American Institute of Mathematical Sciences

Advanced
March  2013, 8(1): 261-273. doi: 10.3934/nhm.2013.8.261

Stochastic control of traffic patterns

Yuri B. Gaididei 1, , Carlos Gorria 2, , Rainer Berkemer 3, , Peter L. Christiansen 4, , Atsushi Kawamoto 5, , Mads P. Sørensen 6, and  Jens Starke 6,

1. 

Bogolyubov Institute for Theoretical Physics, Metrologichna str. 14 B, 01413, Kiev
2. 

Department of Applied Mathematics and Statistics, University of the Basque Country, E-48080 Bilbao
3. 

AKAD University of Applied Sciences, D-70469 Stuttgart, Germany
4. 

Department of Mathematics and Computer Science & Department of Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
5. 

Toyota Central R&D Labs, Nagakute, Aichi, Japan
6. 

Department of Mathematics and Computer Science, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark, Denmark

Received  January 2012 Revised  February 2013 Published  April 2013

A stochastic modulation of the safety distance can reduce traffic jams. It is found that the effect of random modulation on congestive flow formation depends on the spatial correlation of the noise. Jam creation is suppressed for highly correlated noise. The results demonstrate the advantage of heterogeneous performance of the drivers in time as well as individually. This opens the possibility for the construction of technical tools to control traffic jam formation.
Keywords: optimal velocity model, spatial and temporal colored noise, stochastic system, discrete system, Fokker-Planck equation., Traffic model, Ornstein-Uhlenbeck process.
Mathematics Subject Classification: Primary: 60H10, 34F05, 90B20; Secondary: 35Q84, 35B36, 34E05, 42A1.
Citation: Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen, Jens Starke. Stochastic control of traffic patterns. Networks & Heterogeneous Media, 2013, 8 (1) : 261-273. doi: 10.3934/nhm.2013.8.261
References:
[1]

M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions,", Dover Publications, (1972).  doi: 10.1119/1.1972842.  Google Scholar

[2]

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

[3]

A. Bose and P. Ioannou, Mixed manual/semi-automated traffic: A macroscopic analysis,, Trasp. Res., 11 (2003).  doi: 10.1016/j.trc.2002.04.001.  Google Scholar

[4]

A. H. Cohen, P. J. Holmes and R. H. Rand, The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model,, J. Math. Biol., 13 (1982), 345.  doi: 10.1007/BF00276069.  Google Scholar

[5]

L. C. Davis, Effect of adaptive cruise control systems on traffic flow,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.066110.  Google Scholar

[6]

Y. Gaididei, C. Gorria, R. Berkemer, A. Kawamoto, A. Kawamoto, T. Shiga, P. L. Christiansen, M. P. Sørensen and J. Starke, Traffic jam control by time-modulating the safety distance,, Submitted., ().   Google Scholar

[7]

Yu. B. Gaididei, R. Berkemer, J. G. Caputo, P. L. Christiansen, A. Kawamoto, T. Shiga, M. P. Sørensen and J. Starke, Analytical solutions of jam pattern formation on a ring for a class of optimal velocity traffic models,, New Journal of Phys., 11 (2009), 1.  doi: 10.1088/1367-2630/11/7/073012.  Google Scholar

[8]

Yu. B. Gaididei, R. Berkemer, C. Gorria, P. L. Christiansen, A. Kawamoto, T. Shiga, M. P. Sørensen and J. Starke, Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements,, Discrete and Continuous Dynamical Systems. Series S., 4 (2011), 1167.  doi: 10.3934/dcdss.2011.4.1167.  Google Scholar

[9]

C. W. Gardiner, "Handbook of Stochastic Method,'', 2nd ed. (Springer, (1989).  doi: 10.1007/978-3-662-02377-8.  Google Scholar

[10]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys., 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[11]

V. In, A. Kho, J. D. Neff, A. Palacios, P. Longhini and B. K. Meadows, Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators,, Phys. Rev. Lett., 91 (2003), 244101.  doi: 10.1103/PhysRevLett.91.244101.  Google Scholar

[12]

