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Computation of Dynamic Equilibria in Series-Parallel Networks

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  • Marcus Kaiser

    (Lehrstuhl für Operations Research, Technische Universität München, 80333 München, Germany)

Abstract

We consider dynamic equilibria for flows over time under the fluid queuing model. In this model, queues on the links of a network take care of flow propagation. Flow enters the network at a single source and leaves at a single sink. In a dynamic equilibrium, every infinitesimally small flow particle reaches the sink as early as possible given the pattern of the rest of the flow. Although this model has been examined for many decades, progress has been relatively recent. In particular, the derivatives of dynamic equilibria have been characterized as thin flows with resetting, which allows for more structural results. Our two main results are based on the formulation of thin flows with resetting as a linear complementarity problem and its analysis. We present a constructive proof of existence for dynamic equilibria if the inflow rate is right-monotone. The complexity of computing thin flows with resetting, which occurs as a subproblem in this method, is still open. We settle it for the class of two-terminal, series-parallel networks by giving a recursive algorithm that solves the problem for all flow values simultaneously in polynomial time.

Suggested Citation

  • Marcus Kaiser, 2022. "Computation of Dynamic Equilibria in Series-Parallel Networks," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 50-71, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:50-71
    DOI: 10.1287/moor.2020.1108
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    References listed on IDEAS

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    Cited by:

    1. Lukas Graf & Tobias Harks, 2023. "The Price of Anarchy for Instantaneous Dynamic Equilibria," Mathematics of Operations Research, INFORMS, vol. 48(4), pages 2167-2195, November.
    2. Oosterwijk, Tim & Schmand, Daniel & Schröder, Marc, 2024. "Bicriteria Nash flows over time," Games and Economic Behavior, Elsevier, vol. 147(C), pages 19-37.
    3. Leon Sering & Laura Vargas Koch & Theresa Ziemke, 2023. "Convergence of a Packet Routing Model to Flows over Time," Mathematics of Operations Research, INFORMS, vol. 48(3), pages 1741-1766, August.

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