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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 |
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). 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. |
[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. |
[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. 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. |
[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. 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. |
[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). 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. |
[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). 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. |
[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. 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. |
[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). 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. |
[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). 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. |
[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. 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. |
[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, (). 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. |
[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. 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. |
[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] |
F. Siebel and W. Mauser, On the fundamental diagram of traffic flow,, SIAM J. Appl. Math., 66 (2006), 1150.
doi: 10.1137/050627113. |
[71] |
B. Temple, Systems of conservation laws with coinciding shock and rarefaction curves,, Contemp. Math., 17 (1983), 143. Google Scholar |
[72] |
R. Underwood, Speed, Volume, and Density Relationships: Quality and Theory of Traffic Flow,, Technical Report, (1961). Google Scholar |
[73] |
G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974).
|
[74] |
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.
|
[75] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transp. Res. B, 36 (2002), 275.
doi: 10.1016/S0191-2615(00)00050-3. |
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). 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. |
[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. |
[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. 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. |
[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. 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. |
[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). 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. |
[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). 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. |
[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. 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. |
[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). 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. |
[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). 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. |
[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. 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. |
[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, (). 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. |
[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. 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. |
[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. |
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