American Institute of Mathematical Sciences

Advanced
September  2013, 8(3): 745-772. doi: 10.3934/nhm.2013.8.745

Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

Benjamin Seibold 1, , Morris R. Flynn 2, , Aslan R. Kasimov 3, and  Rodolfo R. Rosales 4,

1. 

Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122
2. 

Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada
3. 

4700 King Abdullah University of, Science and Technology, Thuwal 23955-6900, Saudi Arabia
4. 

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

Received  April 2012 Revised  June 2013 Published  October 2013

Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.
Keywords: traffic model, sensor data, Payne-Whitham, jamiton, Aw-Rascle-Zhang, Second order, traveling wave, effective flow rate., fundamental diagram.
Mathematics Subject Classification: Primary: 35B40, 35C07, 35L65, 35Q91; Secondary: 76L05, 91C9.
Citation: Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745
References:
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[2]

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

[3]

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

[4]

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

[5]

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

[6]

R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model,, Traffic flow—modelling and simulation. Math. Comput. Modelling, 35 (2002), 683.  doi: 10.1016/S0895-7177(02)80029-2.  Google Scholar

[7]

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

[8]

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

[9]

C. F. Daganzo, The cell transmission model, part II: Network traffic,, Transp. Res. B, 29 (1995), 79.  doi: 10.1016/0191-2615(94)00022-R.  Google Scholar

[10]

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

[11]

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

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S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model,, in preparation, (2012).   Google Scholar

[14]

W. Fickett and W. C. Davis, "Detonation,", Univ. of California Press, (1979).   Google Scholar

[15]

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

[16]

H. Greenberg, An analysis of traffic flow,, Oper. Res., 7 (1959), 79.  doi: 10.1287/opre.7.1.79.  Google Scholar

[17]

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

[18]

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

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

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D. Helbing, Video of traffic waves,, Website. , ().   Google Scholar

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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

[22]

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

[23]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1.   Google Scholar

[24]

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, (2013).   Google Scholar

[25]

B. S. Kerner, Experimental features of self-organization in traffic flow,, Phys. Rev. Lett., 81 (1998), 3797.  doi: 10.1103/PhysRevLett.81.3797.  Google Scholar

[26]

B. S. Kerner, S. L. Klenov and P. Konhäuser, Asymptotic theory of traffic jams,, Phys. Rev. E, 56 (1997), 4200.  doi: 10.1103/PhysRevE.56.4200.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion,, Phys. Rev. E, 52 (1995), 5574.  doi: 10.1103/PhysRevE.52.5574.  Google Scholar

[31]

D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection,, Phys. Rev. E, 52 (1995), 218.  doi: 10.1103/PhysRevE.52.218.  Google Scholar

[32]

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

[33]

R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[34]

T. Li, Global solutions and zero relaxation limit for a traffic flow model,, SIAM J. Appl. Math., 61 (2000), 1042.  doi: 10.1137/S0036139999356788.  Google Scholar

[35]

T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation,, Comm. Math. Sci., 3 (2005), 101.   Google Scholar

[36]

T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds,, Discrete Contin. Dyn. Syst., 24 (2009), 511.  doi: 10.3934/dcds.2009.24.511.  Google Scholar

[37]

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

[38]

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

[39]

A. Messmer and M. Papageorgiou, METANET: A macroscopic simulation program for motorway networks,, Traffic Engrg. Control, 31 (1990), 466.   Google Scholar

[40]

K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221.  doi: 10.1051/jp1:1992277.  Google Scholar

[41]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, In A. Ceder, (1999), 51.   Google Scholar

[42]

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

[43]

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

[44]

Minnesota Department of Transportation, Mn/DOT traffic data,, Website. , ().   Google Scholar

[45]

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

[46]

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

[47]

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

[48]

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

[49]

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

[50]

B. Seibold, R. R. Rosales, M. R. Flynn and A. R. Kasimov, Classification of traveling wave solutions of the inhomogeneous Aw-Rascle-Zhang model,, in preparation, (2013).   Google Scholar

[51]

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

[52]

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks - Experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Physics, 10 (2008).  doi: 10.1088/1367-2630/10/3/033001.  Google Scholar

[53]

R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow,, Technical report, (1961).   Google Scholar

[54]

Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM),, Website. , ().   Google Scholar

[55]

P. Varaiya, Reducing highway congestion: An empirical approach,, Eur. J. Control, 11 (2005), 301.  doi: 10.3166/ejc.11.301-309.  Google Scholar

[56]

Y. Wang and M. Papageorgiou, Real-time freeway traffic state estimation based on extended Kalman filter: A general approach,, Transp. Res. B, 39 (2005), 141.  doi: 10.1016/j.trb.2004.03.003.  Google Scholar

[57]

