A number of recent empirical studies of traffic measurements from a variety of working packet networks have convincingly demonstrated that actual network traffic is self-similar or long-range dependent in nature (i.e., bursty over a wide range of time scales) -- in sharp contrast to commonly made traffic modeling assumptions. In this paper, we provide a plausible physical explanation for the occurrence of self-similarity in LAN traffic. Our explanation is based on new convergence results for processes that exhibit high variability (i.e., infinite variance) and is supported by detailed statistical analyses of real-time traffic measurements from Ethernet LAN's at the level of individual sources. This paper is an extended version of [53] and differs from it in significant ways. In particular, we develop here the mathematical results concerning the superposition of strictly alternating ON/OFF sources. Our key mathematical result states that the superposition of many ON/OFF sources (also k...

Recent traffic measurement studies from a wide range of working packet networks have convincingly established the presence of significant statistical features that are characteristic of fractal traffic processes, in the sense that these features span many time scales. Of particular interest in packet traffic modeling is a property called long-range dependence, which is marked by the presence of correlations that can extend over many time scales. In this paper, we demonstrate empirically that, beyond its statistical significance in traffic measurements, long-range dependence has considerable impact on queueing performance, and is a dominant characteristic for a number of packet traffic engineering problems. In addition, we give conditions under which the use of compact and simple traffic models that incorporate long-range dependence in a parsimonious manner (e.g., fractional Brownian motion) is justified and can lead to new insights into the traffic management of high-speed networks. 1...

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 clarified 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. Keywords: LAN traffic, long-range dependence, self-similar, prediction 1 Introduction In this paper we are considering the modelling of traffical phenomena in a connectionless network. The principle of such a network is that all data is sent in relatively small independen...

Recent measurements of local-area and wide-area traffic have shown that network traffic exhibits variability at a wide range of scales. In this paper, we examine a mechanism that gives rise to self-similar network traffic and present some of its performance implications. The mechanism we study is the transfer of files or messages whose size is drawn from a heavy-tailed distribution. First, we show that in a “realistic ” client/server network environment—i.e., one with bounded resources and coupling among traffic sources competing for resources—the degree to which file sizes are heavy-tailed can directly determine the degree of traffic self-similarity at the link level. We show that this causal relationship is robust with respect to changes in network resources (bottleneck bandwidth and

A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono vs multi-fractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.

We consider from a thermodynamic viewpoint queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (a t ; v t ; t 2 R+ ) and a rate function I such that if (W t ; t 2 R+ ) denotes the workload process, then lim t!1 v \Gamma1 t log P (W t =a t ? w) = \GammaI (w) on the continuity set of I . In the case that a t = v t = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt [8], that the queue-length distribution (that is, the distribution of supremum of the workload process Q = sup t0 W t ) decays exponentially: P (Q ? b) ¸ e \Gammaffib and the decay rate ffi is directly related to the rate function I . We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t: we find that the queue-length distribution has an exponential tail only if l...

In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with longrange-dependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing N-point data sets. We study both the second-order and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variance-time plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.

High-resolution traffic measurements from modern communications networks provide unique opportunities for developing and validating mathematical models for aggregate traffic. To exploit these opportunities, we emphasize the need for structural models that take into account specific physical features of the underlying communication network structure. This approach is in sharp contrast to the traditional black box modeling methodology from time series analysis that ignores, in general, specific physical structures. We demonstrate, in particular, how the proposed structural modeling approach provides a direct link between the observed self-similarity characteristic of measured aggregate network traffic, and the strong empirical evidence in favor of heavy-tailed, infinite variance phenomena at the level of individual network connections.

There has been a growing concern about the potential impact of long-term correlations (second-order statistic) in variable-bit-rate (VBR) video traffic on ATM buffer dimensioning. Previous studies have shown that video traffic exhibits long-range dependence (LRD) (Hurst parameter large than 0.5). We investigate the practical implications of LRD in the context of realistic ATM traffic engineering by studying ATM multiplexers of VBR video sources over a range of desirable cell loss rates and buffer sizes (maximum delays). Using results based on large deviations theory, we introduce the notion of Critical Time Scale (CTS). For a given buffer size, link capacity, and the marginal distribution of frame size, the CTS of a VBR video source is defined as the number of frame correlations that contribute to the cell loss rate. In other words, second-order behavior at the time scale beyond the CTS does not significantly affect the network performance. We show that whether the video source model i...

A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.