Conditional limit theorems for regulated fractional Brownian motion

TL;DR

This paper derives the limiting distribution of the time a fractional Brownian motion-based queue spends above a large threshold, revealing that long delays tend to occur in large, clustered periods, with a novel scaling approach.

Contribution

It introduces a new conditional limit theorem for fractional Brownian motion queues, using a finer scaling than traditional fluid analysis to characterize large delay clusters.

Findings

Long delays occur in large clumps of size proportional to b^{2-1/H}

Conditional distribution of busy periods converges as b→∞

Finer scaling reveals detailed structure of large delay events

Abstract

We consider a stationary fluid queue with fractional Brownian motion input. Conditional on the workload at time zero being greater than a large value $b$, we provide the limiting distribution for the amount of time that the workload process spends above level $b$ over the busy cycle straddling the origin, as $b\to\infty$. Our results can be interpreted as showing that long delays occur in large clumps of size of order $b^{2-1/H}$. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.