Convergence of a Packet Routing Model to Flows over Time

Mathematical approaches for modeling dynamic traffic can be roughly divided into two categories: discrete packet routing models and continuous flow over time models. Despite very vital research activities on models in both categories, their connection was poorly understood so far. We build this connection by specifying a (competitive) packet routing model, which is discrete in terms of flow and time, and 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 cannot be guaranteed in the multicommodity setting.

Funding: This work was funded by the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy—The Berlin Mathematics Research Center MATH+ [Grant EXC-2046/1, project ID: 390685689] and the Research Training Group [Grant 2236 UnRAVeL].