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Stability analysis of the classical car-following model

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  • Zhang, Xiaoyan
  • Jarrett, David F.

Abstract

This paper investigates the stability of the classical car-following model (for example, Chandler et al., Operations Research, 6, 165-184, 1958; Herman et al., Operations Research, 7, 86-106, 1959; Wilhelm and Schmidt, Transportation Engineering Journal (ASCE) 99, 923-933, 1973). Conditions for local and asymptotic stability as defined in the references cited are established for the linear model. These differ from those in the literature in two ways. First, it will be shown that, in the autonomous model when the product of the coefficient of proportionality [alpha] and the reaction time [tau] is less than or equal to 1/e, there exist oscillatory solutions with higher frequencies than 2[pi], although there are none with lower frequencies. Secondly, asymptotic stability is considered along with local stability. The derived condition for asymptotic stability is both necessary and sufficient. In addition, the condition depends on the frequency of the forcing term, with the sufficient condition [alpha][tau]

Suggested Citation

  • Zhang, Xiaoyan & Jarrett, David F., 1997. "Stability analysis of the classical car-following model," Transportation Research Part B: Methodological, Elsevier, vol. 31(6), pages 441-462, November.
    • Handle: RePEc:eee:transb:v:31:y:1997:i:6:p:441-462
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    References listed on IDEAS

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    1. Ferrari, Paolo, 1994. "The instability of motorway traffic," Transportation Research Part B: Methodological, Elsevier, vol. 28(2), pages 175-186, April.
    2. Denos C. Gazis & Robert Herman & Richard W. Rothery, 1961. "Nonlinear Follow-the-Leader Models of Traffic Flow," Operations Research, INFORMS, vol. 9(4), pages 545-567, August.
    3. J. G. Bender & R. E. Fenton, 1970. "On the Flow Capacity of Automated Highways," Transportation Science, INFORMS, vol. 4(1), pages 52-63, February.
    4. Ernest A. Unwin & Lucien Duckstein, 1967. "Stability of Reciprocal-Spacing Type Car Following Models," Transportation Science, INFORMS, vol. 1(2), pages 95-108, May.
    5. Robert Herman & Elliott W. Montroll & Renfrey B. Potts & Richard W. Rothery, 1959. "Traffic Dynamics: Analysis of Stability in Car Following," Operations Research, INFORMS, vol. 7(1), pages 86-106, February.
    6. Robert E. Chandler & Robert Herman & Elliott W. Montroll, 1958. "Traffic Dynamics: Studies in Car Following," Operations Research, INFORMS, vol. 6(2), pages 165-184, April.
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    Cited by:

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    2. Erik T. Verhoef, 1998. "An Integrated Dynamic Model of Road Traffic Congestion based on Simple Car-Following Theory," Tinbergen Institute Discussion Papers 98-030/3, Tinbergen Institute.
    3. Sauer, Craig & Andersen, Goerge J. & Saidpour, Asad, 2004. "Detection and Avoidance of Collisions: the REACT Model," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt7st785tt, Institute of Transportation Studies, UC Berkeley.
    4. Jafaripournimchahi, Ammar & Cai, Yingfeng & Wang, Hai & Sun, Lu & Yang, Biao, 2022. "Stability analysis of delayed-feedback control effect in the continuum traffic flow of autonomous vehicles without V2I communication," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 605(C).
    5. Kai Nagel & Peter Wagner & Richard Woesler, 2003. "Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling," Operations Research, INFORMS, vol. 51(5), pages 681-710, October.
    6. Ammar Jafaripournimchahi & Yingfeng Cai & Hai Wang & Lu Sun, 2022. "Environmental Analyses of Delayed-Feedback Control Effects in Continuum-Traffic Flow of Autonomous Vehicles," Sustainability, MDPI, vol. 14(18), pages 1-18, September.
    7. Sun, Jie & Zheng, Zuduo & Sun, Jian, 2018. "Stability analysis methods and their applicability to car-following models in conventional and connected environments," Transportation Research Part B: Methodological, Elsevier, vol. 109(C), pages 212-237.
    8. Verhoef, Erik T., 2001. "An Integrated Dynamic Model of Road Traffic Congestion Based on Simple Car-Following Theory: Exploring Hypercongestion," Journal of Urban Economics, Elsevier, vol. 49(3), pages 505-542, May.
    9. Montanino, Marcello & Punzo, Vincenzo, 2021. "On string stability of a mixed and heterogeneous traffic flow: A unifying modelling framework," Transportation Research Part B: Methodological, Elsevier, vol. 144(C), pages 133-154.

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