J. G. Wardrop and G. Charlesworth, A method of estimating speed and flow of traffic from a moving vehicle,, Proc. Instn. Civ. Engrs., 3 (1954), 158.  doi: 10.1680/ipeds.1954.11628.  Google Scholar

[58]

G. B. Whitham, Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics,, Comm. Pure Appl. Math., 12 (1959), 113.  doi: 10.1002/cpa.3160120107.  Google Scholar

[59]

G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[60]

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.  Google Scholar

show all references

References:
[1]

T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics,, J. Stat. Phys, 133 (2008), 1083.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model,, Traffic flow—modelling and simulation. Math. Comput. Modelling, 35 (2002), 683.  doi: 10.1016/S0895-7177(02)80029-2.  Google Scholar

[7]

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

[8]

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

[9]

C. F. Daganzo, The cell transmission model, part II: Network traffic,, Transp. Res. B, 29 (1995), 79.  doi: 10.1016/0191-2615(94)00022-R.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model,, in preparation, (2012).   Google Scholar

[14]

W. Fickett and W. C. Davis, "Detonation,", Univ. of California Press, (1979).   Google Scholar

[15]

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

[16]

H. Greenberg, An analysis of traffic flow,, Oper. Res., 7 (1959), 79.  doi: 10.1287/opre.7.1.79.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

D. Helbing, Video of traffic waves,, Website. , ().   Google Scholar

[21]

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

[22]

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

[23]

R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1.   Google Scholar

[24]

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, (2013).   Google Scholar

[25]

B. S. Kerner, Experimental features of self-organization in traffic flow,, Phys. Rev. Lett., 81 (1998), 3797.  doi: 10.1103/PhysRevLett.81.3797.  Google Scholar

[26]

B. S. Kerner, S. L. Klenov and P. Konhäuser, Asymptotic theory of traffic jams,, Phys. Rev. E, 56 (1997), 4200.  doi: 10.1103/PhysRevE.56.4200.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion,, Phys. Rev. E, 52 (1995), 5574.  doi: 10.1103/PhysRevE.52.5574.  Google Scholar

[31]

D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection,, Phys. Rev. E, 52 (1995), 218.  doi: 10.1103/PhysRevE.52.218.  Google Scholar

[32]

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

[33]

R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[34]

T. Li, Global solutions and zero relaxation limit for a traffic flow model,, SIAM J. Appl. Math., 61 (2000), 1042.  doi: 10.1137/S0036139999356788.  Google Scholar

[35]

T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation,, Comm. Math. Sci., 3 (2005), 101.   Google Scholar

[36]

T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds,, Discrete Contin. Dyn. Syst., 24 (2009), 511.  doi: 10.3934/dcds.2009.24.511.  Google Scholar

[37]

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

[38]

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

[39]

A. Messmer and M. Papageorgiou, METANET: A macroscopic simulation program for motorway networks,, Traffic Engrg. Control, 31 (1990), 466.   Google Scholar

[40]

K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221.  doi: 10.1051/jp1:1992277.  Google Scholar

[41]

P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, In A. Ceder, (1999), 51.   Google Scholar

[42]

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

[43]

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

[44]

Minnesota Department of Transportation, Mn/DOT traffic data,, Website. , ().   Google Scholar

[45]

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

[46]

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

[47]

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

[48]

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

[49]

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

[50]

B. Seibold, R. R. Rosales, M. R. Flynn and A. R. Kasimov, Classification of traveling wave solutions of the inhomogeneous Aw-Rascle-Zhang model,, in preparation, (2013).   Google Scholar

[51]

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

[52]

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks - Experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Physics, 10 (2008).  doi: 10.1088/1367-2630/10/3/033001.  Google Scholar

[53]

R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow,, Technical report, (1961).   Google Scholar

[54]

Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM),, Website. , ().   Google Scholar

[55]

P. Varaiya, Reducing highway congestion: An empirical approach,, Eur. J. Control, 11 (2005), 301.  doi: 10.3166/ejc.11.301-309.  Google Scholar

[56]

Y. Wang and M. Papageorgiou, Real-time freeway traffic state estimation based on extended Kalman filter: A general approach,, Transp. Res. B, 39 (2005), 141.  doi: 10.1016/j.trb.2004.03.003.  Google Scholar

[57]

J. G. Wardrop and G. Charlesworth, A method of estimating speed and flow of traffic from a moving vehicle,, Proc. Instn. Civ. Engrs., 3 (1954), 158.  doi: 10.1680/ipeds.1954.11628.  Google Scholar

[58]

G. B. Whitham, Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics,, Comm. Pure Appl. Math., 12 (1959), 113.  doi: 10.1002/cpa.3160120107.  Google Scholar

[59]

G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[60]

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.  Google Scholar

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