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The paper focuses on measuring self-similarity using few techniques by an index called Hurst index which is a self-similarity parameter. It has been evident that Internet traffic exhibits self-similarity. Motivated by this fact, real time web users at various centers considered here as traffic and it has been examined by various methods to test the self-similarity. The results from the experiments carried out verify that the traffic examined in the present study is self similar using a new method based on some descriptive measures; for example percentiles have been applied to compute Hurst parameter which gives intensity of the self-similarity. Numerical results and analysis we discussed and presented here play a significant role to improve the services at web centers in the view of quality of service (QOS).

At present one of the major issues to know various traffic flows is in self similar nature to study and design some performance metric as that of Ethernet traffic etc. Until recently Poison approach has been used to model the road traffic irrespective of traffic intensity [

The idea of this paper is, we examine whether web users traffic data has the self similar property. This is to enhancement earlier results using a real time data [

In this section we give a short description of the mathematical basis for second order self-similar processes (long-range dependence).

Exact Second-Order Self-Similar ProcessThe exact second-order self-similar process is defined as follows. Arrival instants are modeled as point process. Divide the time axis into disjoint intervals of unit length and let

For each

This new series

Definition 1: The process “X” is said to be exactly second order self-similar with Hurst parameter

and variance

Definition 2: The process “X” is said to be asymptotically second order self-similar with Hurst parameter

In terms of variance, self-similar process is defined as follows:

Definition 3: The process “X” is said to be exactly second order self-similar with Hurst parameter

and variance

Now we shall differentiate long range dependence (LRD) and short range dependence (SRD) processes. For

The series

tive term series. Accordingly the left hand series

they are convergent. That is, for

As discussed in the introduction, we are primarily interested collecting data from various sources. Real time web users data has been considered. The sample number of users logged on to an Internet server each minute over 100-minutes (see Appendix). In the study web users data can be treated as traffic and verify it is self-similar or not.

The intensity of self-similarity is given by Hurst parameter, H. The parameter H was named after the hydrologist H.E. Hurst who spent many years to investigate the problem of water storage and also to determine the level patterns of the Nile river. The parameter H has range

In the frequency domain, analysis of time series is merely the analysis of a stationary process by means of its spectral representation. The periodogram [

where

In time series analysis [

It has already been observed that slow decay of correlation, which is proportional to

indicates the long-memory process. Therefore, the plot of the sample autocorrelation should exhibit this property. A much better plot for the handling of long-range dependence is the plot of ACF in logarithmic scale. If the asymptotic decay of the correlation is hyperbolic, then the points in the plot should be approximately scattered around a straight line with a negative slope of

Using this method, the obtained value of H in this case is 0.79.

In statistical methodologies, a percentile (or centile) is the value of a variable below which a certain percent of observations fall, like partition values of a process such as quartiles and deciles. There is no exact definition of percentile [

Given data set or time series

Using this method, the H value is computed for the data. The pertaining scattered data and trend line with the slope

In this paper, real time web user’s data has been considered as traffic from various web centers and it has been proved to be self-similar. Various methods to test the self-similarity have been used. The obtained values of Hurst parameter h are reasonably close to each other. This kind of research is useful for future studies to know the performance metrics at web centers.

PushpalathaSarla,Mallikarjuna ReddyDoodipala,ManoharDingari, (2016) Self Similarity Analysis of Web Users Arrival Pattern at Selected Web Centers. American Journal of Computational Mathematics,06,17-22. doi: 10.4236/ajcm.2016.61002

The number of users logged on to an Internet server each minute over 100-minutes.