Queueing Systems ( IF 1.2 ) Pub Date : 2023-09-10 , DOI: 10.1007/s11134-023-09890-y Krzysztof Dȩbicki , Enkelejd Hashorva , Peng Liu
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 \(H=1/2\), we derive the exact asymptotics of
$$\begin{aligned} {\mathbb {P}} \left\{ \int _{T_1}^{T_2}{\mathbb {I}}(Q(t)>u+h(u))d t>x \Big |Q(0) >u \right\} \end{aligned}$$as \(u\rightarrow \infty \), 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
$$\begin{aligned}{} & {} {\mathbb {P}} \left\{ \frac{1}{v(u)} \int _{[T_2(u),T_3(u)]}{\mathbb {I}}(Q(t)\!>\!u\!+\!h(u))dt\!>\!y \Bigl |\frac{1}{v(u)} \int _{[0,T_1(u)]}{\mathbb {I}}(Q(t)\!>\!u)dt\!>\!x \right\} ,\\{} & {} \quad x,y >0 \end{aligned}$$as \(u\rightarrow \infty \), 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.
中文翻译：
分数布朗运动队列的逗留：瞬态渐近
我们研究由赫斯特参数\(H\in (0,1)\ )高于高阈值的分数布朗运动驱动的静态排队过程\(Q(t),t\ge 0\)停留时间的渐近性你。对于布朗运动情况\(H=1/2\)，我们推导出精确的渐近方程
$$\begin{对齐} {\mathbb {P}} \left\{ \int _{T_1}^{T_2}{\mathbb {I}}(Q(t)>u+h(u))d t> x \Big |Q(0) >u \right\} \end{对齐}$$为\(u\rightarrow \infty \)，其中\(T_1,T_2, x\ge 0\)和\(T_2-T_1>x\)，而对于所有\(H\in (0,1)\)，我们获得锐渐近近似
$$\begin{对齐}{} & {} {\mathbb {P}} \left\{ \frac{1}{v(u)} \int _{[T_2(u),T_3(u)]} {\mathbb {I}}(Q(t)\!>\!u\!+\!h(u))dt\!>\!y \Bigl |\frac{1}{v(u)} \ int _{[0,T_1(u)]}{\mathbb {I}}(Q(t)\!>\!u)dt\!>\!x \right\} ,\\{} & {} \quad x,y >0 \end{对齐}$$作为\(u\rightarrow \infty \)，对于适当选择的\(T_i\) 's 和v。考虑了u和h ( u )之间的比率的两种情况，这两种情况会导致质量上不同的近似值。
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