Article

# Long-range dependence ten years of Internet traffic modeling

Dept. of Comput. Sci. & Eng., California Univ., Riverside, CA, USA;

IEEE Internet Computing (Impact Factor: 2.04). 10/2004; 8(5):57- 64. DOI: 10.1109/MIC.2004.46 Source: DBLP

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**ABSTRACT:**It has been previously shown that actual network traffic exhibits longrange dependence. The Hurst parameter captures the degree of longrange dependence; however, because of the nature of computer network traffic, the Hurst parameter may not remain constant over a long period of time. An iterative method to compute the value of the Hurst parameter as a function of time is presented and analyzed. Experimental results show that the proposed method provides a good estimation of the Hurst parameter as a function of time. Additionally, this method allows the detection on changes of the Hurst parameter for long data series. The proposed method is compared with traditional methods for Hurst parameter estimation. Actual and synthetic traffic traces are used to validate our results. The proposed method allows detecting the changing points on the Hurst parameter, and better results can be obtained when modeling selfsimilar series using several values of the Hurst parameter instead of only one for the entire series. A new graphical tool to analyze longrange dependent series is proposed. Because of the nature of this plot, it is called the transitionvariance plot. This tool may be helpful to distinguish between LAN and WAN traffic. Finally, the software LRD Lab* is deployed to analyze and synthesize longrange dependent series. The LRD Lab includes a simple interface to easily generate, analyze, visualize and save longrange dependent series.Computación y Sistemas. 03/2010; 13(3):295-312. -
##### Article: Marcinkiewicz Law of Large Numbers for Outer-products of Heavy-tailed, Long-range Dependent Data

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**ABSTRACT:**The Marcinkiewicz Strong Law, lim n→∞ 1 n 1 p n ∑ k=1 (D k − D) = 0 a.s. with p ∈ (1, 2), is studied for outer products D k = X k X T k , where {X k }, {X k } are both two-sided (multivariate) linear processes (with coefficient matrices (C l), (C l) and i.i.d. zero-mean innovations {Ξ}, {Ξ}). Matrix sequences C l and C l can decay slowly enough (as |l| → ∞) that {X k , X k } have long-range dependence while {D k } can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for {D k } are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy-tails or the long-range dependence, but not the combination. The main result is applied to obtain Marcinkiewicz Strong Law of Large Numbers for stochastic approximation, non-linear functions forms and autocovariances.01/2014; - 01/2014;

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