In the analysis of stochastic systems, estimates for the speed of convergence to stationarity play a crucial role. In many applications one is interested in the distribution of  $M_\infty$, where $M_t := \sup_{s\in [0,t]}X(t) - t$, for some centered stochastic process (for instance fractional Brownian motion (fBm), or a L\'evy process).

One of these applications is the following: in order to estimate ${\mathbb P}(M_\infty>x)$ by simulation, one needs to determine a simulation horizon $T$ such that the difference between $M_\infty$ and $M_T$ is, in some metric, negligible.
In the first part of my talk I present results on the decay rate (in $T$) of several metrics for the special case of fBm. More concretely, I show that the distance behaves as $\exp(\gamma T^{2H2})$, where $\gamma$can be translated in terms of the asymptotics of long busy periods. These busyperiod asymptotics are nontrivial, and are essentially determined by the most likely way in which a long busy period occurs; we show that this path has a rather unexpected shape.

Time permitting, I'll conclude by focusing on the correlation structure of reflected fBm, and corresponding transient characteristics. The main result is that certain correlation measures decay in the same as the input process. This means that, in this respect, the queueing process inherits the longrange dependent properties of the input process.