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A Second Order Stochastic Network Equilibrium Model, I: Theoretical Foundation

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  • David Watling

    (Institute for Transport Studies, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, United Kingdom)

Abstract

Existing models of stochastic network equilibrium route choice in transport networks are able to represent exogenously specified variations in drivers' actual or perceived travel costs, but assume throughout that flows are deterministic. In this paper, a new notion of equilibrium is presented based on stochastic flow variables, in which the impact of variable flows on the variability in travel costs is endogenously handled. Firstly, a very general notion of equilibrium is deduced as a fixed point condition on the joint probability distribution of network flows. Then, an approximation to this condition is derived, which operates by equilibrating moments of order n and below of the joint flow probability distribution, and is termed a Generalized Stochastic User Equilibrium of order n , being denoted GSUE( n ). The GSUE(1) model is seen to be a conventional Stochastic User Equilibrium. The paper goes on to focus on the second order model, GSUE(2). Conditions are presented to guarantee the existence of GSUE(2) solutions. Conditions are deduced to guarantee (a) uniqueness of solutions in networks with a single interzonal movement, and (b) proximity of solutions in networks with multiple interzonal movements. Finally, a simple example is presented.

Suggested Citation

  • David Watling, 2002. "A Second Order Stochastic Network Equilibrium Model, I: Theoretical Foundation," Transportation Science, INFORMS, vol. 36(2), pages 149-166, May.
    • Handle: RePEc:inm:ortrsc:v:36:y:2002:i:2:p:149-166
    DOI: 10.1287/trsc.36.2.149.560
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    References listed on IDEAS

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    Cited by:

    1. Crawford, F. & Watling, D.P. & Connors, R.D., 2017. "A statistical method for estimating predictable differences between daily traffic flow profiles," Transportation Research Part B: Methodological, Elsevier, vol. 95(C), pages 196-213.
    2. Teppei Kato & Kenetsu Uchida & William H. K. Lam & Agachai Sumalee, 2021. "Estimation of the value of travel time and of travel time reliability for heterogeneous drivers in a road network," Transportation, Springer, vol. 48(4), pages 1639-1670, August.
    3. Hazelton, Martin L., 2022. "The emergence of stochastic user equilibria in day-to-day traffic models," Transportation Research Part B: Methodological, Elsevier, vol. 158(C), pages 102-112.
    4. Sumalee, Agachai & Xu, Wei, 2011. "First-best marginal cost toll for a traffic network with stochastic demand," Transportation Research Part B: Methodological, Elsevier, vol. 45(1), pages 41-59, January.
    5. Prakash, A. Arun & Seshadri, Ravi & Srinivasan, Karthik K., 2018. "A consistent reliability-based user-equilibrium problem with risk-averse users and endogenous travel time correlations: Formulation and solution algorithm," Transportation Research Part B: Methodological, Elsevier, vol. 114(C), pages 171-198.
    6. Shanjiang Zhu & David Levinson, 2011. "A Portfolio Theory of Route Choice," Working Papers 000096, University of Minnesota: Nexus Research Group.
    7. Ma, Jie & Meng, Qiang & Cheng, Lin & Liu, Zhiyuan, 2022. "General stochastic ridesharing user equilibrium problem with elastic demand," Transportation Research Part B: Methodological, Elsevier, vol. 162(C), pages 162-194.
    8. Zhang, Chao & Chen, Xiaojun & Sumalee, Agachai, 2011. "Robust Wardrop's user equilibrium assignment under stochastic demand and supply: Expected residual minimization approach," Transportation Research Part B: Methodological, Elsevier, vol. 45(3), pages 534-552, March.
    9. Watling, David, 2006. "User equilibrium traffic network assignment with stochastic travel times and late arrival penalty," European Journal of Operational Research, Elsevier, vol. 175(3), pages 1539-1556, December.
    10. Connors, Richard D. & Sumalee, Agachai, 2009. "A network equilibrium model with travellers' perception of stochastic travel times," Transportation Research Part B: Methodological, Elsevier, vol. 43(6), pages 614-624, July.
    11. Shuang Wang & Jing Lu & Liping Jiang, 2019. "Time Reliability of the Maritime Transportation Network for China’s Crude Oil Imports," Sustainability, MDPI, vol. 12(1), pages 1-18, December.
    12. Li-Jun Tian & Hai-Jun Huang & Zi-You Gao, 2012. "A Cumulative Perceived Value-Based Dynamic User Equilibrium Model Considering the Travelers’ Risk Evaluation on Arrival Time," Networks and Spatial Economics, Springer, vol. 12(4), pages 589-608, December.
    13. Martin L. Hazelton & David P. Watling, 2004. "Computation of Equilibrium Distributions of Markov Traffic-Assignment Models," Transportation Science, INFORMS, vol. 38(3), pages 331-342, August.
    14. David Watling, 2002. "A Second Order Stochastic Network Equilibrium Model, II: Solution Method and Numerical Experiments," Transportation Science, INFORMS, vol. 36(2), pages 167-183, May.
    15. Xie, Chi & Liu, Zugang, 2014. "On the stochastic network equilibrium with heterogeneous choice inertia," Transportation Research Part B: Methodological, Elsevier, vol. 66(C), pages 90-109.

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