A Truncated alpha-Stable Long-Range Dependent Process Model for Modeling Network Traffic

It has been shown that Variable Bit Rate (VBR) video exhibits long-range dependence characteristics. Several models have been proposed to synthesize traces whose autocorrelation match that of the video traffic traces. Given the relationship between self-similarity and long- range dependence, we investigate the application of discrete-time linear scale invariant systems provided by Zhao and Rao to modeling VBR video traffic. This formulation, called a DLSI system, was derived using a continuous dilation operator in discrete-time as a direct analog of the continuous-time linear scale-invariant system formulation of Wornell and differs from other approaches such as wavelet based construction. While simulations had shown that DLSI systems were capable of generating self- similar data such as those found in network traffic, questions regarding their long-range behavior remained. Answers to these questions are dealt in this paper. We present an alternative modeling technique to model VBR video traffic using DLSI system. The proposed model has a fractional pole-zero structure and provides a good model for long-range dependent and self-similar time series. The behavioral pattern of the fractional pole-zero filter for a range of the Hurst parameter is also shown. The time-domain characteristics of the filter are analyzed using power series approximations. Comparison of output autocorrelation function (ACF) to those of the video trace ACF is made. We conclude that some samples of video data are indeed better modeled by the LSI models than the conventional models such as Markovian and Long-range dependent models.

We analyze 20 large sets of actual variable-bit-rate (VBR) video data, generated by a variety of different codecs and representing a wide range of different scenes. Performing extensive statistical and graphical tests, our main conclusion is that long-range dependence is an inherent feature of VBR video traffic, i.e., a feature that is independent of scene (e.g., video phone, video conference, motion picture video) and codec. In particular, we show that the long-range dependence property allows us to clearly distinguish between our measured data and traffic generated by VBR source models currently used in the literature. These findings give rise to novel and challenging problems in traffic engineering for high-speed networks and open up new areas of research in queueing and performance analysis involving long-range dependent traffic models. A small number of analytic queueing results already exist, and we discuss their implications for network design and network control strategies in the presence of long-range dependent traffic. >

The objective of this paper is to demonstrate the limitations of the variance-time (VT) test as a statistical tool for inferring long-range dependence (LRD) in network traffic. Since the early Bellcore studies, LRD has been in the center of a continuous debate within the teletraffic community. The controversy is typically focused on the utility of LRD models to predict the performance at network buffers. Our work here is not meant to advocate one modeling approach over another, but to point out (experimentally and theoretically) to the caveats in using the VT test as a tool for detecting LRD. We do that by deriving simple analytical expressions for the slope of the aggregated variance in three autocorrelated traffic models: M/G//spl infin/ process (short-range dependent (SRD) but non-Markovian), the discrete autoregressive of order one model (SRD Markovian), and the fractional ARIMA process (LRD). Our main result is that the VT test often indicates, falsely, the existence of an LRD structure (i.e., H>0.5) in synthetically generated traces from the two SRD models. The bias in the VT test, however, diminishes monotonically with the length of the trace. We provide some guidelines on selecting the minimum trace length so that the bias is negligible.

Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X—one that does not have long-range dependence. When X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstrate that unlike the case where the input has short-range dependence, the path here is nonlinear.

Recent researches have highlighted the long-range dependence and impulsive nature of the aggregated network traffic, and consequently long-range dependent α-stable processes model has been widely used to capture the network traffic. This paper presents an analytical demonstration of aggregated network traffic converging to long-range dependent α-stable processes, which provides inherent mechanism explanation for the widely used long-range dependent α-stable processes model. Simulations validate the accuracy of the presented theoretical analysis.

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time-series modelling. In this article, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either short- or long-range dependence. It is shown in the short-range dependence setting that this robust estimator is asymptotically normal at the rate , where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared with the asymptotic variance of the classical autocovariance estimator. In the long-range dependence setting, the limiting distribution displays the same behaviour as that of the classical autocovariance estimator, with a Gaussian limit and rate when the Hurst parameter H is less than 3/4 and with a non-Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when H ∈ (3/4,1). Some Monte Carlo experiments are presented to illustrate our claims and the Nile River data are analysed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.

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 (LRD), which is marked by the presence of correlations that can extend over many time scales. 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.

