We study the asymptotics of sojourn time of the stationary queueing process Q(t),t≥0\documentclass[12pt]{minimal}
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\begin{document}$$Q(t),t\ge 0$$\end{document} fed by a fractional Brownian motion with Hurst parameter H∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$H\in (0,1)$$\end{document} above a high threshold u. For the Brownian motion case H=1/2\documentclass[12pt]{minimal}
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\begin{document}$$H=1/2$$\end{document}, we derive the exact asymptotics of P∫T1T2I(Q(t)>u+h(u))dt>x|Q(0)>u\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathbb {P}} \left\{ \int _{T_1}^{T_2}{\mathbb {I}}(Q(t)>u+h(u))d t>x \Big |Q(0) >u \right\} \end{aligned}$$\end{document}as u→∞\documentclass[12pt]{minimal}
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\begin{document}$$u\rightarrow \infty $$\end{document}, where T1,T2,x≥0\documentclass[12pt]{minimal}
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\begin{document}$$T_1,T_2, x\ge 0$$\end{document} and T2-T1>x\documentclass[12pt]{minimal}
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\begin{document}$$T_2-T_1>x$$\end{document}, whereas for all H∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$H\in (0,1)$$\end{document}, we obtain sharp asymptotic approximations of P1v(u)∫[T2(u),T3(u)]I(Q(t)>u+h(u))dt>y|1v(u)∫[0,T1(u)]I(Q(t)>u)dt>x,x,y>0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{} & {} {\mathbb {P}} \left\{ \frac{1}{v(u)} \int _{[T_2(u),T_3(u)]}{\mathbb {I}}(Q(t)\!>\!u\!+\!h(u))dt\!>\!y \Bigl |\frac{1}{v(u)} \int _{[0,T_1(u)]}{\mathbb {I}}(Q(t)\!>\!u)dt\!>\!x \right\} ,\\{} & {} \quad x,y >0 \end{aligned}$$\end{document}as u→∞\documentclass[12pt]{minimal}
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\begin{document}$$u\rightarrow \infty $$\end{document}, for appropriately chosen Ti\documentclass[12pt]{minimal}
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\begin{document}$$T_i$$\end{document}’s and v. Two regimes of the ratio between u and h(u), that lead to qualitatively different approximations, are considered.
