Traffic Flow Theory

  • Pushkin Kachroo
  • Kaan M. A. Özbay
Part of the Advances in Industrial Control book series (AIC)


This chapter presents the basic traffic flow theory which is used in the following chapters for control problem formulations. The theory develops the Lighthill–Whitham–Richards (LWR) model that uses the conservation law for traffic. Additionally, a density-dependent speed formula is used. There are many relationships available for this fundamental diagram, the chapter uses Greenshields’ formula for further analysis. Elementary partial differential equations (PDE) theory is also presented including the method of characteristics needed for the analysis of the traffic model. Shockwaves and weak solutions are discussed followed by a brief discussion of traffic measurements.


  1. 1.
    Haberman R (1977) Mathematical models. Prentice-Hall, New JerseyzbMATHGoogle Scholar
  2. 2.
    Kachroo P (2009) Pedestrian dynamics: mathematical theory and evacuation control. CRC PressGoogle Scholar
  3. 3.
    Lighthill MJ, Whitham GB (1955) On kinematic waves II. A theory of traffic flow on long crowded roads. Proc Royal Soci London. Ser A. Math Phys Sci 229(1178):317–345. Scholar
  4. 4.
    Richards PI (1956) Shockwaves on the highway. Oper Res 4:42–51CrossRefGoogle Scholar
  5. 5.
    Greenshields BD, Channing WS, Miller HH (1935) A study of traffic capacity. In: Highway research board proceedings, volume 1935. National Research Council (USA), Highway Research BoardGoogle Scholar
  6. 6.
    Greenberg H (1959) An analysis of traffic flow. Oper Res 7(1):79–85MathSciNetCrossRefGoogle Scholar
  7. 7.
    Underwood R (1961) Speed, volume, and density relationships. Paper presented at the In: Quality and Theory of Traffic Flow; Bureau of Highway Traffic, Yale University, New HavenGoogle Scholar
  8. 8.
    Drake JS, Schofer JL, May Jr AD (1967) A statistical analysis of speed-density hypotheses. in vehicular traffic science. Highway Research RecordGoogle Scholar
  9. 9.
    Drew DR (1968) Traffic flow theory and control. McGraw-HillGoogle Scholar
  10. 10.
    Pipes LA (1967) Car following models and the fundamental diagram of road traffic. Transp Res 1(1):21–29CrossRefGoogle Scholar
  11. 11.
    May AD (1990) Traffic flow fundamentals. Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  12. 12.
    Musha T, Higuchi H (1978) Traffic current fluctuation and the burgers equation. Jpn J Appl Phys 17(5):811CrossRefGoogle Scholar
  13. 13.
    Burns JA, Kang S (1991) A control problem for burgers’ equation with bounded input/output. Nonlinear Dyn 2(4):235–262CrossRefGoogle Scholar
  14. 14.
    Cole JD (1951) On a quasi-linear parabolic equation occurring in aerodynamics. Quart Appl Math 9(3):225–236MathSciNetCrossRefGoogle Scholar
  15. 15.
    Glimm J, Lax PD (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am Math SocCrossRefGoogle Scholar
  16. 16.
    Hopf E (1950) The partial differential equation ut\(+\) uux \(=\)\(\mu \)xx. Commun Pure Appl Math 3(3):201–230CrossRefGoogle Scholar
  17. 17.
    Lax PD (1973) Hyperbolic systems of conservation laws and the mathematical theory of shock waves. In: Society for Industrial and Applied Mathematics, vol 11–16CrossRefGoogle Scholar
  18. 18.
    Maslov VP (1987) On a new principle of superposition for optimization problems. Rus Math Surv 42(3):43–54MathSciNetCrossRefGoogle Scholar
  19. 19.
    Maslov VP (1987) A new approach to generalized solutions of nonlinear systems. In: Soviet Math. Dokl, vol 1, pp 29–33Google Scholar
  20. 20.
    Curtain RF (1984) Stability of semilinear evolution equations in hilbert space. J Math Pures et Appl 63:121–128zbMATHGoogle Scholar
  21. 21.
    Papageorgiou M (1983) Applications of automatic control concepts to traffic flow modeling and control (Lecture Notes in Control and Information Sciences). SpringerGoogle Scholar
  22. 22.
    Garber NJ, Hoel LA (2014) Traffic and highway engineering. Cengage LearningGoogle Scholar
  23. 23.
    Gazis DC, Herman R, Potts RB (1959) Car-following theory of steady-state traffic flow. Oper Res 7(4):499–505MathSciNetCrossRefGoogle Scholar
  24. 24.
    Farlow SJ (2012) Partial differential equations for scientists and engineers. Courier Dover PublicationsGoogle Scholar
  25. 25.
    Zachmanoglou EC, Thoe DW (1986) Introduction to partial differential equations with applications. Courier Dover PublicationsGoogle Scholar
  26. 26.
    Logan JD (2010) An introduction to nonlinear partial differential equations. WileyGoogle Scholar
  27. 27.
    LeVeque RJ (1990) Numerical methods for conservation laws. Birkhäuser Verlag AGCrossRefGoogle Scholar
  28. 28.
    Lebacque J-P (1996) The godunov scheme and what it means for first order traffic flow models. In: Proceedings of the 13th International symposium on transportation and traffic theory, Lyon, France, pp 647–677Google Scholar
  29. 29.
    Bressan A (2000) Hyperbolic systems of conservation laws: the one-dimensional cauchy problem. Oxford University PressGoogle Scholar
  30. 30.
    S Strub I, M Bayen A (2006) Weak formulation of boundary conditions for scalar conservation laws: an application to highway modeling. Int J Robust Nonlinear Control 16:733–748Google Scholar
  31. 31.
    Wardrop JG (1952) Some theoretical aspects of road traffic research. In: Proceedings of the Institution of Civil Engineers, Part II, vol 1, pp 325–378Google Scholar
  32. 32.
    Gerlough DL, Huber MJ (1975) Traffic flow theory: a monograph. Transportation Research Board, National Research CouncilGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of NevadaLas VegasUSA
  2. 2.Department of Civil and Urban EngineeringNew York UniversityBrooklynUSA

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