  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.
