  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.
