Sojourns of fractional Brownian motion queues: transient asymptotics

2023/08/29 by Dȩbicki, Krzysztof, Hashorva, Enkelejd, Liu, Peng · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2308.15662

Abstract

We study the asymptotics of sojourn time of the stationary queueing process Q(t),t≥0 fed by a fractional Brownian motion with Hurst parameter H∈(0,1) above a high threshold u. For the Brownian motion case H=1/2, we derive the exact asymptotics of P(∫_T1^T₂ 1(Q(t)gt;u+h(u))d tgt;x |Q(0) gt;u ) as u→∞, where T₁,T₂, x≥ 0 and T₂-T₁>x, whereas for all H∈(0,1), we obtain sharp asymptotic approximations of P( \frac 1 v(u) ∫_{[T2(u),T3(u)]}1(Q(t)gt;u+h(u))dtgt;y | \frac 1 v(u) ∫_[0,T1(u)]1(Q(t)gt;u)dtgt;x), x,y gt;0 as u→∞, 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.

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