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.
In contrast, the derivation of performance bounds using the stochastic network calculus builds on sample path arguments, such as the backlog bound ), and requires a bound for @math for all @math , i.e. a sample path envelope of the form Such sample path envelopes are constructed in @cite_21 using Boole's inequality under the assumption of a time scale limit @math , i.e. by summing the constant point-wise overflow probabilities @math over all @math . The time scale in this context can be regarded as a constraint on the duration of busy periods. In case of fBm the duration of busy periods has, however, been found to grow extremely fast with @math @cite_8 .
@cite_21 @cite_8
1987582408 2158018483
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. Several types of network traffic have been shown to exhibit long-range dependence (LRD). In this work, we show that the busy period of an ATM system driven by a long-range dependent process can be very large. We introduce a new traffic model based on a fractional Brownian motion envelope process. We show that this characterization can be used to predict queueing dynamics. Furthermore, we derive a new framework for computing delay bounds in ATM networks based on this traffic model. We show that it agrees with results given by large deviation theory with less computational complexity.