Convergence of a Packet Routing Model to Flows Over Time

Abstract

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.

Convergence of a Packet Routing Model to Flows over Time

“…Positive experimental evidence for convergence was found in [ZSV + 21]. In [SVZ21], convergence was shown for a fixed choice of paths for all packets (in an appropriate sense). This already involves some significant technicalities, but their result does not say anything about the relationship between equilibria in the two settings.…”

Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

“…This has been studied from the perspective of algorithms and optimization (e.g., [LMR94,LMR99]) and in various variants for self-interested strategic users. While e.g., [CCCW17,Ism17,SST18,SVKZ21] examine the discrete version of the deterministic queueing model, Hoefer et al [HMRT11] (see also [KM15]) introduce temporal congestion games, where arcs represent machines that must process a packet (job) before it can continue onto a next machine and [HPS + 18, SSVK18] examine different rules to handle the interaction of the packets. The latter two variants do not have a first-in-first-out requirement, and instead the emphasis is on obtaining good scheduling rules.…”

“…No such rigorous justification has been made so far, however: empirical evidence has been provided [ZSV + 21], and rigorous results have been obtained for the static model [CSSSM21]. As some first steps, the convergence of the underlying packet model to the fluid queueing model has been proven (without considering equilibrium behavior) [SVKZ21]. We anticipate our work being a main ingredient in proving convergence of equilibria; it should be clear that without continuity, there is little hope for such a convergence result.…”