# American Institute of Mathematical Sciences

September  2011, 6(3): 485-519. doi: 10.3934/nhm.2011.6.485

## Handling congestion in crowd motion modeling

 1 Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex, France 2 Laboratoire de Mathématiques d'Orsay, UMR CNRS 8628, Faculté des Sciences, Université Paris-Sud XI, 91405 Orsay Cedex 3 LAMAV, Université de Valenciennes et du Hainaut-Cambrésis Mont Houy, 59313 Valenciennes Cedex 9, France

Received  December 2010 Revised  July 2011 Published  August 2011

We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontaneous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description.
Citation: Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, Juliette Venel. Handling congestion in crowd motion modeling. Networks & Heterogeneous Media, 2011, 6 (3) : 485-519. doi: 10.3934/nhm.2011.6.485
##### References:
 [1] A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications,, Trudy Mat. Inst. Steklov., 38 (1951), 5. [2] L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur., 19 (1995), 191. [3] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Lectures in Mathematics, (2005). [4] N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scalling to hyperbolic macroscopic models,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1317. doi: 10.1142/S0218202508003054. [5] F. Bernicot and J. Venel, Differential inclusions with proximal normal cones in Banach spaces,, J. Convex Anal., 17 (2010), 451. [6] F. Bernicot and J. Venel, Convergence order of a numerical scheme for sweeping process, submitted., Available from: \url{http://arxiv.org/abs/1009.2837}., (). [7] V. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways,, Transportation Research B, 35 (2001), 293. doi: 10.1016/S0191-2615(99)00052-1. [8] A. Borgers and H. Timmermans, A model of pedestrian route choice and demand for retail facilities within inner-cityshopping areas,, Geographycal Analysis, 18 (1986), 115. doi: 10.1111/j.1538-4632.1986.tb00086.x. [9] A. Borgers and H. Timmermans, City centre entry points, store location patterns and pedestrian route choice behavior: A microlevel simulation model,, Socio-Economic Planning Sciences, 20 (1986), 25. doi: 10.1016/0038-0121(86)90023-6. [10] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space,, J. Nonlinear Convex Anal., 6 (2005), 359. [11] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. [12] A. Canino, On $p$-convex sets and geodesics,, J. Differential Equations, 75 (1988), 118. [13] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows,, SIAM J. Sci. Comput., 29 (2007), 539. doi: 10.1137/050641211. [14] C. Chalons, "Transport-Equilibrium Schemes for Pedestrian Flows with Nonclassical Shocks,", Traffic and Granular Flows'05, (2007), 347. doi: 10.1007/978-3-540-47641-2_31. [15] F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property,, J. Convex Anal., 2 (1995), 117. [16] G. Colombo and V. V. Goncharov, The sweeping processes without convexity,, Set-Valued Anal., 7 (1999), 357. doi: 10.1023/A:1008774529556. [17] G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set,, J. Differential Equations, 187 (2003), 46. [18] R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. [19] V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. [20] J. Dambrine, B. Maury, N. Meunier and A. Roudneff-Chupin, A congestion model for cell migration,, to appear in Communications in Pure and Applied Analysis., (). [21] E. De Giorgi, New problems on minimizing movements,, in, 29 (1993), 81. [22] P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness,, J. Stat. Phys., 138 (2010), 85. doi: 10.1007/s10955-009-9879-x. [23] M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model of pedestrian flow: The one-dimensional case,, J. Diff. Eq., 250 (2011), 1334. [24] C. Dogbé, On the numerical solutions of second order macroscopic models of pedestrian flows,, Comput. Appl. Math., 56 (2008), 1884. doi: 10.1016/j.camwa.2008.04.028. [25] A. Donev, S. Torquato, F. H. Stillinger and Robert Connelly, Jamming in hard sphere and disk packings,, J. Appl. Phys., 95 (2004). doi: 10.1063/1.1633647. [26] J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984). [27] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program, 104 (2005), 347. doi: 10.1007/s10107-005-0619-y. [28] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation,, J. Differential Equations, 226 (2006), 135. [29] H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418. doi: 10.1090/S0002-9947-1959-0110078-1. [30] P. G. Gipps