B. S. Kerner, "The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory,", Heidelberg: Springer, (2004).   Google Scholar

[13]

B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory,", Berlin: Springer, (2009).   Google Scholar

[14]

S. Kikuchi, N. Uno and M. Tanaka, Impacts of shorter perception-reaction time of adapted cruise controlled vehicles on traffic flow and safety,, J. Trans. Eng., 129 (2003).   Google Scholar

[15]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", 3rd ed., (2004).   Google Scholar

[16]

K. Konishi, H. Kokame and K. Hirata, Coupled map car-following model and its delayed-feedback control,, Phys. Rev. E 60 4000;, 60 (4000).   Google Scholar

[17]

H. K. Lee, H.-W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevA.70.059902.  Google Scholar

[18]

ChiYing Liang and Huei Peng, String stability analysis of adaptive cruise controlled vehicles,, JSME Int. J., 43 (2000).   Google Scholar

[19]

P. Y. Li and A. Shrivastava, Traffic flow stability induced by constant time headway policy for adaptive cruise control vehicles,, Transp. Res., 10 (2002).   Google Scholar

[20]

S. Maerivoet and B. De Moor, Cellular automata models of road traffic,, Phys. Reps., 419 (2005).   Google Scholar

[21]

T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331.   Google Scholar

[22]

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

[23]

K. Nishinari, K. Sugawara, T. Kazama, A. Schadschneider and D. Chowdhury, Modelling of self-driven particles: Foraging ants and pedestrians,, Physica A, 372 (2006).   Google Scholar

[24]

C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion,, J. Math. Biol., 53 (2006), 474.  doi: 10.1007/s00285-006-0021-2.  Google Scholar

[25]

A. Schadschneider, D. Chowdhury and K. Nishinari, "Stochastic Transport in Complex Systems,", Elsevier, (2011).   Google Scholar

[26]

N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats,, Phys. Rev. E, 81 (2010).   Google Scholar

[27]

Yu. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, Sh. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Phys., 10 (2008), 1.   Google Scholar

[28]

A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators,, Phys. Rev. Lett., 92 (2004), 228102.   Google Scholar

[29]

A. Takamatsu, R. Tanaka, T. Nakagaki, T. Fujii and I. Endo, Spatiotemporal symmetry in rings of coupled biological oscillators of Physarum Plasmodial slime mold,, Phys. Rev. Lett., 87 (2001), 078102.   Google Scholar

[30]

D. Tanaka, General chemotactic model of oscillators,, Phys. Rev. Lett., 99 (2007).   Google Scholar

[31]

D. E. Wolf, M. Schreckenberg and A. Bachem, "Traffic and Granular Flow,", Word Scientific, (1996).   Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions,", Dover Publications, (1972).  doi: 10.1119/1.1972842.  Google Scholar

[2]

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

[3]

A. Bose and P. Ioannou, Mixed manual/semi-automated traffic: A macroscopic analysis,, Trasp. Res., 11 (2003).  doi: 10.1016/j.trc.2002.04.001.  Google Scholar

[4]

A. H. Cohen, P. J. Holmes and R. H. Rand, The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model,, J. Math. Biol., 13 (1982), 345.  doi: 10.1007/BF00276069.  Google Scholar

[5]

L. C. Davis, Effect of adaptive cruise control systems on traffic flow,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.066110.  Google Scholar

[6]

Y. Gaididei, C. Gorria, R. Berkemer, A. Kawamoto, A. Kawamoto, T. Shiga, P. L. Christiansen, M. P. Sørensen and J. Starke, Traffic jam control by time-modulating the safety distance,, Submitted., ().   Google Scholar

[7]

Yu. B. Gaididei, R. Berkemer, J. G. Caputo, P. L. Christiansen, A. Kawamoto, T. Shiga, M. P. Sørensen and J. Starke, Analytical solutions of jam pattern formation on a ring for a class of optimal velocity traffic models,, New Journal of Phys., 11 (2009), 1.  doi: 10.1088/1367-2630/11/7/073012.  Google Scholar

[8]

