Statistical End-to-end Performance Bounds for Networks under Long Memory FBM Cross Traffic

TL;DR

This paper derives rigorous end-to-end performance bounds for networks with long-range dependent fBm cross traffic, revealing how variability and correlation affect network performance and providing insights into overflow probabilities.

Contribution

It introduces a rigorous sample path envelope for fBm traffic and applies stochastic network calculus to derive new end-to-end bounds in tandem networks.

Findings

Overflow probabilities have Weibullian tails.

Bounds grow as O(n (log n)^(1/(2-2H))) with number of systems.

Variability and correlation significantly impact network performance.

Abstract

Fractional Brownian motion (fBm) emerged as a useful model for self-similar and long-range dependent Internet traffic. Approximate performance measures are known from large deviations theory for single queuing systems with fBm through traffic. In this paper we derive end-to-end performance bounds for a through flow in a network of tandem queues under fBm cross traffic. To this end, we prove a rigorous sample path envelope for fBm that complements previous approximate results. We find that both approaches agree in their outcome that overflow probabilities for fBm traffic have a Weibullian tail. We employ the sample path envelope and the concept of leftover service curves to model the remaining service after scheduling fBm cross traffic at a system. Using composition results for tandem systems from the stochastic network calculus we derive end-to-end statistical performance bounds for individual flows in networks under fBm cross traffic. We discover that these bounds grow in O(n (log n)^(1/(2-2H))) for n systems in series where H is the Hurst parameter of the fBm cross traffic. We show numerical results on the impact of the variability and the correlation of fBm traffic on network performance.