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 construction of a sample path envelope for fBm traffic is, however, not straightforward since no simple overflow profile exists, see Sect. . A simplifying approach is proposed in @cite_33 where it is argued that the Weibull tail ) implies an envelope for fBm traffic. Such an envelope is, however, based on the approximation by the largest term ). A rigorous sample path envelope for fBm as well as end-to-end performance bounds under fBm cross traffic have not been derived.
@cite_33
2022940918
A network calculus is developed for processes whose burstiness is stochastically bounded by general decreasing functions. This calculus is useful for a large class of input processes, including important processes exhibiting "subexponentially bounded burstiness" such as fractional Brownian motion. Moreover, it allows judicious capture of the salient features of real-time traffic, such as the "cell" and "burst" characteristics of multiplexed traffic. This accurate characterization is achieved by setting the bounding function as a sum of exponentials.