TKK | Tietoverkkolaboratorio | Tutkimus
The following individuals have contributed to the library: Attila Vidács and Jorma Virtamo.
Reports of bugs in the program will be gratefully received by vidacs@ttt-atm.ttt.bme.hu
A more detailed description is given within the source codes of the routines. Furthermore, the following papers deal with the subject:
where X(t) represents the amount of traffic arrived in (0,t). The model has three parameters, m is the mean input rate, a is a variance parameter and 0.5 < H < 1 is the self-similarity parameter of Z(t), which is a normalized fractional Brownian motion.
Assume the traffic has been observed at n time instants forming the vector t = (t_{1}, t_{2}, ..., t_{n})^{t}. Since X(t) is Gaussian, the joint distribution of the observed traffic values (X_{1}, X_{2}, ..., X_{n}) is n dimensional Gaussian with mean m·t and covariance matrix G(n x n). Thus, a Gaussian Maximum Likelihood Estimation (MLE) can be applied (in the time domain) to estimate the model parameters m, a and H. Explicit expressions for the ML estimates of m and a in terms of H can be given, as well as the expression for the log-likelihood function from which the estimate of H is obtained as the minimizing argument.
The Hurst parameter H describes the scaling behavior of the traffic. Therefore, in order to determine its value from measured traffic, the sample points have to cover several time scales. With the ordinary linear sampling this leads to the requirement of very large number of sample points. In order to use the measurements more efficiently we introduce a geometric sequence of sampling points:
In addition to distributing the sample points in a better way on different time scales, geometric sampling fits neatly with the self-similar behavior of the fBm traffic. By a proper descaling the traffic process is stationary on this grid leading to a Toeplitz-type covariance matrix, and allows us to develop approximations to the inverse and determinant of the covariance matrix G.
In practice, exact MLE poses computational problems. To avoid these problems, one can use approximate methods to calculate the inverse and the determinant of the covariance matrix G. For example, a simple linear approximation can be used to compute an approximate inverse and its determinant for G. Nevertheless, the use of this approximation in the likelihood function yields a good estimate for H, while the accuracy of the estimate of a sufers more. (An improved approximation can be achieved if we approximate the inverse of G by a band-matrix.)
However, the geometrical sampling does indeed give a better estimate for H than the traditional linear sampling. For the estimation of a the geometrical sampling does not give any direct advantage, but as its estimator actually depends on the estimator of H, the overall accuracy obtained with geometrical sampling is better. (As for m, in fact, the ML estimate gives almost negligible reduction in its variance when compared to the sample mean as an estimate for m.)
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