Teletrac modeling and estimation

Transcription

1 Teletrac modeling and estimation File 2 José Roberto Amazonas jra@lcs.poli.usp.br Telecommunications and Control Engineering Dept. - PTC Escola Politécnica University of São Paulo - USP São Paulo 11/2008 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

2 Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

3 Fractals Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

4 Fractals Introduction The shapes of classical geometry - triangles, circles, spheres, etc. - loose their structures when magnied. For example, a person on the Earth's surface has the impression that it is at. On the other hand, an astronaut in orbit sees a round Earth. Suppose that someone has not been informed of being on a point of a circle which has a radius of hundreds of kilometers. This observer realizes the circle as a straight line even though it is not true. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

5 Fractals Mandelbrot set Benoit B. Mandelbrot proposed in 1975 the term fractal (from latin fractus, that means fractured, broken) to describe mathematical objects having a details rich structure along several observation scales. The Mandelbrot set is a mathematical fractal with a detailed structure (i. e., highly irregular) along an innite series of scales. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

6 Fractals Mandelbrot set 2 y interval: [ 1.5, 1.5] x interval: [ 2.5, 1.5] y interval: [ , ] x interval: [ , ] Self-similarity: a self-similar object contains smaller copies of itself in all scales. The Mandelbrot set is a deterministic fractal as it is exactly self-similar. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

7 Fractals Random fractals Physical and human sciences provide several examples of random fractals, in which self-similarity occurs in a statistical sense: climatology and hydrology time series; functional magnetic resonance of the human brain; uids turbulent movements; nancial data series; Sweden coast; cauliower. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

8 Fractals Sweden coast Figure: Resolution = 500 m Figure: Resolution = 1.4 km Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

9 Fractals Sweden coast 2 Figure: Resolution = 250 m Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

10 Fractals Cauliower Figure: Cauliower self-similarity Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

11 Fractals Cantor set Figure: Five initial iterations of the Cantor set Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

12 Fractals Ethernet trac Fig. 6 shows the Ethernet trac collected at the Drexel University's local network; It is self-similar and highly impulsive at four aggregation time scales: (10 ms, 100 ms, 1 s and 10 s).; The 100 ms scale series was obtained by aggregating the 10 ms scale series, i. e., a point at the 100 ms scale corresponds to the bytes addition in 10 consecutive bins of the 10 ms scale; It is surprising that successive aggregations do not smooth the trac; Smoothing would happen if the trac could be well modeled by the Poisson process. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

13 Fractals Ethernet trac 2 Figure: Drexel University's local area network Ethernet trac Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

14 Fractals Figure: (l) Ethernet real trac; (c) Poisson simulated trac; (r) Self-similar simulated trac Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61 Ethernet real trac vs. simulated trac

15 The Hurst Exponent Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

16 The Hurst Exponent General remarks Due to historical reasons, the persistence degree (LRD) of a time series is characterized by means of the Hurst parameter H, 0 < H < 1; A time series is LRD (self-similar) when 1/2 < H < 1; It is SRD (Short Range Dependence) when 0 < H 1/2; Closer is H to 1, higher is the series persistence degree; We say the series is monofractal if H is time invariant; We say the series is multifractal, if H varies in time, both in a deterministic or random way. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

17 The Hurst Exponent General remarks 2 It has been shown that the WAN trac may be multifractal, with non-gaussian marginal distribution, at rened time scales; On the other hand, a monofractal behavior has been observed for the LAN trac; Fig. 8 shows a realization produced by the Riedi's MWM model. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

18 The Hurst Exponent MWM realization 7 MWM c/ a mesma AC do FGN, H= Figure: Multifractal trac simulated by means of the MWM model. The time series impulsiveness varies in time: hateroscedasticity Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

19 The Hurst Exponent A long-memory process ACF Series : lrd1h09noiir.df ACF Lag Figure: A LRD series' ACF with H = 0.9 and N = 4096 samples. The continuous line is the autocorrelation function of the AR model that could be adjusted by the ar function of the S+FinMetrics software according to the AIC criterion. An AR(15) model was obtained. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

