^{1}

^{2}

It is convincingly demonstrated by numerous studies that the self-similarity of modern multimedia network traffic is presented by Hurst parameter (H). The specific performance is that the similar degree is higher along with the increase of H when H is between 0.5 and 1. However, it is doubtable that whether the complicated process of self-similarity can be described comprehensively by the parameter H only. Therefore, another important parameter cf has been proposed based on the discrete wavelet decomposition in this paper. The significance of the parameters is provided and the performance of the self-similarity process is described better.

It is a truth universally acknowledged that the Self-Similarity of network traffic has demonstrated that network data exhibit two major attributes: scale-invariant and the slow power-law decrease of the autocorrelation function [

Recently, people have designed a number of approaches to evaluate the Hurst parameter, such as R/S method, variance-time analysis, periodogram method [

Therefore, unlike most existing studies that primarily focus on the estimating of H, we not only improve the estimator for the H, but also make a comparison and analysis between Hurst parameter and c_{f} parameter in this paper. What’s more, some detail comparison figure between H and c_{f} is displayed under the reasonable and additional technical idealization through the wavelet method and discrete wavelet decomposition. Based on the results, it is shown that the c_{f} parameter also plays a key role in measuring the self-similarity of networking traffic.

The remainder of the paper is set out as follows. Section 2 presents the mathematical definitions and properties of LRD. And we present the proposed method for estimating the Hurst parameter and c_{f} parameter of second-order self-similar process. In Section 3, the simulation results and deductions of experiments on the basis of wavelet method are analyzed. And concluding remarks and further research directions are finally presented in Section 4.

The network traffic in mathematics can be characterized as a random process, reflecting the self-similarity in the structure of network traffic on different time scales. The self-similarity process has complicated qualities and one of the important character is long-range dependence (LRD) [_{f} parameter, which can be described as follows:

Equivalently, it can be defined as the power-law divergence at the origin of its spectrum:

where,

and

Apparently, each of these definitions includes two parameters：

where,

According to the reference of [

Consequently, the expression of

The long-range dependence of networking traffic was analyzed by the wavelet decomposition coefficients in wavelet estimation methods [

where, the

And with the spectrum estimation method, the energy spectrum of

where,

where, α comes from the equation (5) and

Therefore, we can get the curve graph about j and

The networking traffic model simulates the actual networking traffic, which is the basis for analyzing networking performance, predicting networking traffic and designing networking destruction. Recently, there are many Self-Similarity networking traffic models [

In

In _{f} parameter.

In

From Figures 3-6 the wavelet decomposition for H = 0.6 and H = 0.7 are presented. The sequence length is reduced to the half of the original sequence and the decomposition coefficients will be used in estimating the self-similarity parameter in the next.

According to the relevant definition in section 2,

In _{f} parameter through the curve’s slope and related expression in

In _{f} parameter by the slop of curve are estimated based on the wavelet decomposition. When the theoretical value of H is 0.6 and the actual value is 0.602, the actual value of c_{f} is 5.614. Likewise, when the theoretical value of H is 0.7 and the actual value is 0.680, the actual value of c_{f} is 2.112. Obviously, the difference between theoretical value and actual value is small. The wavelet decomposition is an unbiased and efficient method.

In _{f} parameter are analyzed. According to eight different

c_{f} | |||||
---|---|---|---|---|---|

0.60 | 0.602 | 0.002 | 5.770 | 5.614 | 0.156 |

0.65 | 0.639 | 0.011 | 2.955 | 3.365 | 0.410 |

0.70 | 0.680 | 0.020 | 1.712 | 2.112 | 0.400 |

0.75 | 0.728 | 0.022 | 1.064 | 1.301 | 0.237 |

0.80 | 0.808 | 0.008 | 0.692 | 0.647 | 0.045 |

0.85 | 0.857 | 0.007 | 0.464 | 0.440 | 0.024 |

0.90 | 0.879 | 0.021 | 0.319 | 0.372 | 0.053 |

0.95 | 0.924 | 0.026 | 0.224 | 0.270 | 0.046 |

parameters, the different results of eight group data are shown in

Where, m means the difference between the theoretical and actual value of H parameter and v means the difference between the theoretical and actual value of c_{f} parameter. We can also get

In

In _{f} vividly. Where the abscissa means the group numbers of comparing is 8 and the ordinate describe the range of c_{f} value. And the dotted line represents theoretical value, the solid line represents the estimate value. Obviously, the c_{f} value is declining with the change of self-similarity of network traffic.

It can be seen the estimated result of H and c_{f} by wavelet method shows high accuracy. _{f} parameter, namely, the value of H parameter is increasing and the value of c_{f}parameter is decreasing with the change of self-similarity of network traffic.

The accurate estimation of self-similar characteristic parameters is the basis to improve Internet analysis and design. This paper makes a lot of theoretical analysis and numerical calculation on the basis of wavelet decomposition method. Not only estimating the H parameter, but also researching another important parameter c_{f}. Moreover, it reveals the fact that c_{f} parameter has important relationship with the Self-Similarity of networking traffic. The Self-Similarity degree is growing with the increasing of H when_{f} parameter plays a crucial role for estimating the Self-Similarity degree of networking traffic. Therefore, we should considerate both H parameter and c_{f} parameter in the future research. So it is not very accurate to take c_{f} as a constant in some article. This paper adopts FGN model and wavelet estimating method, so we also need further research to show the character of c_{f} parameter for other calculation.

In the end, this paper verified a novel parameter c_{f} for estimating the self-similarity of networking traffic. Through the fractional Gaussian Noise model and wavelet method, we get lots of simulation results that show the better performance of c_{f} parameter in networking traffic. In addition, it show the important role of c_{f} parameter for describing self-similarity. In the future research, we should estimate H parameter and c_{f} parameter together to measure the self-similarity of networking traffic better.

Peng Luo,Juan Wang, (2016) Another Important Parameter’s Research on Estimating Self-Similarity. International Journal of Communications, Network and System Sciences,09,338-345. doi: 10.4236/ijcns.2016.98030