Flows Over Time as Continuous Limits of Packet-Based Network Simulations

Ziemke, Theresa; Sering, Leon; Koch, Laura Vargas; Zimmer, Max; Nagel, Kai; Skutella, Martin; Gastaldi, G. · 2021 · Transportation research procedia

DOI: 10.1016/j.trpro.2021.01.014

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Summary

This paper investigates the theoretical relationship between agent-based transport simulations, specifically MATSim, and the mathematical model of Nash flows over time (FOT). The research is motivated by the complementary strengths and weaknesses of these two approaches: MATSim effectively models large-scale, real-world traffic scenarios using discrete vehicles and time steps but yields only approximate user equilibria, whereas the FOT model provides exact user equilibria (Nash flows) with provable mathematical properties but relies on continuous flow and time, limiting its ability to represent detailed traffic features. The central research question is whether Nash flows over time represent the continuous limit of the discrete MATSim simulation as vehicle size and time step size decrease coherently. To address this, the authors compare the structural dynamics of both models, noting that while both utilize deterministic FIFO queues and spatial spillback mechanisms, MATSim operates on discrete agents and time steps, while FOT treats flow as continuous. The study introduces a refinement process using two variables: $\alpha$ for the duration of a time step and $\beta$ for the volume of a single vehicle. The authors hypothesize that for fixed routes, MATSim’s flow model converges to a flow over time when $\beta = \alpha^2$ as both approach zero. This coupling ensures that vehicles are distributed evenly over time and allows for the merging of different commodities, avoiding isolation caused by capacity exhaustion. The analysis focuses on the convergence from discrete to continuous parameters, examining how rounding errors and discretization affect the simulation's alignment with the continuous model. The experiments, conducted on illustrative scenarios and a small real-world instance, demonstrate a strong connection between the two models. The results indicate that as the time step and vehicle size decrease according to the proposed coupling, the discrete MATSim model converges toward the continuous Nash flow over time. This convergence suggests that the discrete simulation can be interpreted as a stochastic realization where the average distribution corresponds to the Nash flow. The findings provide a theoretical foundation for MATSim, justifying its use through the lens of continuous flow theory, and offer a justification for the FOT model by linking it to practical simulation outcomes. The significance of this work lies in bridging the gap between computational simulation and mathematical theory in traffic modeling. By establishing that Nash flows over time are the limit of the MATSim convergence process, the study allows structural insights from the analytical FOT model to be transferred to the user equilibria observed in MATSim. This connection validates the approximate equilibria produced by large-scale simulations and enhances the understanding of dynamic traffic behavior, offering a unified perspective on discrete and continuous traffic modeling approaches.

Key finding

Discrete agent-based transport simulations converge to continuous Nash flows over time as the time step and vehicle size parameters are simultaneously decreased.

Methodology

simulation_modeling

Provenance

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Topics

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• gap acceptance