The Associative Multifractal Process: A Novel Model for Computer Network Traffic Flows

A novel constructive mathematical model based on the multifractal formalism in order to accurately characterizing the localized fluctuations present in the course of traffic flows today high-speed computer networks is presented. The proposed model has the target to analyze self-similar second-order time series representative of traffic flows in terms of their roughness and impulsivity.

On the Time Series Length for an Accurate Fractal Analysis in Network Systems

It is well-known that fractal signals appear in many fields of science. LAN and WWW traces, wireless traffic, VBR resources, etc. are among the ones with this behavior in computer networks traffic flows. An important question in these applications is how long a measured trace should be to obtain reliable estimates of de Hurst index (H). This paper addresses this question by first providing a thorough study of estimator for short series based on the behavior of bias, standard deviation (s), Root-Mean-Square Error (RMSE), and convergence when using Gaussian H-Self-Similar with Stationary Increments signals (H-sssi signals). Results show that Whittle-type estimators behave the best when estimating H for short signals. Based on the results, empirically derived the minimum trace length for the estimators is proposed. Finally for testing the results, the application of estimators to real traces is accomplished. Immediate applications from this can be found in the real-time estimation of H which is useful in agent-based control of Quality of Service (QoS) parameters in the high-speed computer network traffic flows.

Estimation of Hurst Exponent in Self-similar Traffic Flows

In this paper it presents, develops and discusses the existence of a process with long scope memory structure, representing of the independence between the degree of randomness of the traffic generated by the sources and flow pattern exhibited by the network. The process existence is presented in term of a new algorithmic that is a variant of the maximum likelihood estimator (MLE) of Whittle, for the calculation of the Hurst exponent (H) of self-similar stationary second order time series of the flows of the individual sources and their aggregation. Also, it is discussed the additional problems introduced by the phenomenon of the locality of the Hurst exponent, that appears when the traffic flows consist of diverse elements with different Hurst exponents. The instance is exposed with the intention of being considered as a new and alternative approach for modeling and simulating traffic in existing computer networks.

On the Generation of Self-similar with Long-range Dependent Traffic Using Piecewise Affine Chaotic One-dimensional Maps

A qualitative and quantitative extension of the chaotic models used to generate self-similar traffic with long-range dependence (LRD) is presented by means of the formulation of a model that considers the use of piecewise affine one-dimensional maps. Based on the disaggregation of the temporal series generated, a valid explanation of the behavior of the values of Hurst exponent is proposed and the feasibility of their control from the parameters of the proposed model is shown.

A Simplified Multifractal Model for Self-Similar Traffic Flows in High-Speed Computer Networks Revisited

In the context of the simulations carried out using a simplified multifractal model that is proposed to give an explanation to the locality phenomenon that appears in the estimation of the Hurst exponent in the second-order stationary series that represent the self-similar traffic flows in high-speed computer networks, its formulation is perfected to reduce the variability in the singularity limits and it is demonstrated through by its wavelet variant that this modification leads to a higher resolution in the interval of interest under study.

Development of Multifractal Models for Self-Similar Traffic Flows

This paper presents a simple technique of multifractal traffic modeling. It proposes a method of fitting model to a given traffic trace. A comparison of simulation results obtained for an exemplary trace, multifractal model and Markov Modulated Poisson Process models has been performed.

On the Hurst Exponent, Markov Processes, and Fractional Brownian Motion

There is much confusion in the literature over Hurst exponent (H). The purpose of this paper is to illustrate the difference between fractional Brownian motion (fBm) on the one hand and Gaussian Markov processes where H is different to 1/2 on the other. The difference lies in the increments, which are stationary and correlated in one case and nonstationary and uncorrelated in the other. The two- and one-point densities of fBm are constructed explicitly. The two-point density does not scale. The one-point density for a semi-infinite time interval is identical to that for a scaling Gaussian Markov process with H different to 1/2 over a finite time interval. We conclude that both Hurst exponents and one-point densities are inadequate for deducing the underlying dynamics from empirical data. We apply these conclusions in the end to make a focused statement about nonlinear diffusion.

Traffic Flows Analysis in High-Speed Computer Networks Using Time Series

This article explores the required amount of time series points from a high-speed traffic network to accurately estimate the Hurst exponent. The methodology consists in designing an experiment using estimators that are applied to time series, followed by addressing the minimum amount of points required to obtain accurate estimates of the Hurst exponent in real-time. The methodology addresses the exhaustive analysis of the Hurst exponent considering bias behavior, standard deviation, mean square error, and convergence using fractional gaussian noise signals with stationary increases. Our results show that the Whittle estimator successfully estimates the Hurst exponent in series with few points. Based on the results obtained, a minimum length for the time series is empirically proposed. Finally, to validate the results, the methodology is applied to real traffic captures in a high-speed network based on the IEEE 802.3ab standard.

On the LRD of the Aggregated Traffic Flows in High-Speed Computer Networks

This paper studies and analyses the behavior of the Long-Range Dependence in network traffic after classifying traffic flows in aggregated time series. Following Differentiated Services architecture principles, the generic Quality of Service applications that requirements and use the transport control protocol, a basic classification criterion of time series is established. Using the fractal theory, the resulting time series are analyzed. The Hurst exponent is estimated and used as a measure of traffic burstiness and Long-Range Dependency in each traffic class. The traffic volume per class is also measured. The study uses traffic traces collected at the core switch at the Electric Engineering Department at Universidad de Santiago de Chile in different periods of network activity.