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]

Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021004
[2]

Mahir Demir, Suzanne Lenhart. A spatial food chain model for the Black Sea Anchovy, and its optimal fishery. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 155-171. doi: 10.3934/dcdsb.2020373
[3]

Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047
[4]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362
[5]

Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264
[6]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[7]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[8]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[9]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[10]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357
[11]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003
[12]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034
[13]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003
[14]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012
[15]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175
[16]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
[17]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[18]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[19]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263
[20]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences