# Traffic Behavior Analysis with Poisson Sampling on High-speed Network 1

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3 factors such as time of day, day of the week, etc. These factors may stay in effect for different time periods during the sampling process. Changes in the effective parameters create different stages of the stochastic process, such as morning stages, Spring Festival stages, etc. The advantage of the Poisson sampling process is that the duration length ratio of stages is captured in the ratio of number of observations taken during the stages, allowing proportional sampling of the different stages of the observed stochastic process [8]. Thirdly, the heterogeneous Poisson sampling arrivals are comparable. If { (t), t 0} and { (t), t 0} are both Poisson processes with rates and respectively ( ), then the averages of the samples obtained by (t) and (t) can be compared or combined depending on the time interval they have been applied [8]. This property allows utilization of different studies that apply Poisson sampling even if they use different parameters. Finally, there is flexibility on the stochastic arrival process from which the sample is selected. Poisson sampling can be applied to any kind of stochastic arrival process [8]. This is the most important property for applicability of Poisson sampling on the traffic traces, since there is no information on the type of stochastic arrival process of the traffic packets. Poisson sampling can be applied in two different cases: continuous process sampling and discrete process sampling. For continuous process sampling, selection of the next sample point is comparatively easy. The random number of the next sample is generated according to an exponential distribution with parameter (inter-arrival number of the next sample x Exp( )). The formulation of the random number generator for exponential distribution can be derived from the cumulative density function (cdf) of the exponential distribution, given in Equation. F( x) = x λx e f ( y) dy = 0,, x 0 x < 0 If x is generated according to an exponential distribution, then the outcome of cdf, F(x) for x () 0, has a Uniform(0, ) distribution. By calculating the analytical inverse of the exponential cdf in Equation, we can develop the desired formula, which is stated in Equation. After each sample point, a new uniform number u has to be generated to calculate the next exponentially distributed inter-arrival time using Equation. F (/ λ) *ln( u), 0 u ( u) = 0, u < 0 or u > The other case of the Poisson sampling, discrete process sampling is used where the stochastic process under observations has discrete arrivals. For discrete stochastic arrival traffic packets, sampling is done by randomly generating a number u Uniform(0, ) and then find the corresponding n, the number of arrivals to skip before the next sample, using Poisson process with parameter >0, { (t), t 0}. The probability mass function of the Poisson process is given in Equation 3. k λ exp( λ) F( x) =, >0, k=0,,, (3) k! However, the analytical inverse of the Equation 3 is not available. Therefore the following algorithm is used to generate the Poisson variant n [7]. First step: set j = 0 and y j = u 0, where u j Uniform(0, ), j = 0,,, Second step: if y j exp(- ), return n = j and stop. Last step: j = j+, and y j = u j y j-, Goto second step. As in the continuous sampling case, another random n is generated using the algorithm stated above. () 5. Sampling traffic performance analysis 5. sampling traffic results In order to illuminate the algorithm of Poisson sampling, now we study the relation between the mean length of packets of the Poisson sampling data sets and of the test data set. The entire traffic population consists of all traffic packets that pass through the monitored link all the time. For the analysis simplification, 0,000 continuous traffic packets are chosen as the test data set, and the selected samples will be Table : 0 sample set of Poisson sampling Sample Set λ n x S

4 used to estimate the characteristics of this data set. In the traffic packets traces structure, the entries are given in the order they arrive. Ten different sample sets were generated from the test data set using different sampling rates for Poisson sampling. The sampling rate is equivalent to the mean used for Poisson sampling, hence is the number of arrivals between two sample points. As the mean used for Poisson sampling increases, the number of observations skipped between the data units that contribute to the sample set increases either, implying a decrease in sample size. Table shows the sampling rate applied for Poisson sampling and the sample size for each individual sample. Therefore, each individual sample set is compared with the entire test data set. The mean length of packets obtained from the data set is 78.8 with a standard deviation of The mean length of packets obtained from each sample and the standard deviation of the sample of traffic packets are listed in Table. Suppose the traffic packet population mean packet length is 0, and its standard deviation 0, they can be shown as Equation 4. µ 0 = x i = 0 ( x i µ 0) σ (4) In Equation 4, x i is the length of the packet, and is the size of the traffic packet population. We also can get the mean packet length x of the sample set and its standard deviation S according to the Equation 5 that is the unbiased parameter estimation of the traffic population. x = n n x i s = n n ( x i x) In Equation 5, n is the size of the traffic packet sample set. From the Table, we can know that when (5) <00, the number of the sample set doesn't decrease continuously. Because the function rand() of linux return a pseudo-random integer between 0 and RAD_MAX, based on the problem, the maximum that we will choose for the sampling study, should be less than 00. value 5. Mean Length of Packets Hypothesis Tests When obtained the estimated value of parameter, hypothesis testing at the 5% and % significance levels must be used to test the statistical significance of the difference in mean length of packets between each sample set and the entire data set. The hypotheses are as follows Equation 6, H 0 = µ = µ H = µ µ i i where is the mean length of packets obtained from the data set and i is the mean length of packets obtained from sample set i, i =,, 0. Each sample is tested with respect to the entire data set, considering the mean length from the data set as a constant. Hence, the tests are based on comparing a sample mean to a specific value. Because the standard deviation sample size is larger than 30, so we choose (0, ) as the test statistics. The test statistics is Equation 7. X µ 0 U = ~ (0,) (7) σ / n (6) 0 is known and the Given 5% and % significance levels, we can get the u 0.05 =.95 and u =.575 according to additive Table A in [8]. Table displays the test statistics testing the statistical significance of the difference between the mean length of packets from each sample and the mean length of packets from the entire data set. The critical values for the relevant tests and the results of the hypothesis testing on the 5% and % significance levels. It can be observed that the difference between the mean of packets of each sample set and the test data set are statistically insignificant. In the other words, sample size larger than 00 can be used in traffic statistical study. Table : the hypothesis tests of mean length of packets Sampling Test Results of hypothesis testing set Value 95% 99% cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H cannot reject H 0 Cannot reject H Theoretical Sample Size for Mean Length