For a long time, the area of network traffic research has lacked adequate traffic measurements. During the past years, large amounts of network traffic measurements have become available, collected in the Web and high-speed networks. Measurements of real traffic indicate that burstiness is present on a wide range of time scales and show evidence of self-similarity and long-range dependency. Many traditional models originating in conventional voice networks either assumed that the underlying time series are short-range dependent or do not focus on the various, interrelated network components affected by bursty traffic. In order to provide explanations for empirically observed phenomena, such as burstiness in terms of physical network parameters, we need to develop models that can allow for the details of the complex architecture of today's networks and can analyze the interrelated network parameters that ultimately determine the performance and operation of a network. In the paper, we investigate the harmful consequences of burstiness on various network components using a model implemented by a discrete event simulation methodology. The methodology based on a modified version of the M/Pareto model can measure network performance parameters, such as the momentary utilization of links, response time, and the queuing performance of interconnected switches and routers along the traffic paths. Our paper intends to narrow the gap between existing, well-known theoretical results and their applicability in everyday, practical network analysis and modeling. Our methodology can help network designers and engineers, the ultimate users of traffic modeling, understand the dynamic nature of network traffic and assist them to design, monitor, and control complex, high-speed networks in everyday practice.

Chapter 16 - Long range dependence in heavy tailed stochastic processes

Recent results from packet measurement analysis have shown that packet traffic exhibits fractal properties such as self-similarity (or its concomitant, long-range dependence) which are fundamentally different from features found in circuit switched voice traffic and captured by commonly used packet traffic models such as batch-Poisson and Markov modulated Poisson process (MMPP). These fractal properties are associated with the well-known burstiness of packet traffic, and self-similar traffic models (e.g., fractional Brownian motion (FBM)) permit parsimonious descriptions of packet traffic. The FBM model requires only 3 parameters: mean rate, peakedness (a measure of the fluctuations about the mean rate), and the Hurst parameter (characterizing long-range dependence). From the viewpoint of applying such models in practice, the estimation of these parameters is of crucial importance. We use a semi-parametric periodogram-based algorithm (SPA) to estimate the Hurst parameter and the peakedness. The algorithm is based on the fact that the power spectral density (PSD) of long-range dependent processes obey a power-law near the origin (i.e., 1/f-noise). We apply SPA to estimate the traffic parameters of both synthetically generated FBM traces as well as real Ethernet traces. Our analysis indicates that, compared with other methods SPA offers several advantages: (i) the marginals of the traffic time series are not required to be Gaussian (ii) it is computationally efficient (iii) it can estimate the peakedness factor as well as the Hurst parameter.

A simple first-order recurrence in a (max, +) dynamic system is numerically investigated and shown to exhibit statistical long-range dependence, characterized by slowly decaying aggregated variances and power-law evolutions of the autocorrelation and spectrum. We propose this model as a basis for a very parsimonious modeling of some long-range dependent processes such as data traffic.

The paper provides central limit theorems on multivariate stationary processes with long-range dependence as a natural extension of the corresponding theory on short-range dependent processes. In order to establish those theorems, the paper imposes weak assumptions on conditional moments of innovation processes, dispensing with the usual assumptions of exact Martingale difference or the contemporaneously transformed Gaussianity.

The article introduces spatial long-range dependent models based on the fractional difference operators associated with the Gegenbauer polynomials. The results on consistency and asymptotic normality of a class of minimum contrast estimators of long-range dependence parameters of the models are obtained. A methodology to verify assumptions for consistency and asymptotic normality of minimum contrast estimators is developed. Numerical results are presented to confirm the theoretical findings.

We show that the periodogram behaves differently for a weakly dependent process with a small monotonic trend and a stationary strongly dependent process. In the former case it has a non-central-distribution with noncentrality parameter tending to 0 uniformly outside intervals. In the latter case it has λα−1times a central-distribution, 0 <α< 1.

Self-similar processes based on fractal point processes (FPPs) provide natural and attractive network traffic models. We show that the point process formulation yields a wide range of FPPs which in turn yield a diversity of parsimonious, computationally efficient, and highly practical asymptotic second-order self-similar processes. Using this framework, we show that the relevant second-order fractal characteristics such as long-range dependence (LRD), slowly-decaying variance, and 1/f noise are completely characterized by three fundamental quantities: mean arrival rate, Hurst parameter, and fractal onset time. Four models are proposed, and the relationship between their model parameters and the three fundamental quantities are analyzed. By successfully applying the proposed models to Bellcore's Ethernet traces, we show that the FPP models prove useful in evaluating and predicting the queueing performance of various types of fractal traffic sources