B. B. Mandelbrot, "Long-run linearity, locally Gaussian processes, Hspectra and infinite variances," International Economic Review 10, pp. 82-- 113, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it sent a signal that Internet technology would have to be quite different from telephone network technology. The discovery of long range dependence was confirmed in many other studies (e.g. 3] 4] 5] The work on longrange dependence drew heavily on the brilliant work of Mandelbrot [6], both for basic concepts and for methodology. Models of source traffic were put forward to explain the traffic characteristics [3] 7] 8] 9] The sizes of transferred files utilizing a link vary immensely; to a good approximation, the upper tail of the file size distribution is Pareto with a ....

B. B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances," International Economic Review, vol. 10, pp. 82--113, 1969.

....assume finite variance distributions for the sojourn time in ON and OFF periods. As a result, the aggregation of large number of such sources will not have significant correlation, except possibly in the short range [35] An extension to such traditional ON OFF models was first introduced by [36, 37] (as cited in [35] by allowing the ON and OFF periods to have infinite variance (high variability or Noah Effect) The superposition of many such sources produces aggregate traffic that exhibits long range dependence (also called the Joseph Effect [35, 38] The source model used in [38] can be ....

B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variance," Int'l. Economic Rev., vol. 10, 1969, pp. 82--113.

....renewal reward processes with heavy tailed (infinite variance) interarrival times and finite variance rewards. If the number of summands grows to infinity, then, after rescaling, the limit turns out to be a fractional Brownian motion (fBm) This has been already observed in an economics context by Mandelbrot (1969). The result is also described in Taqqu and Levy (1986) An on o# version of the model has been adapted to the telecommunications context by Taqqu et al. 1997) Recall that fBm BH with H # (0, 1) is a Gaussian self similar process with stationary increments. The self similarity means that BH ....

Mandelbrot, B. B. (1969), `Long-run linearity, locally Gaussian processes, H-spectra and infinite variances', International Economic Review 10, 82--113.

....level form the basis for the self similar features observed in the WAN traffic traces collected at Berkeley [35, 36] It is the (occasional) sustained inactivity or activity implied by power law distributions that manifests itself as the 1=f noise spectra or LRD. Extending the results in [30, 37], it can be shown (see [41] that aggregating a large number of such independent ON OFF sources (with ON OFF periods characterized by P (T t) t Gammaff ) will in fact generate FBM of index H = 3 Gamma ff) 2. Thus for 1 H 1=2, we have 2 ff 1, so that the distribution has a finite ....

B.B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances", Intern. Econom. Review 10, pp. 82-113, 1969.

....and introduced mathematically in a remarkably large number of fields, such as agronomy, astronomy, chemistry, economics, engineering, enviromental sciences, geosciences, hydrology, mathematics, physics, and statistics. Pioneering mathematical work on them was done by Kolmogorov [9] and Mandelbrot [14]. Recently, these processes were considered in modeling cell traffic in modern communications networks. In particular, second order self similar processes were used to model telecommunication traffic in high resolution Ethernet local area networks, wide area networks and also for variable bit rate ....

....2 hold for continuous value and discretetime processes as well. For exactly self similar process, some of its main properties are reviewed and commented. The definitions of exactly and strictly self similar processes are compared. These considerations are known and can be found in [6] 9] [14] and [17] 20] For an asymptotically self similar process, we give a necessary and sufficient condition of its self similarity in terms of variance of its averaged version. Also, we give a new sufficient condition of self similarity in terms of correlation coefficient of the process itself. We ....

B. B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra, and Infinite Variancies". International Economic Review, Vol. 10, pp. 82-113, 1996.

....amenable to physical interpretation. Two such models, the exactly (second order) self similar fractional Gaussian noise process and the asymptotically (secondorder) fractional ARIMA process, will be presented next. We also discuss a construction of self similar models, originally due to Mandelbrot [71] and later extended in [92, 60] which appears to be promising in terms of providing a physical explanation for the self similarity property in high speed packet traffic (see [97] 2.6.4 Fractional Gaussian Noise A fractional Gaussian noise [70] is a stationary Gaussian process, X = fX k g ....

