Long term behavior of dynamic equilibria in fluid queuing networks

TL;DR

This paper proves that under certain conditions, fluid queuing networks reach a steady state in finite time, providing insights into long-term behavior and characterizing equilibrium states as solutions to linear programs.

Contribution

It establishes finite-time convergence to steady state in fluid queuing networks and links equilibrium behavior to linear programming solutions.

Findings

Queue lengths remain bounded under capacity conditions.

Steady state can be characterized as a linear program solution.

Long-term behavior is predictable with unique linear program solutions.

Abstract

A fluid queuing network constitutes one of the simplest models in which to study flow dynamics over a network. In this model we have a single source-sink pair and each link has a per-time-unit capacity and a transit time. A dynamic equilibrium (or equilibrium flow over time) is a flow pattern over time such that no flow particle has incentives to unilaterally change its path. Although the model has been around for almost fifty years, only recently results regarding existence and characterization of equilibria have been obtained. In particular the long term behavior remains poorly understood. Our main result in this paper is to show that, under a natural (and obviously necessary) condition on the queuing capacity, a dynamic equilibrium reaches a steady state (after which queue lengths remain constant) in finite time. Previously, it was not even known that queue lengths would remain bounded. The proof is based on the analysis of a rather non-obvious potential function that turns out to be monotone along the evolution of the equilibrium. Furthermore, we show that the steady state is characterized as an optimal solution of a certain linear program. When this program has a unique solution, which occurs generically, the long term behavior is completely predictable. On the contrary, if the linear program has multiple solutions the steady state is more difficult to identify as it depends on the whole temporal evolution of the equilibrium.