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 IR(+)(J), with drift theta is an element of IR(J) and Hurst parameter H is an element of (0, 1), and let a is an element of IR(+)(J), a not equal 0, be a vector of weights. We define M(t) = max(0 <= s <= t) a(T) W (s) and prove that M(t) grows like t if mu = a(T) theta > 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 t(H) instead of t in the driftless case (theta = 0). If mu < 0 we prove that the increase of M(t) is smaller than that of any function growing faster than t and also that (log t)(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.