Sojourns of fractional Brownian motion queues: transient asymptotics

We study the asymptotics of sojourn time of the stationary queueing process $$Q(t),t\ge 0$$ Q ( t ) , t ≥ 0 fed by a fractional Brownian motion with Hurst parameter $$H\in (0,1)$$ H ∈ ( 0 , 1 ) above a high threshold u. For the Brownian motion case $$H=1/2$$ H = 1 / 2 , we derive the exact asymptotics of $$\begin{aligned} {\mathbb {P}} \left\{ \int _{T_1}^{T_2}{\mathbb {I}}(Q(t)>u+h(u))d t>x \Big |Q(0) >u \right\} \end{aligned}$$ P ∫ T 1 T 2 I ( Q ( t ) > u + h ( u ) ) d t > x | Q ( 0 ) > u as $$u\rightarrow \infty $$ u → ∞ , where $$T_1,T_2, x\ge 0$$ T 1 , T 2 , x ≥ 0 and $$T_2-T_1>x$$ T 2 - T 1 > x , whereas for all $$H\in (0,1)$$ H ∈ ( 0 , 1 ) , we obtain sharp asymptotic approximations of $$\begin{aligned}{} & {} {\mathbb {P}} \left\{ \frac{1}{v(u)} \int _{[T_2(u),T_3(u)]}{\mathbb {I}}(Q(t)\!>\!u\!+\!h(u))dt\!>\!y \Bigl |\frac{1}{v(u)} \int _{[0,T_1(u)]}{\mathbb {I}}(Q(t)\!>\!u)dt\!>\!x \right\} ,\\{} & {} \quad x,y >0 \end{aligned}$$ P 1 v ( u ) ∫ [ T 2 ( u ) , T 3 ( u ) ] I ( Q ( t ) > u + h ( u ) ) d t > y | 1 v ( u ) ∫ [ 0 , T 1 ( u ) ] I ( Q ( t ) > u ) d t > x , x , y > 0 as $$u\rightarrow \infty $$ u → ∞ , for appropriately chosen $$T_i$$ T i ’s and v. Two regimes of the ratio between u and h(u), that lead to qualitatively different approximations, are considered.