Spillback Changes the Long-Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

Authors Theresa Ziemke , Leon Sering , Kai Nagel

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Theresa Ziemke
  • Combinatorial Optimization and Graph Algorithms, Technische Universität Berlin, Germany
  • Transport Systems Planning and Transport Telematics, Technische Universität Berlin, Germany
Leon Sering
  • Institute for Operations Research, ETH Zürich, Switzerland
Kai Nagel
  • Transport Systems Planning and Transport Telematics, Technische Universität Berlin, Germany


We thank Max Zimmer for technical work on the Nash flow over time computation tool.

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Theresa Ziemke, Leon Sering, and Kai Nagel. Spillback Changes the Long-Term Behavior of Dynamic Equilibria in Fluid Queuing Networks. In 23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023). Open Access Series in Informatics (OASIcs), Volume 115, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the long-term behavior of dynamic traffic equilibria and find that it heavily depends on whether spillback is captured in the traffic model or not. We give an example where no steady state is reached. Although the example consists of a single-commodity instance with constant inflow rate, the Nash flow over time consists of infinitely many phases. This is in contrast to what has been proven for Nash flows over time without spillback [Cominetti et al., 2021; N. Olver et al., 2021]. Additionally, we show that similar phase oscillations as in the Nash flow over time with spillback can be observed in the co-evolutionary transport simulation MATSim. This reaffirms the robustness of the findings as the simulation does (in contrast to Nash flows over time) not lead to exact user equilibra and, moreover, models discrete time steps and vehicles.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Agent / discrete models
  • Mathematics of computing → Network flows
  • Applied computing → Transportation
  • flows over time
  • transport simulation
  • Nash flow
  • dynamic equilibrium
  • long-term behavior
  • steady state
  • oscillation
  • spillback
  • MATSim


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