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Dynamic Systems, Variational Inequalities and Control Theoretic Models for Predicting Time-Varying Urban Network Flows

Author

Listed:
  • Terry L. Friesz

    (George Mason University, Fairfax, Virginia 22030)

  • David Bernstein

    (Princeton University, Princeton, New Jersey 08544)

  • Roger Stough

    (George Mason University, Fairfax, Virginia 22030)

Abstract

In this paper, we set forth certain axioms for a positive theory of dynamic urban network flows. We then show that mathematical models which fulfill these axioms may be created by adapting and extending certain fundamental notions from microeconomics and nonlinear systems theory. We further show that models created in this fashion, using concepts of fast and slow dynamic processes, may be manipulated into a variety of mathematical forms, thereby providing a synthesis of dynamic systems, variational inequality and control theoretic perspectives for predicting dynamic urban network flows. We close with a discussion of the implications of this synthesis for route guidance and intelligent vehicle infrastructure. Throughout, our presentation is at a conceptual level; the mathematical arguments are purposely not rigorous to embrace the widest possible readership.

Suggested Citation

  • Terry L. Friesz & David Bernstein & Roger Stough, 1996. "Dynamic Systems, Variational Inequalities and Control Theoretic Models for Predicting Time-Varying Urban Network Flows," Transportation Science, INFORMS, vol. 30(1), pages 14-31, February.
    • Handle: RePEc:inm:ortrsc:v:30:y:1996:i:1:p:14-31
    DOI: 10.1287/trsc.30.1.14
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    Cited by:

    1. Li, Jun & Fujiwara, Okitsugu & Kawakami, Shogo, 2000. "A reactive dynamic user equilibrium model in network with queues," Transportation Research Part B: Methodological, Elsevier, vol. 34(8), pages 605-624, November.
    2. Friesz, Terry L. & Mookherjee, Reetabrata, 2006. "Solving the dynamic network user equilibrium problem with state-dependent time shifts," Transportation Research Part B: Methodological, Elsevier, vol. 40(3), pages 207-229, March.
    3. Friesz, Terry L. & Suo, Zhong-Gui & Bernstein, David H., 1998. "A dynamic disequilibrium interregional commodity flow model," Transportation Research Part B: Methodological, Elsevier, vol. 32(7), pages 467-483, September.
    4. Friesz, Terry L. & Mookherjee, Reetabrata & Yao, Tao, 2008. "Securitizing congestion: The congestion call option," Transportation Research Part B: Methodological, Elsevier, vol. 42(5), pages 407-437, June.
    5. Avinash Unnikrishnan & Steven Waller, 2009. "User Equilibrium with Recourse," Networks and Spatial Economics, Springer, vol. 9(4), pages 575-593, December.
    6. G. E. Cantarella & D. P. Watling, 2016. "Modelling road traffic assignment as a day-to-day dynamic, deterministic process: a unified approach to discrete- and continuous-time models," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 69-98, March.
    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.
    8. A. Stathopoulos & T. Tsekeris, 2003. "Framework for analysing reliability and information degradation of demand matrices in extended transport networks," Transport Reviews, Taylor & Francis Journals, vol. 23(1), pages 89-103, January.
    9. Hsun-Jung Cho & Yow-Jen Jou & Chien-Lun Lan, 2009. "Time Dependent Origin-destination Estimation from Traffic Count without Prior Information," Networks and Spatial Economics, Springer, vol. 9(2), pages 145-170, June.
    10. 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.
    11. Zhang, Pengcheng & Peeta, Srinivas, 2014. "Dynamic and disequilibrium analysis of interdependent infrastructure systems," Transportation Research Part B: Methodological, Elsevier, vol. 67(C), pages 357-381.
    12. Wei Nai & Zan Yang & Dan Li & Lu Liu & Yuting Fu & Yuao Guo, 2024. "Urban Day-to-Day Travel and Its Development in an Information Environment: A Review," Sustainability, MDPI, vol. 16(6), pages 1-29, March.
    13. Zhong, R.X. & Xie, X.X. & Luo, J.C. & Pan, T.L. & Lam, W.H.K. & Sumalee, A., 2020. "Modeling double time-scale travel time processes with application to assessing the resilience of transportation systems," Transportation Research Part B: Methodological, Elsevier, vol. 132(C), pages 228-248.
    14. Y. S. Xia & J. Wang, 2000. "On the Stability of Globally Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 106(1), pages 129-150, July.
    15. Friesz, Terry L. & Shah, Samir, 2001. "An overview of nontraditional formulations of static and dynamic equilibrium network design," Transportation Research Part B: Methodological, Elsevier, vol. 35(1), pages 5-21, January.
    16. Farhat Suhel & S. K. Srivastava & Suhel Ahmad Khan, 2014. "A Wiener-Hopf Dynamical System for Mixed Equilibrium Problems," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-8, April.
    17. Bellei, Giuseppe & Gentile, Guido & Papola, Natale, 2005. "A within-day dynamic traffic assignment model for urban road networks," Transportation Research Part B: Methodological, Elsevier, vol. 39(1), pages 1-29, January.
    18. Friesz, Terry L. & Rigdon, Matthew A. & Mookherjee, Reetabrata, 2006. "Differential variational inequalities and shipper dynamic oligopolistic network competition," Transportation Research Part B: Methodological, Elsevier, vol. 40(6), pages 480-503, July.
    19. Y. S. Xia, 2004. "Further Results on Global Convergence and Stability of Globally Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 627-649, September.

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