20 The Hurst Exponent Long memory behavior Aggregate trac can also show an ACF with long and short memory mixed characteristics. This behavior is typical of the ARFIMA model class. Long memory is characterized in the frequency domain by an 1/f α singularity, 0 < α < 1 (α = 2H 1), for f 0. Fig. 10 shows that the SDF of a FD model class with d = 0.4 (d = H 1/2) has an 1/f α behavior at the spectrum origin, while the AR(4) has not. FD(d) corresponds to an ARFIMA(p, d, q) with p = q = 0. The literature refers the LRD processes as 1/f noise. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

21 The Hurst Exponent Long memory FD(0.4) AR(4) 20 PSD (db) frequency Figure: SDF for same power AR(4) and FD(0.4) models. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

22 LRD Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

23 LRD Sample mean variance Consider a stationary random process x t, t Z, with mean µ x and variance σ 2. x Let x 1, x 2,..., x N be the observations of a realization of x t. If the random variables x 1, x 2,..., x N are independent or uncorrelated, then, the sample mean x variance is given by σ 2 x = σ2 x N. (1) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

24 LRD Condence interval If the sample is large enough, the x estimator's sample distribution is normal. The expression for the µ x condence interval, at (1 β) condence level is given by σ x x z β/2 N µ x x + z β/2 N, (2) σ x in which z β/2 denotes the quantile q (1 β/2) of the standard normal distribution. The quantile q α of a distribution function F x is the value for which F qα = α. The median, for example, corresponds to q 0,5. Given a probability 1 β, we nd z β/2 for which P{ z β/2 < Z < z β/2 } = 1 β (z β/2 = 1, 96 para 1 β = 95%). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

25 LRD Long range dependence Denition (Long range dependence) x t is a long range dependence or long memory process if there are constants α and C P, satisfying 0 < α < 1 and C P > 0, for which P x (f ) lim = 1, (3) f 0 C P f α where P x (f ) denotes the SDF of x t and f represents the normalized frequency ( 1/2 f 1/2), in cycles/sample. This is an asymptotic denition because the SDF is not specied for frequencies far from the origin. H = α + 1, 1/2 < H < 1. (4) 2 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

26 LRD Long range dependence 2 An alternate denition may be given in the time domain. x t is a 1/f α process if its autocorrelation R x (τ), for suciently large values of the lag τ, decreases according to a power function: in which C R > 0. lim τ The singularity at the spectrum's origin implies R x (τ) = 1, (5) C R τ (1 α) τ= R x (τ) =, (6) for 1/2 < H < 1, i. e., the autocorrelations decay towards zero so slowly that they are not summable. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

27 LRD Long range dependence 3 This behavior is drastically dierent from that presented by an ARMA process in which the autocorrelation's decay is fast, i. e., exponential: in which C > 0 and 0 < r < 1. R x (τ) Cr τ, τ = 1, 2,..., (7) ARMA processes have short range dependence. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

28 LRD Long range dependence 4 If x t is LRD, the variance of x decreases with the sample size N more slowly than in the traditional case( independent or uncorrelated variables): σ 2 x σ 2 x c(ρ x)n α 1, (8) in which c(ρ x ) is dened by lim N (1+α) ρ x (i, j). (9) N i j In this case, the x distribution is asymptotically Gaussian with E[ x] = µ x. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

29 LRD Long range dependence 5 The LRD behavior of x t makes parameters estimation, as x, more dicult than that of uncorrelated observations. In this case, the condence interval equation for µ x (given by (2)) is no longer valid. In fact, for a given condence level (1 β), the condence interval must be streched multiplying it by a factor F : F = N α/2 c(ρ x ). (10) This correction factor F increases with N, and in the limit goes to innity. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