and B. Marksjö, A micro-simulation model for pedestrian flows,, Mathematics and Computers in Simulation, 27 (1985), 95. doi: 10.1016/0378-4754(85)90027-8. [31] B. Gustafsson and M. Sakai, Properties of some balayage operators, with applications to quadrature domains and moving boundary problems,, Nonlinear Analysis, 22 (1994), 1221. doi: 10.1016/0362-546X(94)90107-4. [32] S. Gwynne, E. R. Galea, P. J. Lawrence and L. Filippidis, Modelling occupant interaction with fire conditions using the buildingEXODUS evacuation model,, Fire Safety Journal, 36 (2001), 327. doi: 10.1016/S0379-7112(00)00060-6. [33] D. Helbing, A fluid dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. [34] D. Helbing, P. Molnar and F. Schweitzer, "Computer Simulations of Pedestrian Dynamics and Trail Formation,", Evolution of Natural Structures, 230 (1994), 229. [35] D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. [36] R. L. Hughes, A continuum theory for the flow of pedestrian,, Transport. Res. Part B, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. [37] R. L. Hughes, "The Flow of Human Crowds,", Ann. Rev. Fluid Mech., 35 (2003), 169. [38] A. D. Ioffe and J. V. Outrata, On metric and calmness qualification conditions in subdifferential calculus,, Set-Valued Anal., 16 (2008), 199. doi: 10.1007/s11228-008-0076-x. [39] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. [40] L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources,, J. Anal. Math., 11 (2010), 151. doi: 10.1007/s11854-010-0015-2. [41] G. G. Løvås, Modelling and simulation of pedestrian traffic flow,, Transportation Research B, 28 (1994), 429. doi: 10.1016/0191-2615(94)90013-2. [42] B. Maury, A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint,, Numerische Mathematik, 102 (2006), 649. doi: 10.1007/s00211-005-0666-6. [43] B. Maury and J. Venel, A discrete contact model for crowd motion,, ESAIM Mathematical Modelling and Numerical Analysis, 45 (2011), 145. doi: 10.1051/m2an/2010035. [44] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1787. doi: 10.1142/S0218202510004799. [45] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 346. [46] J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,, C. R. Acad. Sci. Paris, 255 (1962), 238. [47] K. Nagel, From particle hopping models to traffic flow theory,, Transportation Research Record, 1644 (1998), 1. doi: 10.3141/1644-01. [48] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707. doi: 10.1007/s00205-010-0366-y. [49] B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. [50] R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis,, Trans. Amer. Math. Soc., 348 (1996), 1805. doi: 10.1090/S0002-9947-96-01544-9. [51] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions,, Trans. Amer. Math. Soc., 352 (2000), 5231. doi: 10.1090/S0002-9947-00-02550-2. [52] R. T. Rockafellar and R. Wets, "Variational Analysis,", Grundlehren der Mathematischen, 317 (1998). [53] A. Roudneff-Chupin, "Modélisation Macroscopique des Mouvements de Foules,", Ph.D thesis, (). [54] Y. Saisho and H. Tanaka, Stochastic differential equations for mutually reflecting Brownian balls,, Osaka J. Math., 23 (1986), 725. [55] A. Schadschneider, Cellular automaton approach to pedestrian dynamics-theory,, in, (2001), 75. [56] A. Schadschneider, A. Kirchner and K. Nishinari, From ant trails to pedestrian dynamics,, Applied Bionics and Biomechanics, 1 (2003), 11. doi: 10.1533/abib.2003.1.1.11. [57] L. Thibault, Sweeping process with regular and nonregular sets,, J. Differential Equations, 193 (2003), 1. [58] S. Torquato and F. H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond,", Reviews of Modern Physics, 82 (2010). [59] S. Torquato, T. M. Truskett and P. G. Debenedetti, Is random close packing of spheres well defined?,, Phys. Rev. Lett., 84 (2000), 2064. doi: 10.1103/PhysRevLett.84.2064. [60] J. Venel, A numerical scheme for a class of sweeping processes,, Numerische Mathematik, 118 (2011), 367. doi: 10.1007/s00211-010-0329-0. [61] J. Venel, "Integrating Strategies in Numerical Modelling of Crowd Motion,", Pedestrian and Evacuation Dynamics '08, (2010), 641. doi: 10.1007/978-3-642-04504-2_59. [62] J. Venel, "Modélisation Mathématique et Numérique des Mouvements de Foule,", Ph.D thesis, (2008). [63] C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003). [64] S. J. Yuhaski and J. M. Smith, Modeling circulation systems in buildings using state dependent queueing models,, Queueing Systems Theory Appl., 4 (1989), 319. doi: 10.1007/BF01159471.

