On the γ-reflected processes with fBm input

Define a γ-reflected process W γ(t) = Y H (t) − γ inf s ∈ [0. t] Y H (s), t ≽ 0, γ ∈ [0, 1], with {Y H (t), t ≽ 0} a fractional Brownian motion with Hurst index H ∈ (0, 1)and negative linear trend. In risk theory, R γ (t)=u-Wγ(t), t ≽ 0, is the risk process with tax of a loss-carry-forward type and initial reserve u ≽ 0 whereas in queueing theory, W 1 is referred to as the queue length process. In this paper, we investigate the ruin probability and the ruin time of R γ over a reserve-dependent time interval.