30 LRD Aggregate process The M order aggregate process of x t, denoted by X (M) t, corresponds to a moving average of M size blocks without overlap of x t, i. e., X (M) i = 1 M Mi t=m(i 1)+1 x t. (11) The following property is valid for a long memory process x t : in which c is a constant. lim M Var X (M) Aggregation is equivalent to a time scale change. t = c, (12) M (2H 2) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

31 LRD Aggregate process 2 It can be realized that the aggregate process is statistically similar to the original process, in the sense that a nite number of successive aggregations does not destroy the original proccess' impulsive character. Therefore, (12) suggests that long range dependence and self-similarity properties are closely related. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

32 Self-similarity Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

33 Self-similarity Denition Denition (Processo H-ss) A stochastic process {y t } t R is self-similar with parameter 0 < H < 1, i. e., it is H-ss if, for any a > 0, {y(t)} d = {a H y(at)}, (13) in which d = denotes equality between the nite-dimensions distributions. A H-ss process is LRD if 1/2 < H < 1. The Brownian Movement (continuous time), also known as Wiener's process, is self-similar with H = 1/2 (but it is not LRD). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

34 Self-similarity Stationary increments If the process x t = y t, called process of y t increments or rst dierence of y t, is stationary, then y t is called H-sssi (H self-similar with stationary increments). In this case, the process H-sssi y t is a rst order integrated process, y t I (1). If the moments of y(t) of order lower or equal to q exist, from (13) we conclude that E y t q = E y 1 q t qh. (14) So, the process y t I (1) cannot be stationary. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

35 Self-similarity Autocovariance of a H-ss process Assuming E[y t ] = 0 for the sake of simplifying notation 1, it can be shown that the y t autocovariance is given by C y (t, s) = E[y t y s ] = σ2 x 2 [t2h + s 2H (t s) 2H ]. (15) in which is σ 2 x = E[(y t y t 1 ) 2 ] = E[y 2 (1)] is the variance of the x t process of increments. 1 So, E[x t] = E[y t y t 1 ] = 0 Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

36 Self-similarity DFBM and FGN Consider the sampled version y t, t Z, of a y t H-sssi process, with unit sampling interval. There are many y t H-sssi non-gaussian processes. However, for value of H (0, 1) there is exactly just one y t H-sssi Gaussian process, called Discrete-time Fractional Brownian Motion or DFBM. The Fractional Gaussian Noise - FGN, proposed by Mandelbrot and van Ness in 1968, corresponds to the DFBM process of increments. The FGN is a model widely used in LRD trac simulations. The DFBM and FGN models are non-parametric and are not used in trac future values predictions. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

37 Self-similarity Exact second order self-similarity Denition (Exact second order self-similarity) Consider the discrete time stationary process x t = y t y t 1. x t is an exact second order self-similar process with Hurst parameter H (1/2 < H < 1) if its autocovariance exists and is given by C x (τ) = σ2 x 2 [ τ +1 2H 2 τ 2H + τ 1 2H ], τ =..., 1, 0, 1,.... (16) It can be shown that the autocovariance given by (16) satises lim τ i. e., C x (τ) has a hyperbolic decay. C x (τ) = 1, (17) σxτ 2 2H 2 H(2H 1) Second order self-similarity implies LRD when 1/2 < H < 1. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

38 Self-similarity Exact second order self-similarity 2 Consider the aggregate process X (M) t of an exact second order self-similar process x t, at the M aggregation level. It can be shown that C (M) x (τ) = C x (τ), M = 2, 3,... (18) Eq. (18) says that the second order statistics of the original process with a scale change, justifying the term exact second order self similar. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

39 Self-similarity Asymptotic second order self-similarity Denition (Asymptotic second order self-similarity) A process x t is asymptotically second order self-similar with Hurst parameter H (1/2 < H < 1) if its autocovariance and its aggregate process autocovariance are related by lim C (M) x (τ) = C x (τ). (19) M Tsybarov and Georganas have shown that (12) implies (19). Therefore, a LRD process is asymptotically second order self similar too. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

