We consider a d-dimensional reflected fractional Brownian motion (RFBM)
process on the positive orthant
S = R+d, with drift
r0 ∈ Rd and Hurst parameter
H ∈ (½, 1). Under a natural stability condition on the
drift vector r0 and reflection directions, we establish a
return time result for the RFBM process Z; that is, for some
δ, κ > 0,
supx∈BEx[τB(δ)] < ∞,
where B = {x ∈ S : |x| ≤ κ}
and
τB(δ) = inf{t ≥ δ : Z(t) ∈ B}.
Similar results are known for reflected processes driven by standard Brownian
motions, and our result can be viewed as their FBM counterpart. Our motivation
for this study is that RFBM appears as a limiting workload process for fluid
queueing network models fed by a large number of heavy-tailed ON/OFF sources in
heavy traffic.