Internet Traffic and Multiresolution Analysis
Ying Zhang, Zihui Ge, Suhas Diggavi, Z. Morley Mao, Matthew Roughan, Vinay Vaishampayan, Walter Willinger, Yin Zhang
Abstract
Traditional Internet traffic studies have primarily focused on the temporal characteristics of packet traces as observed on a single link within an ISP’s network. They have contributed to advances in the areas of self-similar stochastic processes, long-range dependence, and heavy-tailed distributions and have demonstrated the benefits of applying a wavelet-based multireso- lution analysis (MRA) approach when analyzing these traces. However, an ISP’s physical infrastructure typically consists of 100s or 1000s of such links which are connected by routers or switches, and the Internet as a whole is made up of about 20,000 such ISPs. When viewed within this bigger context, the importance of the traffic’s spatial characteristics becomes evident, and traffic matrices—compact and succinct descriptions of the traffic exchanges between nodes in a given network structure—are used in practice to capture and explore critical aspects of this spatial component of Internet traffic. In this paper, we first review some of the known results about the observed multi- faceted scaling behavior of Internet traffic as seen on a single link. Next, we give a detailed account of how the architectural design of the Internet gives rise to natural representation of traffic matrices at different scales or levels of resolution. Moreover, we discuss the development of a MRA-like framework of traffic matrices that respects the different physically or logically meaningful Internet connectivity structures and provides new insights into Internet traffic as a spatio-temporal object.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.imsc/1233152944
Digital Object Identifier: doi:10.1214/074921708000000390
References
[1] Abilene Network. http://www.internet2.edu/abilene.
[2] Abry, P. and Veitch, D. (1998). Wavelet analysis of long-range dependent traffic. IEEE Transactions on Information Theory 44 (1) 2–15.
Mathematical Reviews (MathSciNet):
MR1486645
Digital Object Identifier: doi:10.1109/18.650984
[3] Abry, P, Taqqu, M. S., Flandrin, P. and Veitch, D. (2000). Wavelets for the analysis, estimation, and synthesis of scaling data. In Self-Similar Network Traffic and Performance Evaluation (K. Park and W. Willinger, eds.) 39–88. Wiley, New York.
[4] Abry, P., Flandrin, P., Taqqu, M. S. and Veitch, D. (2003). Self-similarity and long-range dependence through the wavelet lens. In Long-range Dependence: Theory and Applications (P. Doukhan, G. Oppenheim and M. S. Taqqu, eds.) 527–556. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1956041
[5] Adler, R. J., Feldman, R. E. and Taqqu, M. S. (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1652283
[6] Alderson, D., Chang, H., Roughan, M., Uhlig, S. and Willinger, W. (2006). The many facets of Internet topology and traffic. Networks and Heterogeneous Media 1 (4) 569–600.
Mathematical Reviews (MathSciNet):
MR2276254
[7] Appenzeller, G., Keslassy, I. and McKeown, N. (2004). Sizing router buffers. Computer Communication Review (Proc. of ACM/Sigcomm’04, Portland, OR) 34 (4) 281–292.
[8] Beran, J. (1994). Statistics for Long-Memory Processes. Chapman & Hall, New York.
Mathematical Reviews (MathSciNet):
MR1304490
Zentralblatt MATH:
0869.60045
[9] Cao, J., Cleveland, W. S. and Sun, D. X. (2002). Internet traffic tends toward Poisson and independent as the load increases. In Nonlinear Estimation and Classification (C. Holmes, D. Dennison, M. Hansen, B. Yu and B. Mallick, eds.) 83–109. Springer-Verlag, New York.
[10] Cox, D. R. (1984). Long-range dependence: A review. In Statistics: An Appraisal (H. A. David and H. T. David, eds.) 55–74. Iowa State University Press, Ames, Iowa.
[11] Crovella, M. E. and Bestavros, A. (1996). Self-similarity in World Wide Web traffic—evidence and possible causes. Proc. ACM/Sigmetrics’96 160–169. Philadelphia, PA.
[12] Crovella, M. E. and Bestavros, A. (1997). Self-similarity in World Wide Web traffic—evidence and possible causes. IEEE/ACM Transactions on Networking 5 835–846.
[13] Crovella, M. E. and Kolaczyk, E. (2003). Graph wavelets for spatial traffic analysis. Proc. IEEE Infocom.
[14] Crovella, M. E., Taqqu, M. S. and Bestavros, A. (1998). Heavy-tailed probability distributions in the World Wide Web. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications (R. Adler, R. Feldman and M. S. Taqqu, eds.) 27–53. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1652283
[15] Crovella, M. E. and Krishnamurthy, B. (2006). Internet Measurements: Infrastructure, Traffic, and Applications. J. Wiley & Sons, New York.
[16] Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. SIAM.
Mathematical Reviews (MathSciNet):
MR1162107
[17] Jiang, H. and Dovrolis, C. (2005). Why is Internet traffic bursty in short (sub-RTT) time scales? Proc. ACM/Sigmetrics’05. Banff, Canada.
[18] Erramilli, A., Narayan, O. and Willinger, W. (1996). Experimental queueing analysis with long-range dependent packet traffic. IEEE/ACM Transactions on Networking 4 (2) 209–223.
[19] Erramilli, A., Roughan, M., Veitch, D., and Willinger, W. (2002). Self-similar traffic and network dynamics. Proceedings of the IEEE 90 (5) 800–819.
[20] Feldmann, A. (2000). Characteristics of TCP connection arrivals. In Self-Similar Network Traffic and Performance Evaluation (K. Park and W. Willinger, eds.) 367–399. Wiley, New York.
