We present a detailed statistical analysis of a 2-hour long empirical sample of VBR video. The sample was obtained by applying a simple intraframe video compression code to an action movie. The main findings of our analysis are (1) the tail behavior of the marginal bandwidth distribution can be accurately described using "heavy-tailed" distributions (e.g., Pareto); (2) the autocorrelation of the VBR video sequence decays hyperbolically (equivalent to long-range dependence) and can be modeled using self-similar processes. We combine our findings in a new (non-Markovian) source model for VBR video and present an algorithm for generating synthetic traffic. Trace-driven simulations show that statistical multiplexing results in significant bandwidth efficiency even when long-range dependence is present. Simulations of our source model show long-range dependence and heavy-tailed marginals to be important components which are not accounted for in currently used VBR video traffic models. 1 I...

A storage model with self-similar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given. Keywords: Self-similar, fractional Brownian motion, Local Area Network traffic 1 Introduction In a series of papers (e.g. Leland [8], Leland and Wilson [7], Fowler and Leland [4], Leland et al. [9]), researchers from Bellcore have reported and analyzed remarkable Local Area Network (LAN) traffic measurements challenging traditional data traffic modelling. The Bellcore data are both very accurate and extensive in time, and their most striking feature is the tremendous burstiness of LAN traffic at, practically, any timescale. More than that, the statistical analysis has shown that the traffic is self-similar with a surprising accuracy (see Leland et al. [9]). Traditional traffic models based on the Poisson process or, more gener...

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...

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.

Absfruet-A new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence. Standard formulations of this problem require the computationally intensive solution of an elliptic partial differential equation that arises from the often used “smoothness constraint” ’yl”. regularization. The interpretation of the smoothness constraint is utilized as a “fractal prior ” to motivate regularization based on a recently introduced class of multiscale stochastic models. The solution of the new problem formulation is computed with an efficient multiscale algorithm. Experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size. The new approach also has a number of other important advantages. Specifically, multiresolution flow field estimates are available, allowing great flexibility in dealing with the tradeoff between resolution and accuracy. Multiscale error covariance information is also available, which is of considerable use in assessing the accuracy of the estimates. In particular, these error statistics can be used as the basis for a rational procedure for determining the spatially-varying optimal reconstruction resolution. Furthermore, if there are compelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation. Finally, the usefulness of our approach should extend to a wide variety of ill-posed inverse problems in which variational techniques seeking a “smooth ” solution are generally Used. 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.

An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it will be seen how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of “ multiscale stationarity ” in which scale plays the role of a time-like variable. Focus is primarily on the investigation of models on homogenous trees. In particular, the elements of a dynamic system theory on trees are described

We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.