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

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 the authors' 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, the authors 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, the authors 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

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    • Abstracts reprinted with permission of INFORMS (Institute for Operations Research and the Management Sciences,
  • Authors:
    • Xiao, Feng
    • Shen, Minyu
    • Xu, Zhengtian
    • Li, Ruijie
    • Yang, Hai
    • Yin, Yafeng
  • Publication Date: 2019-5


  • English

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  • Accession Number: 01714935
  • Record Type: Publication
  • Files: TRIS
  • Created Date: Jun 17 2019 3:36PM