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##### References:
 [1] A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications,, Trudy Mat. Inst. Steklov., 38 (1951), 5. [2] L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur., 19 (1995), 191. [3] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,", Lectures in Mathematics, (2005). [4] N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scalling to hyperbolic macroscopic models,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1317. doi: 10.1142/S0218202508003054. [5] F. Bernicot and J. Venel, Differential inclusions with proximal normal cones in Banach spaces,, J. Convex Anal., 17 (2010), 451. [6] F. Bernicot and J. Venel, Convergence order of a numerical scheme for sweeping process, submitted., Available from: \url{http://arxiv.org/abs/1009.2837}., (). [7] V. Blue and J. L. Adler, Cellular automata microsimulation for modeling bi-directional pedestrian walkways,, Transportation Research B, 35 (2001), 293. doi: 10.1016/S0191-2615(99)00052-1. [8] A. Borgers and H. Timmermans, A model of pedestrian route choice and demand for retail facilities within inner-cityshopping areas,, Geographycal Analysis, 18 (1986), 115. doi: 10.1111/j.1538-4632.1986.tb00086.x. [9] A. Borgers and H. Timmermans, City centre entry points, store location patterns and pedestrian route choice behavior: A microlevel simulation model,, Socio-Economic Planning Sciences, 20 (1986), 25. doi: 10.1016/0038-0121(86)90023-6. [10] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space,, J. Nonlinear Convex Anal., 6 (2005), 359. [11] C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. [12] A. Canino, On $p$-convex sets and geodesics,, J. Differential Equations, 75 (1988), 118. [13] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows,, SIAM J. Sci. Comput., 29 (2007), 539. doi: 10.1137/050641211. [14] C. Chalons, "Transport-Equilibrium Schemes for Pedestrian Flows with Nonclassical Shocks,", Traffic and Granular Flows'05, (2007), 347. doi: 10.1007/978-3-540-47641-2_31. [15] F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property,, J. Convex Anal., 2 (1995), 117. [16] G. Colombo and V. V. Goncharov, The sweeping processes without convexity,, Set-Valued Anal., 7 (1999), 357. doi: 10.1023/A:1008774529556. [17] G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set,, J. Differential Equations, 187 (2003), 46. [18] R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. [19] V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. [20] J. Dambrine, B. Maury, N. Meunier and A. Roudneff-Chupin, A congestion model for cell migration,, to appear in Communications in Pure and Applied Analysis., (). [21] E. De Giorgi, New problems on minimizing movements,, in, 29 (1993), 81. [22] P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness,, J. Stat. Phys., 138 (2010), 85. doi: 10.1007/s10955-009-9879-x. [23] M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model of pedestrian flow: The one-dimensional case,, J. Diff. Eq., 250 (2011), 1334. [24] C. Dogbé, On the numerical solutions of second order macroscopic models of pedestrian flows,, Comput. Appl. Math., 56 (2008), 1884. doi: 10.1016/j.camwa.2008.04.028. [25] A. Donev, S. Torquato, F. H. Stillinger and Robert Connelly, Jamming in hard sphere and disk packings,, J. Appl. Phys., 95 (2004). doi: 10.1063/1.1633647. [26] J. L. Doob, "Classical Potential Theory and Its Probabilistic Counterpart,", Grundlehren der Mathematischen Wissenschaften, 262 (1984). [27] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program, 104 (2005), 347. doi: 10.1007/s10107-005-0619-y. [28] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation,, J. Differential Equations, 226 (2006), 135. [29] H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418. doi: 10.1090/S0002-9947-1959-0110078-1. [30] P. G. Gipps and B. Marksjö, A micro-simulation model for pedestrian flows,, Mathematics and Computers in Simulation, 27 (1985), 95. doi: 10.1016/0378-4754(85)90027-8. [31] B. Gustafsson and M. Sakai, Properties of some balayage operators, with applications to quadrature domains and moving boundary problems,, Nonlinear Analysis, 22 (1994), 1221. doi: 10.1016/0362-546X(94)90107-4. [32] S. Gwynne, E. R. Galea, P. J. Lawrence and L. Filippidis, Modelling occupant interaction with fire conditions using the buildingEXODUS evacuation model,, Fire Safety Journal, 36 (2001), 327. doi: 10.1016/S0379-7112(00)00060-6. [33] D. Helbing, A fluid dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. [34] D. Helbing, P. Molnar and F. Schweitzer, "Computer Simulations of Pedestrian Dynamics and Trail Formation,", Evolution of Natural Structures, 230 (1994), 229. [35] D. Helbing and P. Molnár, Social force model for pedestrian dynamics,, Phys. Rev E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. [36] R. L. Hughes, A continuum theory for the flow of pedestrian,, Transport. Res. Part B, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. [37] R. L. Hughes, "The Flow of Human Crowds,", Ann. Rev. Fluid Mech., 35 (2003), 169. [38] A. D. Ioffe and J. V. Outrata, On metric and calmness qualification conditions in subdifferential calculus,, Set-Valued Anal., 16 (2008), 199. doi: 10.1007/s11228-008-0076-x. [39] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1. doi: 10.1137/S0036141096303359. [40] L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources,, J. Anal. Math., 11 (2010), 151. doi: 10.1007/s11854-010-0015-2. [41] G. G. Løvås, Modelling and simulation of pedestrian traffic flow,, Transportation Research B, 28 (1994), 429. doi: 10.1016/0191-2615(94)90013-2. [42] B. Maury, A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint,, Numerische Mathematik, 102 (2006), 649. doi: 10.1007/s00211-005-0666-6. [43] B. Maury and J. Venel, A discrete contact model for crowd motion,, ESAIM Mathematical Modelling and Numerical Analysis, 45 (2011), 145. doi: 10.1051/m2an/2010035. [44] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1787. doi: 10.1142/S0218202510004799. [45] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 346. [46] J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,, C. R. Acad. Sci. Paris, 255 (1962), 238. [47] K. Nagel, From particle hopping models to traffic flow theory,, Transportation Research Record, 1644 (1998), 1. doi: 10.3141/1644-01. [48] B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707. doi: 10.1007/s00205-010-0366-y. [49] B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. [50] R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis,, Trans. Amer. Math. Soc., 348 (1996), 1805. doi: 10.1090/S0002-9947-96-01544-9. [51] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions,, Trans. Amer. Math. Soc., 352 (2000), 5231. doi: 10.1090/S0002-9947-00-02550-2. [52] R. T. Rockafellar and R. Wets, "Variational Analysis,", Grundlehren der Mathematischen, 317 (1998). [53] A. Roudneff-Chupin, "Modélisation Macroscopique des Mouvements de Foules,", Ph.D thesis, (). [54] Y. Saisho and H. Tanaka, Stochastic differential equations for mutually reflecting Brownian balls,, Osaka J. Math., 23 (1986), 725. [55] A. Schadschneider, Cellular automaton approach to pedestrian dynamics-theory,, in, (2001), 75. [56] A. Schadschneider, A. Kirchner and K. Nishinari, From ant trails to pedestrian dynamics,, Applied Bionics and Biomechanics, 1 (2003), 11. doi: 10.1533/abib.2003.1.1.11. [57] L. Thibault, Sweeping process with regular and nonregular sets,, J. Differential Equations, 193 (2003), 1. [58] S. Torquato and F. H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond,", Reviews of Modern Physics, 82 (2010). [59] S. Torquato, T. M. Truskett and P. G. Debenedetti, Is random close packing of spheres well defined?,, Phys. Rev. Lett., 84 (2000), 2064. doi: 10.1103/PhysRevLett.84.2064. [60] J. Venel, A numerical scheme for a class of sweeping processes,, Numerische Mathematik, 118 (2011), 367. doi: 10.1007/s00211-010-0329-0. [61] J. Venel, "Integrating Strategies in Numerical Modelling of Crowd Motion,", Pedestrian and Evacuation Dynamics '08, (2010), 641. doi: 10.1007/978-3-642-04504-2_59. [62] J. Venel, "Modélisation Mathématique et Numérique des Mouvements de Foule,", Ph.D thesis, (2008). [63] C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003). [64] S. J. Yuhaski and J. M. Smith, Modeling circulation systems in buildings using state dependent queueing models,, Queueing Systems Theory Appl., 4 (1989), 319. doi: 10.1007/BF01159471.
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