Sojourns of fractional Brownian motion queues: transient asymptotics

TL;DR

This paper analyzes the asymptotic behavior of the sojourn time in fractional Brownian motion queues, deriving precise probability estimates for high thresholds and different Hurst parameters.

Contribution

It provides the first sharp asymptotic approximations for the sojourn times in fractional Brownian motion queues, extending known results from the Brownian case to all Hurst parameters.

Findings

Exact asymptotics for Brownian motion case (H=1/2)

Sharp asymptotic approximations for all H in (0,1)

Identification of regimes based on the ratio of u and h(u)

Abstract

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.