40 Impulsiveness Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

41 Impulsiveness Introduction Several phenomena exhibit impulsive behavior, as low frequency atmospheric noise, man-made noise, submarine acoustic noise, transmission lines transients, seismic activity, nancial series, telephone circuits cluster errors, computer networks trac. In this context, stable probability distributions is a fundamental statistical modeling tool of impulsive signals. Its use is justied by the generalized central limit theorem. This theorem states that: if the limit of the sum of independent and identically distributed (i.i.d) random variables converges, then this limit can only be a stable distributed random variable. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

42 Impulsiveness Heavy tail distributions Denition (Heavy tail distribution) A random variable x has a heavy tail distribution with index α if P(x x) cx α L(x), x, (20) for c > 0 and 0 < α < 2, in which L(x) is a positive function that varies slowly for large values of x, i. e., lim x L(bx)/L(x) = 1 for any positive b. Eq. (20) states that observations of a heavy tail distributed random variables may occur with non-negligible probabilities, with values very dierent from the mean (outliers). So, this kind of random variable has high variability. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

43 Impulsiveness Pareto I distribution A simple example of a heavy tail distribution is the Pareto I distribution, dened by means of its complementary distribution function (survival function): ( ) α x F (x) = P(x x) = x m, x xm, 1, x < x m, (21) Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

44 Impulsiveness Pareto I distribution - cont. Figure: Pareto I probability density functions with xm = 1. Figure: Pareto I probability distribution functions with xm = 1. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

45 Impulsiveness Heavy tail distributions' variance The heavy distributions' p order statistics are nite if, and only if, p α. It is for this reason that such distributions have innite variance. The mean is innite if α < 1. The heavy tail distributions are also known as innite variance probability distributions. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

46 Impulsiveness Stable distributions An important member of the heavy tail distributions class is the stable distribution, discovered by Levy in the 1920 decade. The stable distribution does not have an analytic expression 2. It can be dened in terms of its characteristic function: Φ x (w) = E[e jw x ] = f x (x)e jwx dx = exp{jµw σw α [1 jη sign(w)ϕ(w, α)]}. (22) 2 The exceptions are the limit cases α = 1 (Cauchy) and α = 2 (Gaussian). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

47 Impulsiveness Stable distributions 2 In (22) ϕ(w, α) = { tan (απ/2) if α 1 2 π ln w if α = 1, (23) sign(.) stands for the sign function, α (0 < α 2) is the characteristic exponent, µ (µ R) is the localization parameter η ( 1 η 1) is the asymmetry parameter and σ 0 is the dispersion parameter or scale. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

48 Impulsiveness Stable distribution realization example Figure: Realization of Sα(σ, η, µ) = S 1.2 (1, 1, 0) (256 samples). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

49 Impulsiveness Symmetric stable distribution and density functions examples Figure: Probability density functions Figure: Probability distribution functions. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

50 Impulsiveness Asymmetric stable distribution and density functions examples Figure: Probability density functions. Figure: Probability distribution functions. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

51 Impulsiveness Stable distributions' property It can be shown that, if x S α (σ, η, µ) with 0 < α < 2, then, { P(x>λ) 1+η lim λ = λ α C α 2 σα, P(x< λ) 1 η lim λ = λ α C α 2 σα, in which ( { 1 C α = x α 1 α Γ(2 α) cos(πα/2) if α 1 sin x) = 0 2/π if α = 1. (24) (25) Therefore, (24) shows that the survival function of x decreases according to a power function for large values of λ. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

52 Impulsiveness Stability property Theorem (Stability property) A random variable x is stable if and only if for any independent random variables x 1 and x 2 that have the same distribution as x, and for arbitrary constants a 1, a 2, there are constants a and b as a 1 x 1 + a 2 x 2 d = ax + b, (26) Using the characteristic function of the stable distribution, it can be shown that: if x 1, x 2,..., x N are independent and follow stable distributions with the same (α, η), then all linear combinations N j=1 a j x j are stable with the same parameters α and η. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