Yu. B. Gaididei, R. Berkemer, C. Gorria, P. L. Christiansen, A. Kawamoto, T. Shiga, M. P. Sørensen and J. Starke, Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements,, Discrete and Continuous Dynamical Systems. Series S., 4 (2011), 1167.  doi: 10.3934/dcdss.2011.4.1167.  Google Scholar

[9]

C. W. Gardiner, "Handbook of Stochastic Method,'', 2nd ed. (Springer, (1989).  doi: 10.1007/978-3-662-02377-8.  Google Scholar

[10]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phys., 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[11]

V. In, A. Kho, J. D. Neff, A. Palacios, P. Longhini and B. K. Meadows, Experimental observation of multifrequency patterns in arrays of coupled nonlinear oscillators,, Phys. Rev. Lett., 91 (2003), 244101.  doi: 10.1103/PhysRevLett.91.244101.  Google Scholar

[12]

B. S. Kerner, "The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory,", Heidelberg: Springer, (2004).   Google Scholar

[13]

B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory,", Berlin: Springer, (2009).   Google Scholar

[14]

S. Kikuchi, N. Uno and M. Tanaka, Impacts of shorter perception-reaction time of adapted cruise controlled vehicles on traffic flow and safety,, J. Trans. Eng., 129 (2003).   Google Scholar

[15]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", 3rd ed., (2004).   Google Scholar

[16]

K. Konishi, H. Kokame and K. Hirata, Coupled map car-following model and its delayed-feedback control,, Phys. Rev. E 60 4000;, 60 (4000).   Google Scholar

[17]

H. K. Lee, H.-W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevA.70.059902.  Google Scholar

[18]

ChiYing Liang and Huei Peng, String stability analysis of adaptive cruise controlled vehicles,, JSME Int. J., 43 (2000).   Google Scholar

[19]

P. Y. Li and A. Shrivastava, Traffic flow stability induced by constant time headway policy for adaptive cruise control vehicles,, Transp. Res., 10 (2002).   Google Scholar

[20]

S. Maerivoet and B. De Moor, Cellular automata models of road traffic,, Phys. Reps., 419 (2005).   Google Scholar

[21]

T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331.   Google Scholar

[22]

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

[23]

K. Nishinari, K. Sugawara, T. Kazama, A. Schadschneider and D. Chowdhury, Modelling of self-driven particles: Foraging ants and pedestrians,, Physica A, 372 (2006).   Google Scholar

[24]

C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion,, J. Math. Biol., 53 (2006), 474.  doi: 10.1007/s00285-006-0021-2.  Google Scholar

[25]

A. Schadschneider, D. Chowdhury and K. Nishinari, "Stochastic Transport in Complex Systems,", Elsevier, (2011).   Google Scholar

[26]

N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats,, Phys. Rev. E, 81 (2010).   Google Scholar

[27]

Yu. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, Sh. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Phys., 10 (2008), 1.   Google Scholar

[28]

A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators,, Phys. Rev. Lett., 92 (2004), 228102.   Google Scholar

[29]

A. Takamatsu, R. Tanaka, T. Nakagaki, T. Fujii and I. Endo, Spatiotemporal symmetry in rings of coupled biological oscillators of Physarum Plasmodial slime mold,, Phys. Rev. Lett., 87 (2001), 078102.   Google Scholar

[30]

D. Tanaka, General chemotactic model of oscillators,, Phys. Rev. Lett., 99 (2007).   Google Scholar

[31]

D. E. Wolf, M. Schreckenberg and A. Bachem, "Traffic and Granular Flow,", Word Scientific, (1996).   Google Scholar

[1]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[2]

Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285
[3]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[4]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[5]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic & Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011
[6]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[7]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
[8]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
[9]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871
[10]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[11]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[12]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451
[13]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[14]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[15]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[16]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[17]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[18]

Yoon-Sik Cho, Aram Galstyan, P. Jeffrey Brantingham, George Tita. Latent self-exciting point process model for spatial-temporal networks. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1335-1354. doi: 10.3934/dcdsb.2014.19.1335
[19]

Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651
[20]

Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020058

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences