Structure and parameters of clusters in traffic flow

B. S. Kerner; P. Konhäuser

doi:10.1103/PhysRevE.50.54

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Structure and parameters of clusters in traffic flow

B. S. Kerner and P. Konhäuser

Research Institute, Daimler-Benz AG, F1V/VT, HPC G210, 70546 Stuttgart, Germany

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Phys. Rev. E 50, 54 – Published 1 July, 1994

DOI: https://doi.org/10.1103/PhysRevE.50.54

Abstract

The nonlinear theory of the cluster effect in a traffic flow [B. S. Kerner and P. Konhäuser, Phys. Rev. E 48, 2335 (1993)], i.e., the effect of the appearance of a region of high density and low average velocity of vehicles in an initially homogeneous flow, is presented. The structures of a stationary moving cluster are derived. It is found that the density, the average velocities of vehicles inside and outside the cluster, and also the velocity of the cluster are the characteristic parameters of the traffic flow. The dependencies of the cluster structure and parameters on the density of vehicles in the initially homogeneous flow and on the length of the road are investigated. It is found that the cluster can appear within regions of density of vehicles which correspond to a stable homogeneous flow. It is shown that an appearance of a localized perturbation, having a finite amplitude, in the stable homogeneous flow can lead to a self-formation of a local cluster of vehicles which is surrounded by the homogeneous traffic flow. The parameters of the local cluster do not depend on the amplitude of this perturbation but only on the parameters of the flow.

References (21)

B. S. Kerner and P. Konhäuser, Phys. Rev. E 48, 2335 (1993).
W. Leutzbach, Introduction to the Theory of Traffic Flow (Springer-Verlag, Berlin, 1988).
J. Treiterer, Ohio State University Technical Report PB 246 094, Columbus, Ohio, 1975 (unpublished).
The diffusion term in the kinetic model of traffic flow has been introduced by R. Kühne [see, e.g., R. Kühne, in Highway Capacity and Level of Service, edited by U. Brannolte (A. A. Balkema, Rotterdam, 1991), p. 211].
I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic (American Elsevier, New York, 1971).
H. J. Payne, in Mathematical Models of Public Systems, Simulation Council Proceedings, 1971 (Simulation Councils, La Jolla, CA 1971), Vol. 1, p. 51.
G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
Noise in Nonlinear Dynamical Systems, Vols. 1–3, edited by F. Moss and P. V. E. McClintock (Cambridge University Press, New Rochelle, 1989).
G. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences (Springer-Verlag, Berlin, 1983).
F. L. Hall, L. A. Brian and M. A. Gunter, Transp. Res. A 20, 197 (1986).
H. B. Keller, Soc. Ind. Appl. Math. J. Numer. Anal. 11, 305 (1974).
Notice that the line q bar ( ρ bar ), which corresponds to the clusters (line C in Fig. 12), does not reach exactly the points $ρ_{\min}^{m}$ and $ρ_{\max}^{m}$ . Indeed, the clusters can exist in the interval $ρ_{b 1}$ <= ρ bar <= $ρ_{b 2}$ , but $ρ_{b 1}$ > $ρ_{\min}^{m}$ and $ρ_{b 2}$ < $ρ_{\max}^{m}$ (Secs. IIID and IIIE). Nevertheless, for a simplification of the consideration of the average fundamental diagram this circumstance has not been taken into account in Fig. 12, because a difference between the value $ρ_{\min}^{m}$ and the value $ρ_{b 1}$ (or $ρ_{b 2}$ and $ρ_{\max}^{m}$ ), when the length of the road L is large enough, can be very small.
M. J. Lighthill and B. G. Whitham, Proc. R. Soc. London, Ser. A 229, 317 (1955).
A. D. May, Traffic Flow Fundamental (Prentice Hall, Inc., New Jersey, 1990).
R. Wiedemann, Simulation des Stra\(27enverkehrsflusses (Schriftenreihe des Instituts für Verkerswesens Universität Karlsruhe, Heft 8, 1974).
H.-T. Fritzsche, Traffic Eng. Control 35, 317 (1994).
K. Nagel and M. Schreckenberg, J. Phys. I (France) 2, 2221 (1992).
A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems (Wiley, New York, 1973).
The velocity of a wide and of a narrow cluster does not coincide (see Sec. IVC). For this reason, although the qualitative form of the v- $q^{star}$ curve for both the wide and the narrow clusters has the same ~~~~ shape [Fig. 16(a)], these v- $q^{star}$ curves quantitatively differ one from another. For shorting the quantity of the figures we have not taken into account this circumstance in Figs. 16(a)–16(c) which does not change the qualitative result presented.
The shape of the functions $v^{(0)}$ (x) in Figs. 16(d)–16(i) is formally analogous to the shape of solutions for other physical variables found from the analysis of different nonlinear physical problems, for example, in the investigations of localized dissipative structures (autosolitons) in nonlinear active systems with diffusion. Notice also that, as follows from the properties of the local cluster found (Sec. IIC), some of them are qualitatively similar to general properties of autosolitons in active systems with diffusion (for a review, see B. S. Kerner and V. V. Osipov, Usp. Fiz. Nauk 157, 201 (1989) [Sov. Phys. Usp. 32, 101 (1989)]; Autosolitons: A New Approach to the Problems of Self-Organization and Turbulence (Kluwer Academic, Dordrecht, 1994).
G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1961).

