Sharif E. Guseynov, Alexander V. Berezhnoy


In this paper non-deterministic motion of urban traffic is studied under certain assumptions. Based on those assumptions discrete and continuous mathematical models are developed: continuous model is written as the Cauchy initial-value problem for the integro-differential equation, whence among other things it is obtained the Fokker-Planck equation. Besides, the sufficient condition ensuring the mathematical legitimacy of the developed continuous model is formulated.


traffic flow; mathematical model; Cauchy initial-value problem

Full Text:



F. A. Haight, Mathematical Theories of Traffic Flow. New York, USA: Academic Press, 1963, xi+242 pp.

D. Helbing, "Traffic and related self-driven many-particle systems", Reviews of Modern Physics, vol. 73, pp. 1067-1141, 2001.

R. Mahnke, J. Kaupuzs, and I. Lubashevsky, "Probabilistic description of traffic flow", Physics Reports, vol. 408, pp. 1-130, 2005.

N. H. Gartner, "Traffic Flow Theory", Transportation Research Board, Special Report 165, World Scientific Press, 1992, 365 pp.

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations. New York, USA: Pergamon Press, 1997, 356 pp.

B. S. Kerner, "Three-Phase Traffic Theory and Highway Capacity", Physica A, vol. 333, pp. 379-440, 2004.

K. Nagel, P. Wagner, and R. Woesler, "Still flowing: Approaches to traffic flow and traffic jam modeling", Journal of Operations Research, vol. 51, No. 5, pp. 681-710, 2003.

C. F. Daganzo, "A Behavioral Theory of Multi-Lane Traffic Flow. Part I: Long Homogeneous Freeway Sections", Transportation Research, Part B: Methodological, vol. 36, No. 2, pp. 131-158, 2002.

C. F. Daganzo, "A Behavioral Theory of Multi-Lane Traffic Flow Part II: Merges and the Onset of Congestion", Transportation Research, Part B: Methodological, vol. 36, No. 2, pp. 159-169, 2002.

A. V. Berezhnoy, "Investigation of the traffic flow models managing parameters influence on the efficiency of the urban traffic control", Doctoral Thesis, Transport and Telecommunication Institute, Riga, Latvia, 2008, 256 pp.

N. N. Smirnov, A. B. Kiselev, V. F. Nikitin, and M. V. Yumashev, Mathematical Theory of Traffic Flow. Moscow, Russian Federation: Lomonosov Moscow State University Press, 1999, 30 p.

V. I. Shvetsov and D. Helbing, "Macroscopic dynamics of multilane traffic", Physical Reviews E., vol. 59, 1999, pp. 6328-6339.

H. Risken and T. Frank, The Fokker-Planck Equation: Methods of Solution and Applications. Berlin, Germany: Springer-Verlag, 1989, xiv+472 p.



  • There are currently no refbacks.