Abstract
Let D(n) and H(n) be the fractal dimension and the Hurst parameter of traffic in the nth interval, respectively. Thus, this paper gives the experimental variance analysis of D(n) and H(n) of network traffic based on the generalized Cauchy (GC) process on an interval-by-interval basis. We experimentally infer that traffic has the phenomenon Var[D(n)] > Var[H(n)]. This suggests a new way to describe the multifractal phenomenon of traffic. That is, traffic has local high-variability and global robustness. Verifications of that inequality are demonstrated with real traffic.
Chapter PDF
References
Taqqu, M.S., Teverovsky, V., Willinger, W.: Is Network Traffic Self-Similar or Multifractal? Fractals 5, 63–73 (1997)
Peltier, R.F., Levy-Vehel, J.: Multifractional Brownian Motion: Definition and Preliminaries Results. INRIA TR 2645 (1995)
Lim, S.C., Muniandy, S.V.: On Some Possible Generalizations of Fractional Brownian Motion. Physics Letters A 226, 140–145 (2000)
Willinger, W., Paxson, V., Riedi, R.H., Taqqu, M.S.: Long-Range Dependence and Data Network Traffic. In: Doukhan, P., Oppenheim, G., Taqqu, M.S. (eds.) Long-range Dependence: Theory and Applications, pp. 625–715. Birkhäuser, Basel (2002)
Li, M.: Change Trend of Averaged Hurst Parameter of Traffic under DDOS Flood Attacks. Computers & Security 25, 213–220 (2006)
Beran, J., Sherman, R., Taqqu, M.S., Willinger, W.: Long-Range Dependence in Variable Bit-Rate Video Traffic. IEEE T. Communications 43, 1566–1579 (1995)
Li, M., Zhao, W., et al.: Modeling Autocorrelation Functions of Self-Similar Teletraffic in Communication Networks based on Optimal Approximation in Hilbert Space. Applied Mathematical Modelling 27, 155–168 (2003)
Tsybakov, B., Georganas, N.D.: Self-Similar Processes in Communications Networks. IEEE T. Information Theory 44, 1713–1725 (1998)
Paxson, V., Floyd, S.: Wide Area Traffic: the Failure of Poison Modeling. IEEE/ACM T. Networking 3, 226–244 (1995)
Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman, New York (1982)
Chiles, J.-P., Delfiner, P.: Geostatistics, Modeling Spatial Uncertainty. Wiley, New York (1999)
Gneiting, T., Schlather, M.: Stochastic Models that Separate Fractal Dimension and Hurst Effect. SIAM Review 46, 269–282 (2004)
Li, M., Lim, S.C.: Modeling Network Traffic Using Cauchy Correlation Model with Long-Range Dependence. Modern Physics Letters B 19, 829–840 (2005)
Lim, S.C., Li, M.: Generalized Cauchy Process and Its Application to Relaxation Phenomena. J. Phys. A: Math. Gen. 39, 2935–2951 (2006)
Mandelbrot, B.B.: Multifractals and 1/f Noise. Springer, Heidelberg (1998)
Mandelbrot, B.B.: Gaussian Self-Affinity and Fractals. Springer, Heidelberg (2001)
Feldmann, A., Gilbert, A.C., Willinger, W., Kurtz, T.G.: The Changing Nature of Network Traffic: Scaling Phenomena. Computer Communication Review 28, 5–29 (1998)
Willinger, W., Govindan, R., Jamin, S., Paxson, V., Shenker, S.: Scaling Phenomena in the Internet: Critically Examining Criticality. Proceedings of Natl. Acad. Sci. USA 99(1 Suppl.), 2573–2580 (2002)
Kent, J.T., Wood, T.A.: Estimating the Fractal Dimension of a Locally Self-Similar Gaussian Process by Using Increments. J. R. Statit. Soc. B 59, 579–599 (1997)
Adler, R.J.: The Geometry of Random Fields. Wiley, New York (1981)
Davies, S., Hall, P.: Fractal Analysis of Surface Roughness by Using Spatial Data. Journal of the Royal Statistical Society Series B 61, 3–37 (1999)
Hall, P.: On the Effect of Measuring a Self-Similar Process. SIAM J. Appl. Math. 35, 800–808 (1995)
Constantine, A.G., Hall, P.: Characterizing Surface Smoothness via Estimation of Effective Fractal Dimension. Journal of the Royal Statistical Society Ser. B 56, 97–113 (1994)
Hall, P., Roy, R.: On the Relationship between Fractal Dimension and Fractal Index for Stationary Stochastic Processes. The Annals of Applied Probability 4, 241–253 (1994)
Chan, G., Hall, P., Poskitt, D.S.: Periodogram-Based Estimators of Fractal Properties. The Annals of Statistics 23, 1684–1711 (1995)
Todorovic, P.: An Introduction to Stochastic Processes and Their Applications, p. 100. Springer, New York (1992)
Lim, S.C., Muniandy, S.V.: Generalized Ornstein-Uhlenbeck Processes and Associated Self-Similar Processes. J. Phys. A: Math. Gen. 36, 3961–3982 (2003)
Wolperta, R.L., Taqqu, M.S.: Fractional Ornstein–Uhlenbeck Lévy Processes and the Telecom Process Upstairs and Downstairs. Signal Processing 85, 1523–1545 (2005)
On-line available: http://ita.ee.lbl.gov/html/traces.html
On-line available: http://pma.nlanr.net/Traces/Traces/daily/
Leland, W., Taqqu, M.S., Willinger, W., Wilson, D.: On the Self-Similar Nature of Ethernet Traffic (extended version). IEEE/ACM T. Networking 2, 1–15 (1994)
Li, M., Jia, W.J., Zhao, W.: Correlation Form of Timestamp Increment Sequences of Self-Similar Traffic on Ethernet. Electronics Letters 36, 1168–1169 (2000)
Li, M., Lim, S.C.: A Rigorous Derivation of Power Spectrum of Fractional Gaussian Noise. Fluctuation and Noise Letters 6, C33–C36 (2006)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: the Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Li, M., Lim, S.C., Feng, H. (2007). A Novel Description of Multifractal Phenomenon of Network Traffic Based on Generalized Cauchy Process. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds) Computational Science – ICCS 2007. ICCS 2007. Lecture Notes in Computer Science, vol 4489. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72588-6_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-72588-6_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72587-9
Online ISBN: 978-3-540-72588-6
eBook Packages: Computer ScienceComputer Science (R0)