  We consider a dynamic model of traffic that has received a lot of attention
in the past few years. Infinitesimally small agents aim to travel from a source
to a destination as quickly as possible. Flow patterns vary over time, and
congestion effects are modeled via queues, which form based on the
deterministic queueing model whenever the inflow into a link exceeds its
capacity. Are equilibria in this model meaningful as a prediction of traffic
behavior? For this to be the case, a certain notion of stability under ongoing
perturbations is needed. Real traffic consists of discrete, atomic ''packets'',
rather than being a continuous flow of non-atomic agents. Users may not choose
an absolutely quickest route available, if there are multiple routes with very
similar travel times. We would hope that in both these situations -- a discrete
packet model, with packet size going to 0, and $\epsilon$-equilibria, with
$\epsilon$ going to 0 -- equilibria converge to dynamic equilibria in the flow
over time model. No such convergence results were known. We show that such a
convergence result does hold in single-commodity instances for both of these
settings, in a unified way. More precisely, we introduce a notion of ''strict''
$\epsilon$-equilibria, and show that these must converge to the exact dynamic
equilibrium in the limit as $\epsilon \to 0$. We then show that results for the
two settings mentioned can be deduced from this with only moderate further
technical effort.
