  We study the asymptotics of sojourn time of the stationary queueing process
$Q(t),t\ge0$ fed by a fractional Brownian motion with Hurst parameter
$H\in(0,1)$ above a high threshold $u$. For the Brownian motion case $H=1/2$,
we derive the exact asymptotics of
  \[
  P\left(\int_{T_1}^{T_2} 1(Q(t)>u+h(u))d t>x \Big{|}Q(0) >u \right)
  \]
  as $u\to\infty$, {where $T_1,T_2, x\geq 0$ and $T_2-T_1>x$}, whereas for all
$H\in(0,1)$, we obtain sharp asymptotic approximations of
  \[
  P\left( \frac 1 {v(u)} \int_{[T_2(u),T_3(u)]}1(Q(t)>u+h(u))dt>y \Bigl \lvert
\frac 1 {v(u)} \int_{[0,T_1(u)]}1(Q(t)>u)dt>x\right), \quad x,y >0
  \]
  as $u\to\infty$, for appropriately chosen $T_i$'s and $v$. Two regimes of the
ratio between $u$ and $h(u)$, that lead to qualitatively different
approximations, are considered.
