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Day-to-Day Flow Dynamics for Stochastic User Equilibrium and a General Lyapunov Function

Author

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  • Feng Xiao

    (School of Business Administration, Southwestern University of Finance and Economics, 611130, Sichuan, People’s Republic of China)

  • Minyu Shen

    (Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, People’s Republic of China)

  • Zhengtian Xu

    (Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, Michigan 48109-1316)

  • Ruijie Li

    (School of Transportation and Logistics, Southwest Jiaotong University, 611756, Sichuan, People’s Republic of China)

  • Hai Yang

    (Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, People’s Republic of China)

  • Yafeng Yin

    (Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, Michigan 48109-1316)

Abstract

This study establishes a general framework for continuous day-to-day models to capture the perceptual errors in travelers’ day-to-day route choice behavior. As the counterpart of the Beckmann transformation, which has been widely used as a candidate Lyapunov function to prove the stability of continuous day-to-day traffic evolution models that converge to deterministic user equilibrium, Fisk’s formulation is utilized in our study as a general Lyapunov function for the day-to-day models that converge to stochastic user equilibrium (SUE), so far as the path flow growth rates and the “potentials” of the paths satisfy the condition of negative correlation. A sufficient condition that guarantees the nonnegativity of the path flow is also provided. The logit dynamic, the logit-based Smith dynamic, and the logit-based Brown-von Neumann-Nash (BNN) dynamic are given as three examples under this framework. Moreover, we extend the second-order day-to-day model proposed by Xiao et al. [Xiao F, Yang H, Ye H (2016) Physics of day-to-day network flow dynamics. Transportation Res. Part B: Methodological 86:86–103.] for SUE. Some properties of the new model, such as fixed point and stability, are investigated. Interestingly, we find that even when the model converges to SUE, the path flows could still go negative during the oscillation under extreme situations. A numerical experiment is conducted to demonstrate the existence of negative path flow for the second-order model.The online appendices are available at https://doi.org/10.1287/trsc.2018.0853 .

Suggested Citation

  • Feng Xiao & Minyu Shen & Zhengtian Xu & Ruijie Li & Hai Yang & Yafeng Yin, 2019. "Day-to-Day Flow Dynamics for Stochastic User Equilibrium and a General Lyapunov Function," Transportation Science, INFORMS, vol. 53(3), pages 683-694, May.
    • Handle: RePEc:inm:ortrsc:v:53:y:2019:i:3:p:683-694
    DOI: 10.1287/trsc.2018.0853
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    References listed on IDEAS

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    3. Minyu Shen & Feng Xiao & Weihua Gu & Hongbo Ye, 2024. "Cognitive Hierarchy in Day-to-day Network Flow Dynamics," Papers 2409.11908, arXiv.org.
    4. Lie Han, 2022. "Proportional-Switch Adjustment Process with Elastic Demand and Congestion Toll in the Absence of Demand Functions," Networks and Spatial Economics, Springer, vol. 22(4), pages 709-735, December.
    5. Jiayang Li & Zhaoran Wang & Yu Marco Nie, 2023. "Wardrop Equilibrium Can Be Boundedly Rational: A New Behavioral Theory of Route Choice," Papers 2304.02500, arXiv.org, revised Feb 2024.
    6. Qi, Hang & Jia, Ning & Qu, Xiaobo & He, Zhengbing, 2023. "Investigating day-to-day route choices based on multi-scenario laboratory experiments, Part I: Route-dependent attraction and its modeling," Transportation Research Part A: Policy and Practice, Elsevier, vol. 167(C).
    7. Ye, Hongbo & Xiao, Feng & Yang, Hai, 2021. "Day-to-day dynamics with advanced traveler information," Transportation Research Part B: Methodological, Elsevier, vol. 144(C), pages 23-44.
    8. Qinghui Xu & Xiangfeng Ji, 2020. "User Equilibrium Analysis Considering Travelers’ Context-Dependent Route Choice Behavior on the Risky Traffic Network," Sustainability, MDPI, vol. 12(17), pages 1-25, August.

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