On the Reflected Fractional Brownian Motion Process on the Positive Orthant: Asymptotics for a Maximum with Application to Queueing Networks

Abstract

Let W be a J-dim. reflected fractional Brownian motion process (rfBm) on the positive orthant , with drift θ ∊ ℝ ^J and Hurst parameter H ∊ (0, 1), and let , 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 μ =a ^T θ > 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 (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 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.

Keywords:

• Fluid queue
• Heavy traffic
• Maximum
• Reflected fractional Brownian motion
• State space collapse
• Workload

Mathematics Subject Classification:

• Primary 60G70
• Secondary 60G15, 60G18, 60K25

ACKNOWLEDGMENTS

The author thanks the anonymous referees for careful reading and very helpful comments that resulted in an overall improvement of the article. This work was partially supported by project MEC-FEDER ref. MTM2006-06427.