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An ordinary differential equation formulation of the bottleneck model with user heterogeneity

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  • Wu, Wen-Xiang
  • Huang, Hai-Jun

Abstract

Considering heterogeneous values of time and schedule delay early, in this paper we develop an ordinary differential equation formulation of the bottleneck model without allowance of arriving late. We show that in no-toll equilibrium, the generalized travel cost increases with departure time as the ratio of value of schedule delay to value of time increases with these two values, and commuters with higher values would experience higher generalized costs. We then derive the first-best toll and analyses its efficiency and distributional impacts. We obtain the sufficient Pareto-improving condition for the first-best scheme without imposing specific functional forms on heterogeneity. It is a Pareto improvement when the gap between values of time and schedule delay early increases as the ratio of value of schedule delay early to value of time increases. The proposed approach is applied to deal with such user heterogeneity that both the values of time and schedule delay early follow uniform distributions. In this case, the Pareto-improving condition is not only sufficient but also necessary. The individual gain from tolling is strictly monotonically increasing with respect to values of time and schedule delay early. Outside this condition, the first-best toll scheme makes users with relatively higher values of time and schedule delay early better off and those with relatively lower values worse off. And, users with higher values of time and schedule delay early could gain more or lose less from tolling. In contrasts with previous literature, our approach is robust in the sense that it can deal with not only discrete but also continuous user heterogeneity.

Suggested Citation

  • Wu, Wen-Xiang & Huang, Hai-Jun, 2015. "An ordinary differential equation formulation of the bottleneck model with user heterogeneity," Transportation Research Part B: Methodological, Elsevier, vol. 81(P1), pages 34-58.
  • Handle: RePEc:eee:transb:v:81:y:2015:i:p1:p:34-58
    DOI: 10.1016/j.trb.2015.08.007
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    4. Li, Zhi-Chun & Lam, William H.K. & Wong, S.C., 2017. "Step tolling in an activity-based bottleneck model," Transportation Research Part B: Methodological, Elsevier, vol. 101(C), pages 306-334.
    5. Amirgholy, Mahyar & Gao, H. Oliver, 2017. "Modeling the dynamics of congestion in large urban networks using the macroscopic fundamental diagram: User equilibrium, system optimum, and pricing strategies," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 215-237.
    6. Xiao, Ling-Ling & Liu, Tian-Liang & Huang, Hai-Jun & Liu, Ronghui, 2021. "Temporal-spatial allocation of bottleneck capacity for managing morning commute with carpool," Transportation Research Part B: Methodological, Elsevier, vol. 143(C), pages 177-200.
    7. Li, Chuan-Yao & Huang, Hai-Jun, 2018. "User equilibrium of a single-entry traffic corridor with continuous scheduling preference," Transportation Research Part B: Methodological, Elsevier, vol. 108(C), pages 21-38.
    8. Ren-Yong Guo & Hai Yang & Hai-Jun Huang, 2018. "Are We Really Solving the Dynamic Traffic Equilibrium Problem with a Departure Time Choice?," Transportation Science, INFORMS, vol. 52(3), pages 603-620, June.
    9. Amirgholy, Mahyar & Shahabi, Mehrdad & Gao, H. Oliver, 2017. "Optimal design of sustainable transit systems in congested urban networks: A macroscopic approach," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 103(C), pages 261-285.
    10. Fu, Xinying & van den Berg, Vincent A.C. & Verhoef, Erik T., 2018. "Private road supply in networks with heterogeneous users," Transportation Research Part A: Policy and Practice, Elsevier, vol. 118(C), pages 430-443.
    11. Chen, Zhi & Wu, Wen-Xiang & Huang, Hai-Jun & Shang, Hua-Yan, 2022. "Modeling traffic dynamics in periphery-downtown urban networks combining Vickrey's theory with Macroscopic Fundamental Diagram: user equilibrium, system optimum, and cordon pricing," Transportation Research Part B: Methodological, Elsevier, vol. 155(C), pages 278-303.
    12. Vincent van den Berg, "undated". "Self-financing roads under coarse tolling and heterogeneous preferences," Tinbergen Institute Discussion Papers 22-045/VIII, Tinbergen Institute.
    13. Liu, Wei & Zhang, Fangni & Yang, Hai, 2017. "Modeling and managing morning commute with both household and individual travels," Transportation Research Part B: Methodological, Elsevier, vol. 103(C), pages 227-247.
    14. Li, Zhi-Chun & Huang, Hai-Jun & Yang, Hai, 2020. "Fifty years of the bottleneck model: A bibliometric review and future research directions," Transportation Research Part B: Methodological, Elsevier, vol. 139(C), pages 311-342.
    15. Long, Jiancheng & Tan, Weimin & Szeto, W.Y. & Li, Yao, 2018. "Ride-sharing with travel time uncertainty," Transportation Research Part B: Methodological, Elsevier, vol. 118(C), pages 143-171.
    16. Long, Jiancheng & Szeto, W.Y., 2019. "Congestion and environmental toll schemes for the morning commute with heterogeneous users and parallel routes," Transportation Research Part B: Methodological, Elsevier, vol. 129(C), pages 305-333.
    17. Li, Chuan-Yao & Huang, Hai-Jun, 2017. "Morning commute in a single-entry traffic corridor with early and late arrivals," Transportation Research Part B: Methodological, Elsevier, vol. 97(C), pages 23-49.

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