5 According to the precision, now we consider the choice of the sample size. The statistical determination of the appropriate random sample size for estimating the mean parameter of a population is provided in [9]. The appropriate sample size n can be calculated as Equation 8, σ n (8) ( ε µ ) + σ where is the population size, is the population standard deviation, is the specified accuracy, is the population mean. For our test data set, the packet size distribution had population mean =78 bytes, population standard deviation =687., and population size =0,000. With accuracy of = 5%, according to Equation 8, the appropriate sample size is larger than 300, so the sampling rate for Poisson Sampling is about 3. If we choose = %, then the sample size must large than The population size of the Equation 8 is limited. But for population size that is unlimited, we can induce the Equation 9 that can deal with the unlimited population size on the Equation 8. σ n (9) ( ε µ) The sign of the Equation 9 is the same as the Equation 8. And such as the above, we hypothesis that =78 bytes, and =687., and = 5%, then the sample size must be larger than 309 if the population size is unlimited. 5.4 The Estimation of Traffic Throughput According to the discussion of the above paragraphs, the estimated traffic throughput can be computed by the Equation 0 in the range of the precision of sampling parameters. µ n ( λ + ) 8 throughput = (0) during where is the mean length of sampling packets, n is the sampling size, is the Poisson sample rate, and during is the during time of measuring traffic sample. For example, in the paper, if the chosen precision =0.05, then =769 bytes, n= 303, =3, and during = 0.33s, so the traffic throughput is 64MB/S during the measuring traffic according to the Equation 0. 5 Conclusion Successful modeling of network traffic behavior can be achieved by effective analysis of the traffic data collected from the backbone router. Millions of collected data creates enormous traffic packet traces, the sizes of which are increasing exponentially as the network traffic monitored over the time, so that with current computational tools, even the simplest analytical tasks are difficult to perform. To solve the problem, it is reasonable to develop a sampling methodology for analyzing large traffic data sets. The challenge is how to choose samples effectively from the traffic population, which carry the statistical characteristics of the entire traffic data. Such a sample will reduce the size of the data to be collected and handled by network manager. This paper proposes a random sampling algorithm based on Poisson sampling that can tackle the task of choosing a sample set of manageable size to represent the whole data set. Poisson sampling is chosen because it does not require the traffic packet arrivals to conform to a particular stochastic arrival process and provides unbiased sampling. A test data set is collected from the real traffic from the CERET backbone. Ten different samples are chosen from the entire data set, using difference Poisson sampling rate, and the statistical significance of the difference between the mean length of packets from the 0 samples and the test data set are tested. The tests have shown that the difference between the means of packets from each sample set and the test data set were not statistically significant. we find that when <00, the number of the sample set doesn't decrease continuously. Poisson sampling is a successful sampling strategy that will create the opportunity to analyze large traffic traces using significantly reduced sample sizes without losing the statistical characteristics of the traffic population. More detailed analysis of data can be performed with less computational effort, which can provide the opportunity to apply types of analysis that are impossible due to today's software and hardware capacity limitations. Better and more detailed analysis of traffic traces will lead to better understanding of the high-speed network traffic behavior, and result in managed and devised network services.

6 References [] K. Claffy, G. Polyzos, and H. Braum, "Application of sampling Methodologies to etwork Traffic Characterization", Computer Communication Review, Vol. 3, o. 4, pp ,993. [] Kevin Thompson, gregory J. Miller, and Rick wilder, "Wide-Area Internet Traffic Patterns and characteristics (Extended Version)", IEEE etwork, ovember/december 997. [3] pcap - Packet Capture library, ftp://ftp.ee.lbl.gov/libpcap.tar.z, June 998. [4] V. Paxson, G. Almes, J. Mahdavi, M. Mathis, Framework for IP Performance Metrics, IETF RFC 330, May 998, [5] Jack Drobisz, Kenneth J. Christensen, Adaptive Sampling Methods to Determine etwork Traffic Statistics including the Hurst Parameter, 3 rd. Annual Conference on Local Computer etworks, October -4, 998. [6] Brewington, B.E. and Cybenko G. (000), "how Dynamic is the web?" 9 th world wide web conference, May 000. [7] Li JingChang, "Sampling Investigation and Deduction", the Publisher of China Statistics, 995. (in Chinese) [8] Tang Xiangen, Dai JianHua, "Mathematical Statistics", the Publisher of the Mechanism and Industry, 994. (in Chinese) [9] Xu baolu, "sampling theory", the Publisher of the BeiJing University, 98. (in Chinese)

7 The title: "Traffic Behavior Analysis with Poisson Sampling on High-speed etwork" Author's name: Guang Cheng, Jian Gong Author's organization: Department of Computer Science and Technology, Southeast University Correspondence Address: anjing 0096, P.R.China 0096 Telephone number: (860) address: The paper is intended to submit to the E-09: "Information etwork management" of the COFERECE E: "Progress, Problems and ew Trends in Information etwork"

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