[c1] B. S. Kerner and P. Konhäuser, Phys. Rev. E 48, 2335 (1993).

[c2] W. Leutzbach, Introduction to the Theory of Traffic Flow (Springer-Verlag, Berlin, 1988).

[c3] J. Treiterer, Ohio State University Technical Report PB 246 094, Columbus, Ohio, 1975 (unpublished).

[c4] The diffusion term in the kinetic model of traffic flow has been introduced by R. Kühne [see, e.g., R. Kühne, in Highway Capacity and Level of Service, edited by U. Brannolte (A. A. Balkema, Rotterdam, 1991), p. 211].

[c5] I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic (American Elsevier, New York, 1971).

[c6] H. J. Payne, in Mathematical Models of Public Systems, Simulation Council Proceedings, 1971 (Simulation Councils, La Jolla, CA 1971), Vol. 1, p. 51.

[c7] G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

[c8] Noise in Nonlinear Dynamical Systems, Vols. 1–3, edited by F. Moss and P. V. E. McClintock (Cambridge University Press, New Rochelle, 1989).

[c9] G. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences (Springer-Verlag, Berlin, 1983).

[c10] F. L. Hall, L. A. Brian and M. A. Gunter, Transp. Res. A 20, 197 (1986).

[c11] H. B. Keller, Soc. Ind. Appl. Math. J. Numer. Anal. 11, 305 (1974).

[c12] Notice that the line q bar ( ρ bar ), which corresponds to the clusters (line C in Fig. 12), does not reach exactly the points $ρ_{\min}^{m}$ and $ρ_{\max}^{m}$ . Indeed, the clusters can exist in the interval $ρ_{b 1}$ <= ρ bar <= $ρ_{b 2}$ , but $ρ_{b 1}$ > $ρ_{\min}^{m}$ and $ρ_{b 2}$ < $ρ_{\max}^{m}$ (Secs. IIID and IIIE). Nevertheless, for a simplification of the consideration of the average fundamental diagram this circumstance has not been taken into account in Fig. 12, because a difference between the value $ρ_{\min}^{m}$ and the value $ρ_{b 1}$ (or $ρ_{b 2}$ and $ρ_{\max}^{m}$ ), when the length of the road L is large enough, can be very small.

[c13] M. J. Lighthill and B. G. Whitham, Proc. R. Soc. London, Ser. A 229, 317 (1955).

[c14] A. D. May, Traffic Flow Fundamental (Prentice Hall, Inc., New Jersey, 1990).

[c15] R. Wiedemann, Simulation des Stra\(27enverkehrsflusses (Schriftenreihe des Instituts für Verkerswesens Universität Karlsruhe, Heft 8, 1974).

[c16] H.-T. Fritzsche, Traffic Eng. Control 35, 317 (1994).

[c17] K. Nagel and M. Schreckenberg, J. Phys. I (France) 2, 2221 (1992).

[c18] A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems (Wiley, New York, 1973).

[c19] The velocity of a wide and of a narrow cluster does not coincide (see Sec. IVC). For this reason, although the qualitative form of the v- $q^{star}$ curve for both the wide and the narrow clusters has the same ~~~~ shape [Fig. 16(a)], these v- $q^{star}$ curves quantitatively differ one from another. For shorting the quantity of the figures we have not taken into account this circumstance in Figs. 16(a)–16(c) which does not change the qualitative result presented.

[c20] The shape of the functions $v^{(0)}$ (x) in Figs. 16(d)–16(i) is formally analogous to the shape of solutions for other physical variables found from the analysis of different nonlinear physical problems, for example, in the investigations of localized dissipative structures (autosolitons) in nonlinear active systems with diffusion. Notice also that, as follows from the properties of the local cluster found (Sec. IIC), some of them are qualitatively similar to general properties of autosolitons in active systems with diffusion (for a review, see B. S. Kerner and V. V. Osipov, Usp. Fiz. Nauk 157, 201 (1989) [Sov. Phys. Usp. 32, 101 (1989)]; Autosolitons: A New Approach to the Problems of Self-Organization and Turbulence (Kluwer Academic, Dordrecht, 1994).

[c21] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1961).

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Structure and parameters of clusters in traffic flow

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