# The relation of discrete and continuous competitive flow over time models

To model competitive movement of agents through a network over time, e.g. situations as in a traffic scenario, different approaches are used. One is competitive packet routing (link) and another is Nash flows over time. While competitive packet routing models time and vehicle size in discrete units, flow and time is modeled continuously in Nash flows over time. An other approach which is closely related to both theoretical models mentioned here, is the traffic simulation software MATSIM. It uses a packet-based model, while the arc dynamics are closer to the ones examined in Nash flows over time. To understand the relation of these models is of big interest, first from a theoretical perspective and second also from a practical perspective. Nash flows over time can be analyzed theoretically while MATSim provides a very complete model which is hard to understand and analyze. Thus, a close relation could for instance allow to transfer theoretical properties of Nash flows over time to MATSim. Currently we develop a packet based model where the edge dynamics correspond to those of Nash flows over time and are thus very similar to MATSim. Furthermore, we analyze convergence of this model for decreasing packet and time step size. Beside proving the connection of MATSim and Nash flows over time, we expect that this analysis could lead to further theoretical insights in Nash flows over time. Additionally we analyzed convergence of MATSim to a (Nash) flow over time experimentally. This is joint work of Kai Nagel, Leon Sering, Martin Skutella, Theresa Thunig, Laura Vargas Koch.

One important step in this context was to extend Nash flows over time to a variant including spillback. Spillback is a real-world traffic phenomena and an important feature of MATSim. An extension where kinematic waves are included is in the reviewing process.

[1] Leon Sering, Laura Vargas Koch. Nash flows over time with spillback. 2019