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

Ziemke, Theresa; Sering, Leon; Koch, Laura Vargas; Zimmer, Max; Nagel, Kai; Skutella, Martin

doi:10.1016/j.trpro.2021.01.014

Cited by 10 publications

(5 citation statements)

References 7 publications

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“…Additional properties of this model have been shown by Israel and Sering (2020). Ziemke et al recently presented experiments that indicate a strong connection of the limit of the MATSim flow model for decreasing vehicle and time step size and Nash flows over time with spillback (Ziemke et al 2021). Their article provides a strong justification and motivation for the studies of mathematical traffic models.…”

Section: Introductionmentioning

confidence: 82%

Recent Developments in Mathematical Traffic Models

Schmand

2021

Dynamics in Logistics

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Predictions such as forecasts of congestion effects in transportation networks can be based on complex simulations that include many aspects of actual transportation systems. On the other hand, rigorous mathematical traffic models give rise to theoretical analyses, very general statements, and various traffic optimization opportunities. There has been a huge development in the last years to make mathematical traffic models more realistic. This chapter provides an overview of the mathematical traffic models developed recently and some state-of-the-art results.

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Section: Introductionmentioning

confidence: 82%

Recent Developments in Mathematical Traffic Models

Schmand

2021

Dynamics in Logistics

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“…Our bound on the discretization error increases exponentially in time. Experiments [30] indicate that there might be a time independent bound though. Can a better upper and maybe also a lower bound on the worst case error be shown mathematically?…”

Section: Existence Of Approximate Pure Nash Equilibriamentioning

confidence: 99%

“…A coevolutionary algorithm is used to compute approximate user equilibria. Despite the different perspectives of the two approaches, experiments by Ziemke et al [30] indicate that Nash flows over time are the limits of the convergence processes when decreasing the vehicle size and time step length in the simulation coherently. This motivates the question whether the convergence of the flow models can be verified and proven mathematically.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a Packet Routing Model to Flows Over Time

Sering

Koch

Ziemke

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

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The mathematical approaches for modeling dynamic traffic can roughly be divided into two categories: discrete packet routing models and continuous flow over time models. Despite very vital research activities on models in both categories, the connection between these approaches was poorly understood so far. In this work we build this connection by specifying a (competitive) packet routing model, which is discrete in terms of flow and time, and by proving its convergence to the intensively studied model of flows over time with deterministic queuing. More precisely, we prove that the limit of the convergence process, when decreasing the packet size and time step length in the packet routing model, constitutes a flow over time with multiple commodities. In addition, we show that the convergence result implies the existence of approximate equilibria in the competitive version of the packet routing model. This is of significant interest as exact pure Nash equilibria, similar to almost all other competitive models, cannot be guaranteed in the multi-commodity setting.Moreover, the introduced packet routing model with deterministic queuing is very application-oriented as it is based on the network loading module of the agent-based transport simulation MATSim. As the present work is the first mathematical formalization of this simulation, it provides a theoretical foundation and an environment for provable mathematical statements for MATSim. CCS Concepts: • Theory of computation → Algorithmic game theory; Network games; Network flows; Exact and approximate computation of equilibria; Routing and network design problems; • Mathematics of computing → Network flows.

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“…Theorem 1: For single-sink networks with piecewise constant inflow rates with bounded support, there is an α-extension algorithm computing an IDE 1 Algorithms for DE or IDE computation used in the transportation science literature are numerical, that is, only approximate equilibrium flows are computed given a certain numerical precision using a discretized model, see for example [1,6,11]. While a recent computational study [24] showed some positive results in regards to convergence for DE, Otsubo and Rapoport [18] also reported "significant discrepancies" between the continuous and a discretized solution for the Vickrey model.…”

Section: Our Contribution and Proof Techniquesmentioning

confidence: 99%

A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows

Graf

Harks

2021

Integer Programming and Combinatorial Optimization

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Instantaneous dynamic equilibrium (IDE) is a standard game-theoretic concept in dynamic traffic assignment in which individual flow particles myopically select en route currently shortest paths towards their destination. We analyze IDE within the Vickrey bottleneck model, where current travel times along a path consist of the physical travel times plus the sum of waiting times in all the queues along a path. Although IDE have been studied for decades, no exact finite time algorithm for equilibrium computation is known to date. As our main result we show that a natural extension algorithm needs only finitely many phases to converge leading to the first finite time combinatorial algorithm computing an IDE. We complement this result by several hardness results showing that computing IDE with natural properties is NP-hard.

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

Cited by 10 publications

References 7 publications

Recent Developments in Mathematical Traffic Models

Recent Developments in Mathematical Traffic Models

Convergence of a Packet Routing Model to Flows Over Time

A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows

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