Long term behavior of dynamic equilibria in fluid queuing networks

12/20/2021
by   Roberto Cominetti, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/26/2020

Computation of Dynamic Equilibria in Series-Parallel Networks

We consider dynamic equilibria for flows over time under the fluid queui...
research
11/12/2021

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

We consider a dynamic model of traffic that has received a lot of attent...
research
09/06/2023

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

The steady-state solution of fluid flow in pipeline infrastructure netwo...
research
12/25/2019

Large fork-join networks with nearly deterministic service times

In this paper, we study an N server fork-join queueing network with near...
research
04/20/2019

A Combinatorial Algorithm for the Multi-commodity Flow Problem

This paper researches combinatorial algorithms for the multi-commodity f...
research
12/31/2019

The MAP/M/s+G Call Center Model with General Patience Times: Stationary Solutions and First Passage Times

We study the MAP/M/s+G queuing model with MAP (Markovian Arrival Process...
research
07/15/2020

A Finite Time Combinatorial Algorithm for Instantaneous Dynamic Equilibrium Flows

Instantaneous dynamic equilibrium (IDE) is a standard game-theoretic con...

Please sign up or login with your details

Forgot password? Click here to reset