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Large buffer asymptotics for the queue with fractional Brownian input. (English) Zbl 0955.60096
Let \(\{Z_t\}\) be a standard (centered) fractional Brownian motion with Hurst parameter \(H\in [1/2,1)\), \(W_t= mt+\sigma Z_t\), \(t\geq 0\), \(m>0\), \(\sigma> 0\), and \(V_0= \sup_{t>0} (W_t- ct)\). The distribution of \(V_0\) for \(H\neq 1/2\) has not any explicit expression; however there are some results dealing with an asymptotic of \(P(V_0> x)\), as \(x\to\infty\). The authors derive an exact equivalent of \(P(V_0> x)\), as \(x\to\infty\). They show, in particular, that \(\lim_{x\to\infty} P(V_0> x)/x^{- \gamma} e^{-\kappa^2x^{2(1- H)}/2}\leq L\), where \(\gamma> 0\) if \(1/2< H< 1\), while \(L\) is some constant.

60K25 Queueing theory (aspects of probability theory)
60G15 Gaussian processes
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