2014, 9(2): 239-268. doi: 10.3934/nhm.2014.9.239

Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model

1. 

Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, 205 N. Mathews Ave, Urbana, IL 61801, United States

2. 

Department of Mathematics, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany

3. 

Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122

Received  October 2013 Revised  June 2014 Published  July 2014

The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting data-fitted GARZ model is compared to other traffic models by means of a three-detector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time.
Citation: Shimao Fan, Michael Herty, Benjamin Seibold. Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model. Networks & Heterogeneous Media, 2014, 9 (2) : 239-268. doi: 10.3934/nhm.2014.9.239
References:
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[2]

S. Amin, et al., Mobile century - Using GPS mobile phones as traffic sensors: A field experiment,, in 15th World Congress on Intelligent Transportation Systems, (2008).

[3]

A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099.

[4]

M. Bando, Hesebem K., A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation,, Phys. Rev. E, 51 (1995), 1035.

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A. M. Bayen and C. G. Claudel, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory,, IEEE Trans. Automat. Contr., 55 (2010), 1142. doi: 10.1109/TAC.2010.2041976.

[6]

A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations,, SIAM J. Control Optim., 49 (2011), 383. doi: 10.1137/090778754.

[7]

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

[8]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Ration. Mech. Anal., 187 (2008), 185. doi: 10.1007/s00205-007-0061-9.

[9]

S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach,, Appl. Math. Comput., 210 (2009), 441. doi: 10.1016/j.amc.2009.01.057.

[10]

S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D, (2012), 1421.

[11]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM J. Appl. Math., 71 (2011), 107. doi: 10.1137/090754467.

[12]

R. Borsche, M. Kimathi and A. Klar, A class of multiphase traffic theories for microscopic, kinetic and continuum traffic models,, Comp. Math. Appl., 64 (2012), 2939. doi: 10.1016/j.camwa.2012.08.013.

[13]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling,, Commun. Math. Sci., 5 (2007), 533. doi: 10.4310/CMS.2007.v5.n3.a2.

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R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708. doi: 10.1137/S0036139901393184.

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R. M. Colombo and P. Goatin, Traffic flow models with phase transitions,, Flow Turbulence Combust., 76 (2006), 383.

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R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound,, SIAM J. Appl. Math., 70 (2010), 2652. doi: 10.1137/090752468.

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C. F. Daganzo, In traffic flow, cellular automata = kinematic waves,, Transp. Res. B, 40 (2006), 396. doi: 10.1016/j.trb.2005.05.004.

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S. Fan, Data-fitted Generic Second Order Macroscopic Traffic Flow Models,, Dissertation, (2013).

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S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow,, in preparation, (2014).

[26]

S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data,, in 93rd Annual Meeting of Transportation Research Board, (2013), 13.

[27]

S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models,, , (2013).

[28]

Federal Highway Administration US Department of Transportation, Interstate 80 freeway dataset,, Website, ().

[29]

Federal Highway Administration US Department of Transportation, Next Generation Simulation (NGSIM),, Website, ().

[30]

M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.056113.

[31]

M. Fukui and Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed,, J. Phys. Soc. Japan, 65 (1996), 1868. doi: 10.1143/JPSJ.65.1868.

[32]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006).

[33]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions,, Math. Comput. Modeling, 44 (2006), 287. doi: 10.1016/j.mcm.2006.01.016.

[34]

S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations,, Math. Sbornik, 47 (1959), 271.

[35]

J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657.

[36]

J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175. doi: 10.1137/S0036139903431737.

[37]

B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448.

[38]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,, SIAM Rev., 25 (1983), 35. doi: 10.1137/1025002.

[39]

D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164. doi: 10.1103/PhysRevE.51.3164.

[40]

D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.

[41]

R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic,, Elsevier, (1971).

[42]

M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models,, Kinet. Relat. Models, 3 (2010), 311. doi: 10.3934/krm.2010.3.311.

[43]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models,, Kinet. Relat. Models, 3 (2010), 165. doi: 10.3934/krm.2010.3.165.

[44]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1. doi: 10.4310/CMS.2003.v1.n1.a1.

[45]

R. J. Karunamuni and T. Alberts, A generalized reflection method of boundary correction in kernel density estimation,, Canad. J. Statist., 33 (2005), 497. doi: 10.1002/cjs.5550330403.

[46]

A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow,, in preparation, (2014).

[47]

B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow,, Phys. Rev. E, 48 (1993). doi: 10.1103/PhysRevE.48.R2335.

[48]

B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Phys. Rev. E, 50 (1994), 54. doi: 10.1103/PhysRevE.50.54.

[49]

A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749. doi: 10.1137/S0036139999356181.

[50]

J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24.

