M. S. Taqqu, Self-similar processes, In S. Kotz and N. Johson, editors, Encyclodipia of Statistical Science, vol. 8, Wiley, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....implications of self similarity in network provisioning have already been outlined [9] in terms of congestion control and avoidance. There is a large literature on self similar processes and their applications to various fields including economics, fluid theory and telecommunications traffic (see [18] and references therein) A self similar process [18] has slowly decaying correlations (or a spectral density that tends to infinity in the vicinity of zero) Intuitively, the averaged packet counting process X k (m) does not smooth out to the mean rate with increases in the aggregation level m ....

....have already been outlined [9] in terms of congestion control and avoidance. There is a large literature on self similar processes and their applications to various fields including economics, fluid theory and telecommunications traffic (see [18] and references therein) A self similar process [18] has slowly decaying correlations (or a spectral density that tends to infinity in the vicinity of zero) Intuitively, the averaged packet counting process X k (m) does not smooth out to the mean rate with increases in the aggregation level m as a Poisson process would. This behavior is also ....

M. S. Taqqu, "Self-Similar processes". Encyclopedia of Statistical Sciences, pp. 352-357, Wiley, New York, 1987.

.... (k) as k 1; 9) where 0 fi 1 and L 1 is slowly varying at infinity, that is, lim t 1 L 1 (tx) L 1 (t) 1, for all x 0; examples of such slowly varying functions are L 1 (t) const and L 1 (t) log(t) A stochastic process satisfying relation (9) is said to exhibit long range dependence [6, 16, 93]. In Mandelbrot s terminology [74] long range dependence is also referred to as the Joseph Effect, in reference to the Old Testament figure who had interpreted Pharaoh s dream of the seven lean cows and the seven fat cows to mean the seven fat years and seven lean years that ancient Egypt was ....

Taqqu, M.S., "Self-Similar Processes", Encyclopedia of Statistical Sciences 8, Kotz, S. and Johnson, N. (Eds.), Wiley, new York, 1987.

....deviating far from one would expect from normally distributed signals. A precise Ethernet traffic model must be able to capture both characteristics. For capturing long range dependence, two of the best known classes of stochastic models are the increment processes of self similar models [13], and the family of fractional ARIMA processes [7] On the other hand, the ff stable distribution can be This work was supported by NSF under grant MIP 9553227 used to capture the extreme forms of spatial variability [4] Intergrating these two, the so called ff stable self similar processes ....

M. S. Taqqu, "Self-Similar Processes," in : Encyclopedia a of Statistical Sciences, vol. 8, S. Kotz and N. Johnson, Eds. Wiley, New York, 1987.

....with stationary increments. Self similar processes with their fractal nature have long been attractive for both probabilists and users of stochastic models. Self similar processes with stationary increments have also been used to model the phenomenon of long range dependence. See, for example, Taqqu (1988) for an overview. Much work has been done in describing various classes of SffS H sssi processes and studying their properties. We refer the reader to Chapter 7 of Samorodnitsky and Taqqu (1994) for an extensive discussion. In particular, if (Y (t) t0 is an SffS H sssi process with 1 ff 2, ....

M. Taqqu (1988): Self-similar processes. In Encyclopedia of Statistical Sciences, S. Kotz and N. Johnson, editors. Wiley, New York, pp. 352--357. Volume 8.

....Due to the asymptotic equivalence of differencing and differentiation, 2) can be approximated to r(k; Ts) g(Ts)H(2H Gamma 1)k Gamma(2 Gamma2H) 3) for large k. The discrete time fractional Gaussian Noise (FGN) process has an autocorrelation function of the form (2) with g(Ts ) 1 [21], and therefore is an exact LRD process. A family of fractal stochastic point processes (FSPPs) introduced in [19, 20] also yield exact LRD processes in the sense of (2) with g(Ts ) T 2H Gamma1 s = T 2H Gamma1 s T 2H Gamma1 0 ) T0 is called fractal onset time which is used to control ....

M. S. Taqqu. Self-similar processes. In S. Kotz and N. Johnson, editors, Encyclopedia of Statistical Sciences, volume 8. Wiley, New York, 1987.

....(2) can be approximated to r(k; T s ) g(T s )H(2H Gamma 1)k Gamma(2 Gamma2H ) 3) for large k. But the approximation (3) is usually very good even for small k [5] The discretetime fractional Gaussian noise process has an autocorrelation function of the form (2) with g(T s ) 1 [24], and therefore is of this type. A family of fractal stochastic point processes (FSPPs) introduced in [23, 19] belongs also to this type with g(T s ) T 2H Gamma1 s = T 2H Gamma1 s T 2H Gamma1 0 ) T 0 is called fractal onset time, which we will use to control the variance of the frame ....

M. S. Taqqu. Self-similar processes. In S. Kotz and N. Johnson, editors, Encyclopedia of Statistical Sciences, volume 8. Wiley, New York, 1987.

....of time series. 2 Definition of self similarity The most common way to define self similarity of a process X = X t ; Gamma1 t 1) is by means of its distribution: if (X at ) and a H (X t ) have identical finite dimensional distributions for all a 0 then X is self similar with parameter H [11]. In our case, however, we need a definition which is more related to the properties of time series and which is more appropriate for the development of estimators for the self similarity parameter H [13] Let X = X t ; t = 0; 1; 2; be a covariance stationary stochastic process with mean ....

M. S. Taqqu. Self-similar processes. In Encyclopedia of Statistical Sciences 8, pages 352--357. Wiley, New York, 1988.

....and differentiation, 2) can be approximated to r(k; T s ) g(T s )H(2H Gamma 1)k Gamma(2 Gamma2H ) 3) 2 The symbol denotes an asymptotic relation. for large k. The discrete time fractional Gaussian Noise (FGN) process has an autocorrelation function of the form (2) with g(T s ) 1 [19], and therefore is an exact LRD process. A family of fractal stochastic point processes (FSPPs) introduced in [18] also yield exact LRD processes in the sense of (2) with g(T s ) T 2H Gamma1 s = T 2H Gamma1 s T 2H Gamma1 0 ) T 0 is called fractal onset time; see [18] for detail. We ....

M. S. Taqqu. Self-similar processes. In S. Kotz and N. Johnson, editors, Encyclopedia of Statistical Sciences, volume 8. Wiley, New York, 1987.

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M. S. Taqqu, "Self-Similar Processes", in: Encyclopedia of Statistical Sciences, Vol. 8, S. Kotz and N. Johnson (Eds.), Wiley, New York, 1987.

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M. S. Taqqu, Self-similar processes, In S. Kotz and N. Johson, editors, Encyclodipia of Statistical Science, vol. 8, Wiley, 1987.

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M.S. Taqqu, "Self-similar processes", Encyclopedia of Statistical Sciences, Vol.8, pp. 352-357

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