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.
This paper establishes a link between two principal tools for the analysis of network traffic, namely, effective bandwidth and network calculus. It is shown that a general version of effective bandwidth can be expressed within the framework of a probabilistic version of the network calculus, where both arrivals and service are specified in terms of probabilistic bounds. By formulating well-known effective bandwidth expressions in terms of probabilistic envelope functions, the developed network calculus can be applied to a wide range of traffic types, including traffic that has self-similar characteristics. As applications, probabilistic lower bounds are presented on the service given by three different scheduling algorithms: static priority, earliest deadline first, and generalized processor sharing. Numerical examples show the impact of specific traffic models and scheduling algorithms on the multiplexing gain in a network. The viewpoint is that communication networking is about efficient resource sharing. The focus is on the three building blocks of communication networking, namely, multiplexing, switching and routing. The approach is analytical, with the discussion being driven by mathematical analyses of and solutions to specific engineering problems. The result? A comprehensive, effectively organized treatment of core engineering issues in communication networking. Written for both the networking professional and for the classroom, this book covers fundamental concepts in detail and places design issues in context by drawing on real world examples from current technologies. ·Systematically uses mathematical models and analyses to drive the development of a practical understanding of core network engineering problems. ·Provides in-depth coverage of many current topics, including network calculus with deterministically-constrained traffic, congestion control for elastic traffic, packet switch queuing, switching architectures, virtual path routing, and routing for quality of service. ·Includes over 200 hands-on exercises and class-tested problems, dozens of schematic figures, a review of key mathematical concepts, and a glossary.