On the reflected fractional brownian motion process on the positive orthant: asymptotics for a maximum with application to queueing networks

Let W be a J-dim. reflected fractional Brownian motion process (rfBm) on the positive orthant IRJ+, with drift θ € IRJ and Hurst parameter H € (0, 1), and let a € IR J+ a≠0, be a vector of weights. We define M(t)=max 0≤s≤tat(s) and prove that M(t) grows like t if μ=aTθ>0, in the sense that its increase is smaller than that of any function growing faster than t, and if a restriction on the weights holds, it is also bigger than that of any function growing slower than t. We obtain similar results with tH instead of t in the driftless case (θ=0). If μ <0 we prove that the increase of M(t) is smaller than that of any function growing faster than t and also that 1/2(1-H) is a lower bound for M(t). Motivation for this study is that rfBm appears as the workload limit associated to a fluid queueing network fed by a big number of heavy-tailed On/Off sources under heavy traffic and state space collapse; in this scenario, M(t) can be interpreted as the maximum amount of fluid in a queue at the network over the interval [0, t], which turns out to be an interesting performance process to describe the congestion of the queueing system. Copyright © Taylor & Francis Group, LLC.