Dynamic equilibria in fluid queueing networks

Flows 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 L^p.