Dynamic Equilibria in Fluid Queueing Networks

Abstract

Artículo de publicación ISIFlows over time provide a natural and convenient description for the dynamics of a continuous stream of particles traveling from a source to a sink in a network, allowing to track the progress of each infinitesimal particle along time. A basic model for the propagation of flow is the so-called fluid queue model in which the time to traverse an edge is composed of a flow-dependent waiting time in a queue at the entrance of the edge plus a constant travel time after leaving the queue. In a dynamic network routing game each infinitesimal particle is interpreted as a player that seeks to complete its journey in the least possible time. Players are forward looking and anticipate the congestion and queuing delays induced by others upon arrival to any edge in the network. Equilibrium occurs when each particle travels along a shortest path. This paper is concerned with the study of equilibria in the fluid queue model and provides a constructive proof of existence and uniqueness of equilibria in single origin-destination networks with piecewise constant inflow rate. This is done through a detailed analysis of the underlying static flows obtained as derivatives of a dynamic equilibrium. Furthermore, for multicommodity networks, we give a general nonconstructive proof of existence of equilibria when the inflow rates belong to Lp.Nucleo Milenio Informacion y Coordinacion en Rede

Long Term Behavior of Dynamic Equilibria in Fluid Queuing Networks

“…The results are stated without proofs for which we refer to Koch and Skutella [5] and Cominetti et al [1].…”

“…These solutions are called normalized thin flows with resetting (ntfr) and can be used to reconstruct a dynamic equilibrium by integration, proving the existence of equilibria. We refer to [1] for the existence and uniqueness of ntfr's and to [5] for a description of the integration algorithm and how to find the equilibrium inflows f + e (·). Observe that there are only finitely many options for E * and E .…”

“…Substantial progress was recently achieved by Koch and Skutella [5] by introducing the concept of thin flows with resetting that characterize the time derivatives of a dynamic equilibrium, and which provide in turn a method to compute an equilibrium by integration. A slightly refined notion of normalized thin flows with resetting was considered by Cominetti et al [1], who proved existence and uniqueness, and provided a constructive proof for the existence of a dynamic equilibrium.…”

Algorithms for Flows over Time with Scheduling Costs

“…Dynamic equilibria, which is the flow over time equivalent of Wardrop equilibria for static flows, are key objects of study. Existence, uniqueness, structural and algorithmic issues, and much more have been receiving increasing recent interest from the optimization community [4,5,6,7,16,22,23].…”

Dynamic Flows with Adaptive Route Choice

“…In the past years, however, several new exciting developments have emerged: Koch and Skutella [12] elegantly characterized dynamic equilibria by their derivatives, which gives a template for their computation. Subsequently, Cominetti, Correa and Larré [4] derived alternative characterizations and proved existence and uniqueness in terms of experienced travel times of equilibria even for multi-commodity networks. Very recently, Cominetti, Correa and Olver [5] shed light on the behavior of steady state queues assuming single commodity networks and constant inflow rates.…”