53 Impulsiveness The central limit theorem The central limit states that the normalized sum of (i.i.d) random variables with nite variance σ 2 and mean µ converges to a Gaussian distribution. Formally, x µ σ/ N The relation (27) may be rewritten as d x N(0, 1) for N. (27) a N (x 1 + x x N ) b N d x N(0, 1) for x, (28) in which a N = 1/(σ N) and b N = Nµ/σ. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

54 Impulsiveness The generalized central limit theorem Theorem (Generalized central limit theorem) Let {x 1, x 2, x 3...} be a sequence of i.i.d random variables. There are constants a N > 0, b N R and a random variable x with a N (x 1 + x x N ) b N d x if and only if x is α-stable with 0 < α 2. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

55 Impulsiveness Impulsive stochastic process Denition (Impulsive stochastic process) A stochastic process x t is impulsive if it has a heavy tail marginal probability distribution. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

56 Impulsiveness Impulsive processes dependence measurements Covariances (or correlations) cannot be dened in the stable random variables space. Remember, the stable random variable variance is innite. Two kinds of measurements have been proposed: Co-variation; Co-dierence; Co-variation will not be used in this course. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

57 Impulsiveness Co-dierence The co-dierence of two jointly SαS random variables x 1 and x 2, 0 < α 2, is given by γ x1,x2 = (σ x1 )α + (σ x2 ) α (σ x1 x2 )α, (29) in which σ x is the scale parameter of the SαS x random variable. The co-dierence is symmetric, i. e., γ = γ x1,x2 x2,x1 the covariance when α = 2. If x 1 e x 2 are independent, then γ x1,x2 = 0. and reduces to Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

58 Impulsiveness Generalized co-dierence Denition (Generalized co-dierence) I (w 1, w 2 ; x 1, x 2 ) = ln E[e j(w 1x1+w 2 x2) ] + ln E[e jw 1x1 ] + ln E[e jw 2x2 ], (w 1, w 2 ) R 2 (30) If x 1 and x 2 are independent, then I (w 1, w 2 ; x 1, x 2 ) = 0. For jointly Gaussian random variables I (w 1, w 2 ; x 1, x 2 ) = w 1 w 2 C(x 1, x 2 ), in which C(x 1, x 2 ) is the covariance between x 1 and x 2. For stationary random processes we have I (w 1, w 2 ; τ) = I (w 1, w 2 ; x t+τ, x t ). Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

59 Impulsiveness Long memory - generalized sense Denition (Long memory in the generalized sense) Let {x t } t R be a stationary random process. We say that x t has long memory - generalized sense if its generalized co-diiference I (w 1, w 2 ; τ) w1 = w 2 =1 satises lim I (1, 1; τ)/τ β = L(τ), (31) τ in which L(τ) is a slow varying function for τ and 0 < β < 1. For LRD gaussian processes, the denition given above reduces to the classical denition of a LRD process. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

60 Final remarks Outline 1 Teletrac Fractal Nature Fractals The Hurst Exponent LRD Self-similarity Impulsiveness Final remarks Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61

61 Final remarks Why is the data networks trac fractal? Several authors have been stated that the Internet trac self-similarity is caused by the packets size large variability of the individual sessions (FTP, HTTP, etc.) that make the aggregate trac. Those papers argue that the IP trac is self-similar because the individual sessions sizes that make the Internet trac are generated by a heavy tail probability distribution. On the other hand, the recent study by Gong et al reexamines this question and claims that there is little evidence that the distribution's heavy tail has any impact on the algorithms' design and the Internet's infrastructure. Gong et al propose an on-o Markovian hierarchical model that explain the IP trac's LRD and state that the multiple time scales involved in the trac generation mechanism and transport protocols make the observation of LRD behavior unavoidable. Amazonas (Escola Politécnica da USP) Teletrac modeling and estimation São Paulo 11/ / 61