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
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IP flows have heavy-tailed packet and byte size distributions. This make them poor candidates for uniform sampling---i.e. selecting 1 in N flows---since omission or inclusion of a large flow can have a large effect on estimated total traffic. Flows selected in this manner are thus unsuitable for use in usage sensitive billing. We propose instead using a size-dependent sampling scheme which gives priority to the larger contributions to customer usage. This turns the heavy tails to our advantage; we can obtain accurate estimates of customer usage from a relatively small number of important samples.The sampling scheme allows us to control error when charging is sensitive to estimated usage only above a given base level. A refinement allows us to strictly limit the chance that a customers estimated usage will exceed their actual usage. Furthermore, we show that a secondary goal, that of controlling the rate at which samples are produced, can be fulfilled provided the billing cycle is sufficiently long. All these claims are supported by experiments on flow traces gathered from a commercial network. We introduce the concept of generalized stochastically bounded burstiness (gSBB) for Internet traffic, the tail distribution of whose burstiness can be bounded by a decreasing function in a function class with few restrictions. This new concept extends the concept of stochastically bounded burstiness (SBB) introduced by previous researchers to a much larger extent - while the SBB model can apply to Gaussian self-similar input processes, such as fractional Brownian motion, gSBB traffic contains non-Gaussian self-similar input processes, such as spl alpha -stable self-similar processes, which are not SBB in general. We develop a network calculus for gSBB traffic. We characterize gSBB traffic by the distribution of its queue size. We explore the property of sums of gSBB traffic and the relation of input and output processes. We apply this calculus to a work-conserving system shared by a number of gSBB sources, to analyze the behavior of output traffic for each source and to estimate the probabilistic bounds for delays. We expect this new calculus to be of particular interest in the implementation of services with statistical qualitative guarantees. This paper analyzes the asymptotic behavior of packet-train probing over a multi-hop network path P carrying arbitrarily routed bursty cross-traffic flows. We examine the statistical mean of the packet-train output dispersions and its relationship to the input dispersion. We call this relationship the response curve of path P. We show that the real response curve Z is tightly lower-bounded by its multi-hop fluid counterpart F, obtained when every cross-traffic flow on P is hypothetically replaced with a constant-rate fluid flow of the same average intensity and routing pattern. The real curve Z asymptotically approaches its fluid counterpart F as probing packet size or packet train length increases. Most existing measurement techniques are based upon the single-hop fluid curve S associated with the bottleneck link in P. We note that the curve S coincides with F in a certain large-dispersion input range, but falls below F in the remaining small-dispersion input ranges. As an implication of these findings, we show that bursty cross-traffic in multi-hop paths causes negative bias (asymptotic underestimation) to most existing techniques. This bias can be mitigated by reducing the deviation of Z from S using large packet size or long packet-trains. However, the bias is not completely removable for the techniques that use the portion of S that falls below F. An abstract model for aggregated connectionless traffic, based on the fractional Brownian motion, is presented. Insight into the parameters is obtained by relating the model to an equivalent burst model. Results on a corresponding storage process are presented. The buffer occupancy distribution is approximated by a Weibull distribution. The model is compared with publicly available samples of real Ethernet traffic. The degree of the short-term predictability of the traffic model is studied through an exact formula for the conditional variance of a future value given the past. The applicability and interpretation of the self-similar model are discussed extensively, and the notion of ideal free traffic is introduced. >