A Novel Clock Skew Estimator and Its Performance for the IEEE 1588v2 (PTP) Case in Fractional Gaussian Noise/Generalized Fractional Gaussian Noise Environment

Papers in the literature dealing with the Ethernet network characterize packet delay variation (PDV) as a long-range dependence (LRD) process. Fractional Gaussian noise (fGn) or generalized fraction Gaussian noise (gfGn) belong to the LRD process. This paper proposes a novel clock skew estimator for the IEEE1588v2 applicable for the white-Gaussian, fGn, or gfGn environment. The clock skew estimator does not depend on the unknown asymmetry between the fixed delays in the forward and reverse paths nor on the clock offset between the Master and Slave. In addition, we supply a closed-form-approximated expression for the mean square error (MSE) related to our new proposed clock skew estimator. This expression is a function of the Hurst exponent H, as a function of the parameter a for the gfGn case, as a function of the total sent Sync messages, as a function of the Sync period, and as a function of the PDV variances of the forward and reverse paths. Simulation results confirm that our closed-form-approximated expression for the MSE indeed supplies the performance of our new proposed clock skew estimator efficiently for various values of the Hurst exponent, for the parameter a in gfGn case, for different Sync periods, for various values for the number of Sync periods and for various values for the PDV variances of the forward and reverse paths. Simulation results also show the advantage in the performance of our new proposed clock skew estimator compared to the literature known ML-like estimator (MLLE) that maximizes the likelihood function obtained based on a reduced subset of observations (the first and last timing stamps). This paper also presents designing graphs for the system designer that show the number of the Sync periods needed to get the required clock skew performance (MSE = 10–12). Thus, the system designer can approximately know in advance the total delay or the time the system has to wait until getting the required system’s performance from the MSE point of view.

Supervised Learning of Neural Networks for Active Queue Management in the Internet

The paper examines the AQM mechanism based on neural networks. The active queue management allows packets to be dropped from the router’s queue before the buffer is full. The aim of the work is to use machine learning to create a model that copies the behavior of the AQM PIα mechanism. We create training samples taking into account the self-similarity of network traffic. The model uses fractional Gaussian noise as a source. The quantitative analysis is based on simulation. During the tests, we analyzed the length of the queue, the number of rejected packets and waiting times in the queues. The proposed mechanism shows the usefulness of the Active Queue Management mechanism based on Neural Networks.

Response of Quasi-Integrable and Resonant Hamiltonian Systems to Fractional Gaussian Noise

The major difficulty in studying the response of multi-degrees-of-freedom (MDOF) nonlinear dynamical systems driven by fractional Gaussian noise (fGn) is that the system response is not Markov process diffusion and thus the diffusion process theory cannot be applied. Although the stochastic averaging method (SAM) for quasi Hamiltonian systems driven by fGn has been developed, the response of the averaged systems still needs to be predicted by using Monte Carlo simulation. Later, noticing that fGn has rather flat power spectral density (PSD) in certain frequency band, the SAM for MDOF quasi-integrable and nonresonant Hamiltonian system driven by wideband random process has been applied to investigate the response of quasi-integrable and nonresonant Hamiltonian systems driven by fGn. The analytical solution for the response of an example was obtained and well agrees with Monte Carlo simulation. In the present paper, the SAM for quasi-integrable and resonant Hamiltonian systems is applied to investigate the response of quasi-integrable and resonant Hamiltonian system driven by fGn. Due to the resonance, the theoretical procedure and calculation will be more complicated than the nonresonant case. For an example, some analytical solutions for stationary probability density function (PDF) and stationary statistics are obtained. The Monte Carlo simulation results of original system confirmed the effectiveness of the analytical solutions under certain condition.

Characterizing stochastic resonance in a triple cavity

Many biological systems possess confined structures, which produce novel influences on the dynamics. Here, stochastic resonance (SR) in a triple cavity that consists of three units and is subjected to noise, periodic force and vertical constance force is studied, by calculating the spectral amplification η numerically. Meanwhile, SR in the given triple cavity and differences from other structures are explored. First, it is found that the cavity parameters can eliminate or regulate the maximum of η and the noise intensity that induces this maximum. Second, compared to a double cavity with similar maximum/minimum widths and distances between two maximum widths as the triple cavity, η in the triple one shows a larger maximum. Next, the conversion of the natural boundary in the pure potential to the reflection boundary in the triple cavity will create the necessity of a vertical force to induce SR and lead to a decrease in the maximum of η . In addition, η monotonically decreases with the increase of the vertical force and frequency of the periodic force, while it presents several trends when increasing the periodic force’s amplitude for different noise intensities. Finally, our studies are extended to the impact of fractional Gaussian noise excitations. This article is part of the theme issue ‘Vibrational and stochastic resonance in driven nonlinear systems (part 2)’.

APPROXIMATE SOLUTIONS FOR THE KOLMOGOROV-WIENER FILTER WEIGHT FUNCTION FOR CONTINUOUS FRACTIONAL GAUSSIAN NOISE

Context. We consider the Kolmogorov-Wiener filter for forecasting of telecommunication traffic in the framework of a continuous fractional Gaussian noise model. Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding WienerHopf integral equation. Also the aim of the work is to show the convergence of the proposed method of solution of the corresponding equation. Method. The Wiener-Hopf integral equation for the filter weight function is a Fredholm integral equation of the first kind. We use the truncated polynomial expansion method in order to obtain an approximate solution of the corresponding equation. A set of Chebyshev polynomials of the first kind is used. Results. We obtained approximate solutions for the Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise. The solutions are obtained in the approximations of different number of polynomials; the results are obtained up to the nineteen-polynomial approximation. It is shown that the proposed method is convergent for the problem under consideration, i.e. the accuracy of the coincidence of the left-hand and right-hand sides of the integral equation increases with the number of polynomials. Such convergence takes place due to the fact that the correlation function of continuous fractional Gaussian noise, which is the kernel of the corresponding integral equation, is a positively-defined function. Conclusions. The Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise is obtained as an approximate solution of the corresponding Fredholm integral equation of the first kind. The proposed truncated polynomial expansion method is convergent for the problem under consideration. As is known, one of the simplest telecommunication traffic models is the model of continuous fractional Gaussian noise, so the results of the paper may be useful for telecommunication traffic forecast.