On the return time for a reflected fractional Brownian motion process on the positive orthant

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Abstract

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 ∈ (1/2, 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∈B ExB(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.

Original languageEnglish
Pages (from-to)145-153
Number of pages9
JournalJournal of Applied Probability
Volume48
Issue number1
DOIs
StatePublished - Mar 2011

Keywords

  • Heavy traffic theory
  • Reflected fractional Brownian motion
  • Return time

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