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J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling,, in Transportation and Traffic Theory (eds. R. E. Allsop, (2007), 755.

[52]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. A, 229 (1955), 317. doi: 10.1098/rspa.1955.0089.

[53]

T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707.

[54]

Minnesota Department of Transportation, Mn/DOT Traffic Data,, Website, ().

[55]

Mobile Millennium project, Website,, , ().

[56]

K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221.

[57]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, in Proceedings of the 14th International Symposium on Transportation and Trafic Theory (ed. A. Ceder), (1999), 51.

[58]

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

[59]

G. F. Newell, A simplified theory of kinematic waves in highway traffic II: Queueing at freeway bottlenecks,, Transp. Res. B, 27 (1993), 289. doi: 10.1016/0191-2615(93)90039-D.

[60]

E. Parzen, On estimation of a probability density function and mode,, Ann. Math. Statist., 33 (1962), 1065. doi: 10.1214/aoms/1177704472.

[61]

H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51.

[62]

H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic,, Transp. Res. Rec., 722 (1979), 68.

[63]

W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131. doi: 10.1080/03081067908717157.

[64]

L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274. doi: 10.1063/1.1721265.

[65]

M. Rascle, An improved macroscopic model of traffic flow: Derivation and links with the Lightill-Whitham model,, Math. Comput. Modelling, 35 (2002), 581. doi: 10.1016/S0895-7177(02)80022-X.

[66]

P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42.

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M. Rosenblatt, Remarks on some nonparametric estimates of a density function,, Ann. Math. Statist., 27 (1956), 832. doi: 10.1214/aoms/1177728190.

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S. Sakai, K. Nishinari and S. IIda, A new stochastic cellular automaton model on traffic flow and its jamming phase transition,, J. Phys. A: Math. Gen., 39 (2006), 15327. doi: 10.1088/0305-4470/39/50/002.

[69]

B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models,, Netw. Heterog. Media, 8 (2013), 745. doi: 10.3934/nhm.2013.8.745.

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F. Siebel and W. Mauser, On the fundamental diagram of traffic flow,, SIAM J. Appl. Math., 66 (2006), 1150. doi: 10.1137/050627113.

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D. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Appl. Math. Res. Express., 1 (2010), 1.

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show all references

References:
[1]

T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics,, J. Stat. Phys, 133 (2008), 1083. doi: 10.1007/s10955-008-9652-6.

[2]

S. Amin, et al., Mobile century - Using GPS mobile phones as traffic sensors: A field experiment,, in 15th World Congress on Intelligent Transportation Systems, (2008).

[3]

A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099.

[4]

M. Bando, Hesebem K., A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation,, Phys. Rev. E, 51 (1995), 1035.

[5]

A. M. Bayen and C. G. Claudel, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory,, IEEE Trans. Automat. Contr., 55 (2010), 1142. doi: 10.1109/TAC.2010.2041976.

[6]

A. M. Bayen and C. G. Claudel, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations,, SIAM J. Control Optim., 49 (2011), 383. doi: 10.1137/090778754.

[7]

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

[8]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Ration. Mech. Anal., 187 (2008), 185. doi: 10.1007/s00205-007-0061-9.

[9]

S. Blandin, G. Bretti, A. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach,, Appl. Math. Comput., 210 (2009), 441. doi: 10.1016/j.amc.2009.01.057.

[10]

S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D, (2012), 1421.

[11]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM J. Appl. Math., 71 (2011), 107. doi: 10.1137/090754467.

[12]

R. Borsche, M. Kimathi and A. Klar, A class of multiphase traffic theories for microscopic, kinetic and continuum traffic models,, Comp. Math. Appl., 64 (2012), 2939. doi: 10.1016/j.camwa.2012.08.013.

[13]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling,, Commun. Math. Sci., 5 (2007), 533. doi: 10.4310/CMS.2007.v5.n3.a2.

[14]

G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy,, Comm. Pure Appl. Math., 47 (1994), 787. doi: 10.1002/cpa.3160470602.

[15]

R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708. doi: 10.1137/S0036139901393184.

[16]

R. M. Colombo and P. Goatin, Traffic flow models with phase transitions,, Flow Turbulence Combust., 76 (2006), 383.

[17]

R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound,, SIAM J. Appl. Math., 70 (2010), 2652. doi: 10.1137/090752468.

[18]

R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik,, Mathematische Annalen, 100 (1928), 32. doi: 10.1007/BF01448839.

[19]

C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transp. Res. B, 28 (1994), 269. doi: 10.1016/0191-2615(94)90002-7.

[20]

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277. doi: 10.1016/0191-2615(95)00007-Z.

[21]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations,, Emerald Group Pub Ltd, (1997).

