IDEAS home Printed from https://ideas.repec.org/a/inm/ortrsc/v53y2019i3p683-694.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/trsc.2018.0853
    Download Restriction: no
    File URL: https://libkey.io/10.1287/trsc.2018.0853?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Fisk, Caroline, 1980. "Some developments in equilibrium traffic assignment," Transportation Research Part B: Methodological, Elsevier, vol. 14(3), pages 243-255, September.
    2. Fudenberg, Drew & Levine, David, 1998. "Learning in games," European Economic Review, Elsevier, vol. 42(3-5), pages 631-639, May.
    3. Terry L. Friesz & David Bernstein & Nihal J. Mehta & Roger L. Tobin & Saiid Ganjalizadeh, 1994. "Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems," Operations Research, INFORMS, vol. 42(6), pages 1120-1136, December.
    4. Anna Nagurney & Ding Zhang, 1997. "Projected Dynamical Systems in the Formulation, Stability Analysis, and Computation of Fixed-Demand Traffic Network Equilibria," Transportation Science, INFORMS, vol. 31(2), pages 147-158, May.
    5. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
    6. Guo, Ren-Yong & Yang, Hai & Huang, Hai-Jun & Tan, Zhijia, 2015. "Link-based day-to-day network traffic dynamics and equilibria," Transportation Research Part B: Methodological, Elsevier, vol. 71(C), pages 248-260.
    7. G. E. Cantarella & E. Cascetta, 1995. "Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory," Transportation Science, INFORMS, vol. 29(4), pages 305-329, November.
    8. Kumar, Amit & Peeta, Srinivas, 2015. "A day-to-day dynamical model for the evolution of path flows under disequilibrium of traffic networks with fixed demand," Transportation Research Part B: Methodological, Elsevier, vol. 80(C), pages 235-256.
    9. D. Zhang & A. Nagurney, 1997. "Formulation, Stability, and Computation of Traffic Network Equilibria as Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 417-444, May.
    10. Smith, M. J., 1983. "The existence and calculation of traffic equilibria," Transportation Research Part B: Methodological, Elsevier, vol. 17(4), pages 291-303, August.
    11. Yang, Fan & Zhang, Ding, 2009. "Day-to-day stationary link flow pattern," Transportation Research Part B: Methodological, Elsevier, vol. 43(1), pages 119-126, January.
    12. Jin, Wen-Long, 2007. "A dynamical system model of the traffic assignment problem," Transportation Research Part B: Methodological, Elsevier, vol. 41(1), pages 32-48, January.
    13. Bie, Jing & Lo, Hong K., 2010. "Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation," Transportation Research Part B: Methodological, Elsevier, vol. 44(1), pages 90-107, January.
    14. Watling, David, 1999. "Stability of the stochastic equilibrium assignment problem: a dynamical systems approach," Transportation Research Part B: Methodological, Elsevier, vol. 33(4), pages 281-312, May.
    15. Peeta, Srinivas, 2016. "A marginal utility day-to-day traffic evolution model based on one-step strategic thinkingAuthor-Name: He, Xiaozheng," Transportation Research Part B: Methodological, Elsevier, vol. 84(C), pages 237-255.
    16. Carlos F. Daganzo & Yosef Sheffi, 1977. "On Stochastic Models of Traffic Assignment," Transportation Science, INFORMS, vol. 11(3), pages 253-274, August.
    17. Watling, David P., 2016. "A route-swapping dynamical system and Lyapunov function for stochastic user equilibriumAuthor-Name: Smith, Michael J," Transportation Research Part B: Methodological, Elsevier, vol. 85(C), pages 132-141.
    18. Horowitz, Joel L., 1984. "The stability of stochastic equilibrium in a two-link transportation network," Transportation Research Part B: Methodological, Elsevier, vol. 18(1), pages 13-28, February.
    19. Michael J. Smith, 1984. "The Stability of a Dynamic Model of Traffic Assignment---An Application of a Method of Lyapunov," Transportation Science, INFORMS, vol. 18(3), pages 245-252, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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).
    3. 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.
    4. Xu, Xiangdong & Qu, Kai & Chen, Anthony & Yang, Chao, 2021. "A new day-to-day dynamic network vulnerability analysis approach with Weibit-based route adjustment process," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 153(C).
    5. Zhu, Zheng & Mardan, Atabak & Zhu, Shanjiang & Yang, Hai, 2021. "Capturing the interaction between travel time reliability and route choice behavior based on the generalized Bayesian traffic model," Transportation Research Part B: Methodological, Elsevier, vol. 143(C), pages 48-64.
