ML Estimation of the Parameters of FBM Traffic with Geometrical Sampling

Traac model based on the fractional Brownian motion fBm contains three parameters: the mean rate m, variance parameter a and the Hurst parameter H. The estimation of these parameters by the maximum likelihood ML method is studied. Explicit expressions for the ML estimates ^ m and ^ a in terms of H are given, as well as the expression for the log-likelihood function from which the estimate ^ H is obtained as the minimizing argument. A geometric sequence of sampling points, t i = i , i s i n troduced in order to see the scaling behaviour of the traac with fewer samples. It is shown that by a proper`descaling' the traac process is stationary on this grid leading to a Toeplitz-type covariance matrix. Approximations for the inverted covariance matrix and its determinant are introduced. The accuracy of the estimation algorithm is studied by simulations. Comparisons with corresponding estimates obtained with linear grid show that the geometrical sampling indeed improves the accuracy of the estimate ^ H with a given number of samples.