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.
The proof of the large deviations theory builds on the G "artner-Ellis condition which establishes a direct relation to the effective bandwidth of a traffic flow @cite_29 . The theory of effective bandwidths, e.g. @cite_11 @cite_27 , is a major tool for the analysis of traffic flows as it gives a measure for resource requirements at different time scales. The effective bandwidth of a flow @math lies between its average and peak rate depending on the parameter @math . For fBm it holds that In case of @math the effective bandwidth of fBm traffic exhibits a continuous growth in @math due to LRD @cite_11 .
@cite_27 @cite_29 @cite_11
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From the Publisher: Providing performance guarantees is one of the most important issues for future telecommunication networks. This book describes theoretical developments in performance guarantees for telecommunication networks from the last decade. Written for the benefit of graduate students and scientists interested in telecommunications-network performance this book consists of two parts. We consider queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (at,vt,teE+) and a rate function such that if (Wt,teR+) denotes the workload process, then limi;^ 1 (Wt at > w) = — I(w)