....are invariant under time shifts. By aggregating M iid copies, W (1) W (2) Delta Delta Delta ; W (M) of W , one obtains the process W = fW k (M)g k0 , given by W k (M) 8 : 0; k = 0 k X n=1 M X m=1 W (m) n ; k 0 It can be shown [71, 92] that for k and M both large and k M , the process W behaves like a fractional Brownian motion. More precisely, the process W , properly normalized, converges to the integrated version of fractional Gaussian noise, the notion of convergence being that of finite dimensional distributions. Thus, ....

Mandelbrot, B.B., "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances", Intern. Econom. Review 10 (1969), 82--113.

....checked against the actual data. In the following, we will use the Ethernet example to illustrate how answering this question for self similar traffic models leads to new insight into the dynamics of network traffic. In mathematical terms, developing an approach originally suggested by Mandelbrot [12] (see also [16, 4] it is shown in [19] that the superposition of many ON OFF sources, each of which exhibits a phenomenon called the Noah Effect , results in self similar aggregate traffic. Mapping these results into the well known framework of ON OFF source models (also known as packet train ....

B.B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, HSpectra and Infinite Variances", International Economic Review 10, pp. 82--113, 1969.

....assume finite variance distributions for the sojourn time in ON and OFF periods. As a result, the aggregation of large number of such sources will not have significant correlation, except possibly in the short range [35] An extension to such traditional ON OFF models was first introduced by [36, 37] (as cited in [35] by allowing the ON and OFF periods to have infinite variance ( high variability or Noah Effect ) The superposition of many such sources produces aggregate traffic that exhibits long range dependence (also called the Joseph Effect [38, 35] The source model used in [38] can be ....

B. Mandelbrot, "Long-run linearity, locally Gaussian processes, H-spectra and infinite variance, " International Economic Review, Vol. 10, pp. 82--113, 1969.

....and simple explanation for the observed self similarity of measured aggregate packet traffic in terms of the nature of the traffic generated by the individual sources source destination pairs that contribute to the aggregate packet stream. Developing an approach originally suggested by Mandelbrot [31] (see also Taqqu and Levy [42] and Erramilli, Singh and Pruthi [13] they show that the superposition of many ON OFF sources, each of which exhibits a phenomenon called the Noah Effect , results in self similar aggregate traffic. Being able to phrase the results in the wellknown framework of ....

B.B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances ", International Economic Review 10, pp. 82-113, 1969.

....in a variety of practical problems that include wireless communications, teletraffic, hidrology, geology and economics. These processes can be efficiently modeled by heavy tailed processes with infinite variance for which neither the classical second order theory nor the theory of HOS are useful [13, 16, 20, 22, 25]. It has been shown repeatedly in the literature that infinite variance processes that appear in practice are well modeled by probability distributions with algebraic tails, i.e. random variables for which Pr(jXj x) O(x Gammaff ) for some fixed ff 0. Examples of such noise processes ....

B. Mandelbrot, "Long-run linearity, locally Gaussian processes, H-spectra and infinite variances, " Interant. Econ. Rev., vol. 10, pp. 82--111, 1969.

....Pox plot of Ethernet traffic data the discrepancy between the estimates of H obtained using the two methods. 4.1. 3 Stochastic Modeling of Self Similar Phenomenon Self similarity is not a new phenomenon, and has been studied extensively in several fields including hydrology [MH78] and economics [M69]. Because of this, several formal mathematical models have already been developed which exhibit self similarity. However, because of the abstract nature of these models, it is often difficult to find a physical interpretation for the self similarity. Two such models are presented in this section: ....

B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances," Intern. Econom. Rev. 10, 82-113, 1969.

No context found.

B. B. Mandelbrot, "Long-run linearity, locally Gaussian processes, Hspectra and infinite variances," International Economic Review 10, pp. 82-- 113, 1969.

No context found.

B. B. Mandelbrot, "Long-run linearity, locally Gaussian processes, H-spectra and infinite variances," Int. Economic Rev., vol. 10, pp. 82--113, 1969.

No context found.

B. B. Mandelbrot, "Long-run linearity, locally gaussian processes, h-spectra and infinite variances," Intern. Econom. Rev. 10, pp. 82--113, 1969.

No context found.

B. B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances", Intern. Econom. Rev. 10, 82-113, 1969.

No context found.

B. B. Mandelbrot, "Long-Run Linearity, Locally Gaussian Processes, H-Spectra and Infinite Variances", Intern. Econom. Rev. 10, 82-113, 1969.

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