Stochastic control of traffic patterns

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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

Abstract
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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 and 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, Inc., New York, 1972. doi: 10.1119/1.1972842.
[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), 1035.
[3]	A. Bose and P. Ioannou, Mixed manual/semi-automated traffic: A macroscopic analysis, Trasp. Res., Part C: Emerg. Technol., 11 (2003), 439. doi: 10.1016/j.trc.2002.04.001.
[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), 345-369. doi: 10.1007/BF00276069.
[5]	L. C. Davis, Effect of adaptive cruise control systems on traffic flow, Phys. Rev. E, 69 (2004), 066110. doi: 10.1103/PhysRevE.69.066110.
[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.
[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), 073012 (1-19). doi: 10.1088/1367-2630/11/7/073012.
[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), 1167-1179. doi: 10.3934/dcdss.2011.4.1167.
[9]	C. W. Gardiner, "Handbook of Stochastic Method,'' 2nd ed. (Springer, Berlin), 1989. doi: 10.1007/978-3-662-02377-8.
[10]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[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), 244101-244104. doi: 10.1103/PhysRevLett.91.244101.
[12]	B. S. Kerner, "The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory," Heidelberg: Springer, 2004.
[13]	B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory," Berlin: Springer, 2009.
[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), 146.
[15]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," 3rd ed., Springer, Berlin, 2004.
[16]	K. Konishi, H. Kokame and K. Hirata, Coupled map car-following model and its delayed-feedback control, Phys. Rev. E 60 4000;
[17]	H. K. Lee, H.-W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models, Phys. Rev. E, 64 (2001), 056126. doi: 10.1103/PhysRevA.70.059902.
[18]	ChiYing Liang and Huei Peng, String stability analysis of adaptive cruise controlled vehicles, JSME Int. J., Ser. C, 43 (2000), 671.
[19]	P. Y. Li and A. Shrivastava, Traffic flow stability induced by constant time headway policy for adaptive cruise control vehicles, Transp. Res., Part C: Emerg. Technol., 10 (2002), 275.
[20]	S. Maerivoet and B. De Moor, Cellular automata models of road traffic, Phys. Reps., 419 (2005), 1.
[21]	T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386.
[22]	K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. Phys. I France, 2 (1992), 2221.
[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), 132.
[24]	C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion, J. Math. Biol., 53 (2006), 474-489. doi: 10.1007/s00285-006-0021-2.
[25]	A. Schadschneider, D. Chowdhury and K. Nishinari, "Stochastic Transport in Complex Systems," Elsevier, 2011
[26]	N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Phys. Rev. E, 81 (2010), 056210.
[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), 1-7. 033001.
[28]	A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators, Phys. Rev. Lett., 92 (2004), 228102-228105.
[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), 078102-078105.
[30]	D. Tanaka, General chemotactic model of oscillators, Phys. Rev. Lett., 99 (2007), 134103.
[31]	D. E. Wolf, M. Schreckenberg and A. Bachem, "Traffic and Granular Flow," Word Scientific, Singapore, 1996.

show all references

References:

[1]	M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions," Dover Publications, Inc., New York, 1972. doi: 10.1119/1.1972842.
[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), 1035.
[3]	A. Bose and P. Ioannou, Mixed manual/semi-automated traffic: A macroscopic analysis, Trasp. Res., Part C: Emerg. Technol., 11 (2003), 439. doi: 10.1016/j.trc.2002.04.001.
[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), 345-369. doi: 10.1007/BF00276069.
[5]	L. C. Davis, Effect of adaptive cruise control systems on traffic flow, Phys. Rev. E, 69 (2004), 066110. doi: 10.1103/PhysRevE.69.066110.
[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.
[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), 073012 (1-19). doi: 10.1088/1367-2630/11/7/073012.
[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), 1167-1179. doi: 10.3934/dcdss.2011.4.1167.
[9]	C. W. Gardiner, "Handbook of Stochastic Method,'' 2nd ed. (Springer, Berlin), 1989. doi: 10.1007/978-3-662-02377-8.
[10]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[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), 244101-244104. doi: 10.1103/PhysRevLett.91.244101.
[12]	B. S. Kerner, "The physics of Traffic: Empirical Freeway Pattern Features, Engineering Applications, and Theory," Heidelberg: Springer, 2004.
[13]	B. S. Kerner, "Introduction to Modern Traffic Flow Theory and Control. The Long Road to Three-Phase Traffic Theory," Berlin: Springer, 2009.
[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), 146.
[15]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," 3rd ed., Springer, Berlin, 2004.
[16]	K. Konishi, H. Kokame and K. Hirata, Coupled map car-following model and its delayed-feedback control, Phys. Rev. E 60 4000;
[17]	H. K. Lee, H.-W. Lee and D. Kim, Macroscopic traffic models from microscopic car-following models, Phys. Rev. E, 64 (2001), 056126. doi: 10.1103/PhysRevA.70.059902.
[18]	ChiYing Liang and Huei Peng, String stability analysis of adaptive cruise controlled vehicles, JSME Int. J., Ser. C, 43 (2000), 671.
[19]	P. Y. Li and A. Shrivastava, Traffic flow stability induced by constant time headway policy for adaptive cruise control vehicles, Transp. Res., Part C: Emerg. Technol., 10 (2002), 275.
[20]	S. Maerivoet and B. De Moor, Cellular automata models of road traffic, Phys. Reps., 419 (2005), 1.
[21]	T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386.
[22]	K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. Phys. I France, 2 (1992), 2221.
[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), 132.
[24]	C. M. A. Pinto and M. Golubitsky, Central pattern generators for bipedal locomotion, J. Math. Biol., 53 (2006), 474-489. doi: 10.1007/s00285-006-0021-2.
[25]	A. Schadschneider, D. Chowdhury and K. Nishinari, "Stochastic Transport in Complex Systems," Elsevier, 2011
[26]	N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Phys. Rev. E, 81 (2010), 056210.
[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), 1-7. 033001.
[28]	A. Takamatsu, R. Tanaka and T. Fujii, Hidden symmetry in chains of biological coupled oscillators, Phys. Rev. Lett., 92 (2004), 228102-228105.
[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), 078102-078105.
[30]	D. Tanaka, General chemotactic model of oscillators, Phys. Rev. Lett., 99 (2007), 134103.
[31]	D. E. Wolf, M. Schreckenberg and A. Bachem, "Traffic and Granular Flow," Word Scientific, Singapore, 1996.

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