[22]

C. F. Daganzo, In traffic flow, cellular automata = kinematic waves,, Transp. Res. B, 40 (2006), 396. doi: 10.1016/j.trb.2005.05.004.

[23]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).

[24]

S. Fan, Data-fitted Generic Second Order Macroscopic Traffic Flow Models,, Dissertation, (2013).

[25]

S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow,, in preparation, (2014).

[26]

S. Fan and B. Seibold, A comparison of data-fitted first order traffic models and their second order generalizations via trajectory and sensor data,, in 93rd Annual Meeting of Transportation Research Board, (2013), 13.

[27]

S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models,, , (2013).

[28]

Federal Highway Administration US Department of Transportation, Interstate 80 freeway dataset,, Website, ().

[29]

Federal Highway Administration US Department of Transportation, Next Generation Simulation (NGSIM),, Website, ().

[30]

M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.056113.

[31]

M. Fukui and Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars moving with high speed,, J. Phys. Soc. Japan, 65 (1996), 1868. doi: 10.1143/JPSJ.65.1868.

[32]

M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006).

[33]

P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions,, Math. Comput. Modeling, 44 (2006), 287. doi: 10.1016/j.mcm.2006.01.016.

[34]

S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations,, Math. Sbornik, 47 (1959), 271.

[35]

J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657.

[36]

J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175. doi: 10.1137/S0036139903431737.

[37]

B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448.

[38]

A. Harten, P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,, SIAM Rev., 25 (1983), 35. doi: 10.1137/1025002.

[39]

D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164. doi: 10.1103/PhysRevE.51.3164.

[40]

D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.

[41]

R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic,, Elsevier, (1971).

[42]

M. Herty and R. Illner, Analytical and numerical investigations of refined macroscopic traffic flow models,, Kinet. Relat. Models, 3 (2010), 311. doi: 10.3934/krm.2010.3.311.

[43]

M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models,, Kinet. Relat. Models, 3 (2010), 165. doi: 10.3934/krm.2010.3.165.

[44]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1. doi: 10.4310/CMS.2003.v1.n1.a1.

[45]

R. J. Karunamuni and T. Alberts, A generalized reflection method of boundary correction in kernel density estimation,, Canad. J. Statist., 33 (2005), 497. doi: 10.1002/cjs.5550330403.

[46]

A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow,, in preparation, (2014).

[47]

B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow,, Phys. Rev. E, 48 (1993). doi: 10.1103/PhysRevE.48.R2335.

[48]

B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Phys. Rev. E, 50 (1994), 54. doi: 10.1103/PhysRevE.50.54.

[49]

A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749. doi: 10.1137/S0036139999356181.

[50]

J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24.

[51]

J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling,, in Transportation and Traffic Theory (eds. R. E. Allsop, (2007), 755.

[52]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. A, 229 (1955), 317. doi: 10.1098/rspa.1955.0089.

[53]

T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707.

[54]

Minnesota Department of Transportation, Mn/DOT Traffic Data,, Website, ().

[55]

Mobile Millennium project, Website,, , ().

[56]

K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221.

[57]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, in Proceedings of the 14th International Symposium on Transportation and Trafic Theory (ed. A. Ceder), (1999), 51.

[58]

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

[59]

G. F. Newell, A simplified theory of kinematic waves in highway traffic II: Queueing at freeway bottlenecks,, Transp. Res. B, 27 (1993), 289. doi: 10.1016/0191-2615(93)90039-D.

[60]

E. Parzen, On estimation of a probability density function and mode,, Ann. Math. Statist., 33 (1962), 1065. doi: 10.1214/aoms/1177704472.

[61]

H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51.

[62]

H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic,, Transp. Res. Rec., 722 (1979), 68.

[63]

W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131. doi: 10.1080/03081067908717157.

[64]

L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274. doi: 10.1063/1.1721265.

[65]

M. Rascle, An improved macroscopic model of traffic flow: Derivation and links with the Lightill-Whitham model,, Math. Comput. Modelling, 35 (2002), 581. doi: 10.1016/S0895-7177(02)80022-X.

[66]

P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42.

[67]

M. Rosenblatt, Remarks on some nonparametric estimates of a density function,, Ann. Math. Statist., 27 (1956), 832. doi: 10.1214/aoms/1177728190.

[68]

S. Sakai, K. Nishinari and S. IIda, A new stochastic cellular automaton model on traffic flow and its jamming phase transition,, J. Phys. A: Math. Gen., 39 (2006), 15327. doi: 10.1088/0305-4470/39/50/002.

[69]

B. Seibold, M. R. Flynn, A. R. Kasimov and R. R. Rosales, Constructing set-valued fundamental diagrams from jamiton solutions in second order traffic models,, Netw. Heterog. Media, 8 (2013), 745. doi: 10.3934/nhm.2013.8.745.

[70]

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