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

Feng Xiao (), Minyu Shen (), Zhengtian Xu (), Ruijie Li (), Hai Yang and Yafeng Yin ()
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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
Yafeng Yin: Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, Michigan 48109-1316

Transportation Science, 2019, vol. 53, issue 3, 683-694

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 .

Keywords: day-to-day; stochastic user equilibrium; Lyapunov function; stability (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (8)

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Persistent link: https://EconPapers.repec.org/RePEc:inm:ortrsc:v:53:y:2019:i:3:p:683-694

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