Sojourns of fractional Brownian motion queues: transient asymptotics

We study the asymptotics of sojourn time of the stationary queueing process $Q(t),t\ge 0$ fed by a fractional Brownian motion with Hurst parameter $H\in (0,1)$ above a high threshold $u$. For the Brownian motion case $H=1\mathrm{/}2$, 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 $u\to \mathrm{\infty }$, {where $T_1,T_2,x\ge 0$ and $T_2-T_1>x$}, whereas for all $H\in (0,1)$, we obtain sharp asymptotic approximations of $$P(\frac{1}{v(u)}{\int }_{[T_2(u),T_3(u)]}1(Q(t)>u+h(u))dt>y\mid \frac{1}{v(u)}{\int }_{[0,T_1(u)]}1(Q(t)>u)dt>x),x,y>0$$ as $u\to \mathrm{\infty }$, 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.