    6. 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.
    7. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ren-Yong Guo & Hai Yang & Hai-Jun Huang & Zhijia Tan, 2016. "Day-to-Day Flow Dynamics and Congestion Control," Transportation Science, INFORMS, vol. 50(3), pages 982-997, August.
    2. 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.
    3. Han, Linghui & Wang, David Z.W. & Lo, Hong K. & Zhu, Chengjuan & Cai, Xingju, 2017. "Discrete-time day-to-day dynamic congestion pricing scheme considering multiple equilibria," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 1-16.
    4. Peeta, Srinivas, 2016. "A marginal utility day-to-day traffic evolution model based on one-step strategic thinkingAuthor-Name: He, Xiaozheng," Transportation Research Part B: Methodological, Elsevier, vol. 84(C), pages 237-255.
    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. Xiaomei Zhao & Chunhua Wan & Jun Bi, 2019. "Day-to-Day Assignment Models and Traffic Dynamics Under Information Provision," Networks and Spatial Economics, Springer, vol. 19(2), pages 473-502, June.
    7. Ren-Yong Guo & Hai-Jun Huang & Hai Yang, 2019. "Tradable Credit Scheme for Control of Evolutionary Traffic Flows to System Optimum: Model and its Convergence," Networks and Spatial Economics, Springer, vol. 19(3), pages 833-868, September.
    8. Sun, Mingmei, 2023. "A day-to-day dynamic model for mixed traffic flow of autonomous vehicles and inertial human-driven vehicles," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 173(C).
    9. Guo, Ren-Yong & Yang, Hai & Huang, Hai-Jun & Tan, Zhijia, 2015. "Link-based day-to-day network traffic dynamics and equilibria," Transportation Research Part B: Methodological, Elsevier, vol. 71(C), pages 248-260.
    10. Liu, Wei & Geroliminis, Nikolas, 2017. "Doubly dynamics for multi-modal networks with park-and-ride and adaptive pricing," Transportation Research Part B: Methodological, Elsevier, vol. 102(C), pages 162-179.
    11. Kumar, Amit & Peeta, Srinivas, 2015. "A day-to-day dynamical model for the evolution of path flows under disequilibrium of traffic networks with fixed demand," Transportation Research Part B: Methodological, Elsevier, vol. 80(C), pages 235-256.
    12. He, Xiaozheng & Guo, Xiaolei & Liu, Henry X., 2010. "A link-based day-to-day traffic assignment model," Transportation Research Part B: Methodological, Elsevier, vol. 44(4), pages 597-608, May.
    13. Wen-yi Zhang & Wei Guan & Ji-hui Ma & Jun-fang Tian, 2015. "A Nonlinear Pairwise Swapping Dynamics to Model the Selfish Rerouting Evolutionary Game," Networks and Spatial Economics, Springer, vol. 15(4), pages 1075-1092, December.
    14. Li, Ruijie & Liu, Xiaobo & Nie, Yu (Marco), 2018. "Managing partially automated network traffic flow: Efficiency vs. stability," Transportation Research Part B: Methodological, Elsevier, vol. 114(C), pages 300-324.
    15. 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.
    16. Wang, Jian & He, Xiaozheng & Peeta, Srinivas, 2016. "Sensitivity analysis based approximation models for day-to-day link flow evolution process," Transportation Research Part B: Methodological, Elsevier, vol. 92(PA), pages 35-53.
    17. Li, Pengbo & Tian, Lijun & Xiao, Feng & Zhu, Hongwei, 2022. "Can day-to-day dynamic model be solved analytically? New insights on portraying equilibrium and accommodating autonomous vehicles," Transportation Research Part B: Methodological, Elsevier, vol. 166(C), pages 374-395.
    18. Huijun Sun & Si Zhang & Linghui Han & Xiaomei Zhao & Lu Lou, 2020. "Day-to-Day Evolution Model Based on Dynamic Reference Point with Heterogeneous Travelers," Networks and Spatial Economics, Springer, vol. 20(4), pages 935-961, December.
    19. 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).
    20. Liu, Peng & Liao, Feixiong & Tian, Qiong & Huang, Hai-Jun & Timmermans, Harry, 2020. "Day-to-day needs-based activity-travel dynamics and equilibria in multi-state supernetworks," Transportation Research Part B: Methodological, Elsevier, vol. 132(C), pages 208-227.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ortrsc:v:53:y:2019:i:3:p:683-694. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.