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On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Chihoon Lee
Chihoon Lee*
Affiliation:
Colorado State University
*
∗Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA. Email address: chihoon@stat.colostate.edu
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Abstract

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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.

Keywords

Reflected fractional Brownian motionheavy traffic theoryreturn time

MSC classification

Secondary: 60G22: Fractional processes, including fractional Brownian motion 90B18: Communication networks 60G15: Gaussian processes 60G18: Self-similar processes
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 145 - 153
Copyright
Copyright © Applied Probability Trust 2011 

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