1-4419-9725-8_41.

[36]

A. Kneidl, M. Thiemann, D. Hartmann and A. Borrmann, Combining pedestrian simulation with a network flow optimization to support security staff in handling an evacuation of a soccer stadium, in Proceedings of European Conference Forum 2011 (eds. E. Tobin and M. Otreba), University College Cork, Cork, 2011.

[37]

R. Löhner, On the modeling of pedestrian motion, Appl. Math. Model., 34 (2010), 366-382. doi: 10.1016/j.apm.2009.04.017.

[38]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, 2005. doi: 10.1007/978-3-0348-8629-1.

[39]

T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution, Simulation, 81 (2005), 339-352. doi: 10.1177/0037549705052772.

[40]

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics, IEICE Trans. Inf. Syst., E87D (2004), 726-732.

[41]

M. Oppenhäuser, Realisierung und Potenzialanalyse von wissenschaftlichen Konzepten zur Regionalen Evakuierung aus Polizeilicher Sicht am Beispiel des Projektes REPKA, Master's Thesis, German Police University, Münster, 2011, http://opac.pfa-ms.de/onlinedokumente/masterarbeiten/2011/Oppenhaeuser_Markus.pdf.

[42]

N. Pelechano, J. M. Allbeck and N. Badler, Virtual Crowds: Methods, Simulation, and Control, Morgan & Claypool Publishers, San Rafael, Calif., 2008. doi: 10.2200/S00123ED1V01Y200808CGR008.

[43]

D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions, Physica A, 388 (2009), 3600-3608. doi: 10.1016/j.physa.2009.05.027.

[44]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Flow, Elsevier, New York, 1971.

[45]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 2007.

[46]

, REPKA, Regionale Evakuierung: Planung, Kontrolle und Anpassunghttp://www.repka-evakuierung.de/.

[47]

E. V. RiMEA, Richtlinie für Mikroskopische Entfluchtungsanalysenhttp://www.rimea.de/.

[48]

A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications, Encyclopedia of Complexity and System Science, (2009), 3142-3176.

[49]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling, Netw. Heterog. Media, 6 (2011), 545-560. doi: 10.3934/nhm.2011.6.545.

[50]

A. Varas, M. D. Cornejo, D. Mainemer, B. Toledo, J. Rogan, V. Munoz and J. A. Valdivia, Cellular automaton model for evacuation process with obstacles, Physica A, 382 (2007), 631-642. doi: 10.1016/j.physa.2007.04.006.

[51]

U. Weidmann, Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturauswertung) Schriftenreihe des IVT, University of Zurich in Zurich, 1993.

[52]

Y. Xia, S. C. Wong and C. W. Shu, Dynamic continuum pedestrian flow model with memory effect, Phys. Rev. E, 79 (2009), 066113. doi: 10.1103/PhysRevE.79.066113.

[53]

W. J. Yu, R. Chen, L. Y. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics, Phys. Rev. E, 72 (2005), 026112. doi: 10.1103/PhysRevE.72.026112.

[54]

K. Yamamoto, S. Kokubo and K. Nishinari, Simulation for pedestrian dynamics by real-coded cellular automata, Physica A, 379 (2007), 654. doi: 10.1016/j.physa.2007.02.040.

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