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

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

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

Shimao Fan ^1, , Michael Herty ^2, and Benjamin Seibold ^3,

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

Abstract
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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.

Keywords: second order, Traffic model, fundamental diagram, trajectory, data, Lighthill-Whitham-Richards, Aw-Rascle-Zhang, generalized, sensor, validation..

Mathematics Subject Classification: Primary: 35L65, 35Q91; Secondary: 91B7.

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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D, (2012), 1421. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[16]	R. M. Colombo and P. Goatin, Traffic flow models with phase transitions,, Flow Turbulence Combust., 76 (2006), 383. Google Scholar
[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. Google Scholar
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[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. Google Scholar
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[25]	S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow,, in preparation, (2014). Google Scholar
[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. Google Scholar
[27]	S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models,, , (2013). Google Scholar
[28]	Federal Highway Administration US Department of Transportation, Interstate 80 freeway dataset,, Website, (). Google Scholar
[29]	Federal Highway Administration US Department of Transportation, Next Generation Simulation (NGSIM),, Website, (). Google Scholar
[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. Google Scholar
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[32]	M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006). Google Scholar
[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. Google Scholar
[34]	S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations,, Math. Sbornik, 47 (1959), 271. Google Scholar
[35]	J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657. Google Scholar
[36]	J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175. doi: 10.1137/S0036139903431737. Google Scholar
[37]	B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448. Google Scholar
[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. Google Scholar
[39]	D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164. doi: 10.1103/PhysRevE.51.3164. Google Scholar
[40]	D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. Google Scholar
[41]	R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic,, Elsevier, (1971). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[50]	J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24. Google Scholar
[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. Google Scholar
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[53]	T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707. Google Scholar
[54]	Minnesota Department of Transportation, Mn/DOT Traffic Data,, Website, (). Google Scholar
[55]	Mobile Millennium project, Website,, , (). Google Scholar
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[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. Google Scholar
[58]	G. F. Newell, Nonlinear effects in the dynamics of car following,, Operations Research, 9 (1961), 209. doi: 10.1287/opre.9.2.209. Google Scholar
[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. Google Scholar
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[61]	H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51. Google Scholar
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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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	S. Blandin, A. Coque and A. Bayen, On sequential data assimilation for scalar macroscopic traffic flow models,, Physica D, (2012), 1421. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708. doi: 10.1137/S0036139901393184. Google Scholar
[16]	R. M. Colombo and P. Goatin, Traffic flow models with phase transitions,, Flow Turbulence Combust., 76 (2006), 383. Google Scholar
[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. Google Scholar
[18]	R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik,, Mathematische Annalen, 100 (1928), 32. doi: 10.1007/BF01448839. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	C. F. Daganzo, Fundamentals of Transportation and Traffic Operations,, Emerald Group Pub Ltd, (1997). Google Scholar
[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. Google Scholar
[23]	L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar
[24]	S. Fan, Data-fitted Generic Second Order Macroscopic Traffic Flow Models,, Dissertation, (2013). Google Scholar
[25]	S. Fan, B. Piccoli and B. Seibold, The Collapsed Generalized Aw-Rascle-Zhang Model of Traffic Flow,, in preparation, (2014). Google Scholar
[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. Google Scholar
[27]	S. Fan and B. Seibold, Effect of the choice of stagnation density in data-fitted first- and second-order traffic models,, , (2013). Google Scholar
[28]	Federal Highway Administration US Department of Transportation, Interstate 80 freeway dataset,, Website, (). Google Scholar
[29]	Federal Highway Administration US Department of Transportation, Next Generation Simulation (NGSIM),, Website, (). Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	M. Garavello and B. Piccoli, Traffic Flow on Networks,, American Institute of Mathematical Sciences, (2006). Google Scholar
[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. Google Scholar
[34]	S. K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations,, Math. Sbornik, 47 (1959), 271. Google Scholar
[35]	J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657. Google Scholar
[36]	J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175. doi: 10.1137/S0036139903431737. Google Scholar
[37]	B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448. Google Scholar
[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. Google Scholar
[39]	D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164. doi: 10.1103/PhysRevE.51.3164. Google Scholar
[40]	D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. Google Scholar
[41]	R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic,, Elsevier, (1971). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[50]	J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24. Google Scholar
[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. Google Scholar
[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. Google Scholar
[53]	T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707. Google Scholar
[54]	Minnesota Department of Transportation, Mn/DOT Traffic Data,, Website, (). Google Scholar
[55]	Mobile Millennium project, Website,, , (). Google Scholar
[56]	K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221. Google Scholar
[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. Google Scholar
[58]	G. F. Newell, Nonlinear effects in the dynamics of car following,, Operations Research, 9 (1961), 209. doi: 10.1287/opre.9.2.209. Google Scholar
[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. Google Scholar
[60]	E. Parzen, On estimation of a probability density function and mode,, Ann. Math. Statist., 33 (1962), 1065. doi: 10.1214/aoms/1177704472. Google Scholar
[61]	H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51. Google Scholar
[62]	H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic,, Transp. Res. Rec., 722 (1979), 68. Google Scholar
[63]	W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131. doi: 10.1080/03081067908717157. Google Scholar
[64]	L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274. doi: 10.1063/1.1721265. Google Scholar
[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. Google Scholar
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