[21] Feldmann, A., Gilbert, A. C., Willinger, W., and Kurtz, T. G. (1998). The changing nature of network traffic: Scaling phenomena. Computer Communication Review 28 5–29.
[22] Feldmann, A., Gilbert, A. C., Huang, P., and Willinger, W. (1999). Dynamics of IP traffic: A study of the role of variability and the impact of control. Proc. ACM/Sigcomm’99 301–313. Cambridge, MA.
[23] Floyd, S. and Paxson, V. (2001). Difficulties in simulating the Internet. IEEE/ACM Transactions on Networking 9 (4) 392–403.
[24] Fowler, H. J. and Leland, W. E. (1991). Local area network traffic characteristics, with implications for broadband network congestion management. IEEE Journal on Selected Areas in Communication 9 1139–1149.
[25] Gromoll, H. C. and Williams, R. J. (2006). Fluid limit of a network with fair bandwidth sharing and general document size distributions. Preprint.
[26] Gromoll, H. C. and Williams, R. J. (2008). Fluid model for a data network with α-fair bandwidth sharing and general document size distributions: Two examples of stability. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (S. N. Ethier, J. Feng and R. H. Stockbridge, eds.) 255–267. Institute of Mathematical Statistics, Beachwood, OH.
Mathematical Reviews (MathSciNet):
MR2574235
Zentralblatt MATH:
1166.90312
Digital Object Identifier: doi:10.1214/074921708000000417
[27] Joo, Y., Ribeiro, V., Feldmann, A., Gilbert, A. C. and Willinger, W. (2001). TCP/IP traffic dynamics and network performance: A lesson in workload modeling, flow control, and trace-driven simulations. Computer Commu- nication Review 31 (2) 25–37.
[28] Kaj, I. and Taqqu, M. S. (2005). Convergence to fractional Brownian motion and to the Telecom process: The integral representation approach. Preprint.
Mathematical Reviews (MathSciNet):
MR2477392
Zentralblatt MATH:
1154.60020
Digital Object Identifier: doi:10.1007/978-3-7643-8786-0_19
[29] Kannan, J., Jung, J., Paxson, V. and Koksal, C. E. (2006). Semiautomated discovery of application session structure. ACM/Sigcomm Internet Measurement Conference IMC’06. Rio de Janeiro, Brazil, to appear.
Mathematical Reviews (MathSciNet):
MR2215826
[30] Karagiannis, T., Molle, M. and Faloutsos, M.. Long-range dependence: Ten years of Internet traffic modeling. IEEE Internet Computing 8 (5) 57–64.
[31] Kurtz, T. G. (1996). Limit theorems for workload input models. In Stochastic Networks: Theory and Applications (F.P. Kelly, S. Zachary and I. Ziedins, eds.) 119–139. Oxford University Press, Oxford, UK.
[32] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson. D. V. (1993). On the self-similar nature of Ethernet traffic. Proc. of ACM/Sigcomm’93 183–193. San Francisco, CA.
[33] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson. D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Transactions on Networking 2 1–15.
[34] Levy, J. B. and Taqqu, M. S. (2000). Renewal reward processes with heavytailed interrenewal times and heavy-tailed rewards. Bernoulli 6 (1) 23–44.
Mathematical Reviews (MathSciNet):
MR1781180
Digital Object Identifier: doi:10.2307/3318631
Project Euclid: euclid.bj/1082665378
[35] Low, S. H., Paganini, F. and Doyle, J. C. (2002). Internet congestion control. IEEE Control Systems Magazine (Feb.) 28–43.
[36] Mandelbrot, B. B. (1963). New methods in statistical economics. Journal of Political Economics 71 421–440.
[37] Mandelbrot, B. B. (1969). Long-run linearity, locally Gaussian processes, H-spectra and infinite variances. International Economic Review 10 82–113.
[38] Medina, A., Fraleigh, C., Taft, N., Bhattacharyya, S. and Diot, C. (2002). A taxonomy of IP traffic matrices. Proc. SPIE ITCOM 2002. Boston, MA.
[39] Mikosch, T., Resnick, S., Rootzen, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Annals of Applied Probability, 12 (1) 23–68.
Mathematical Reviews (MathSciNet):
MR1890056
Digital Object Identifier: doi:10.1214/aoap/1015961155
Project Euclid: euclid.aoap/1015961155
[40] Paganini, F., Wang, Z., Doyle, J. C. and Low, S. H. (2005). Conges- tion control for high performance, stability, and fairness in general networks. IEEE/ACM Transactions on Networking 13 (1) 43–56.
[41] Park, K. and Willinger, W. (2000). Self-Similar Network Traffic and Performance Evaluation. J. Wiley & Sons, New York.
[42] Paxson, V. and Floyd, S. (1994). Wide-area traffic: The failure of Poisson modeling. Computer Communication Review (Proc. of ACM/Sigcomm’94, London, UK) 24 (4) 257–268.
[43] Paxson, V. and Floyd, S. (1995). Wide area traffic: The failure of Poisson modeling. IEEE/ACM Transactions on Networking 3 226–244.
[44] Pipiras, V., Taqqu, M. S. and Levy, J. B. (2004). Slow, fast, and arbitrary growth conditions for renewal reward processes when the renewals and the rewards are heavy-tailed. Bernoulli 10 121–163.
Mathematical Reviews (MathSciNet):
MR2044596
Digital Object Identifier: doi:10.3150/bj/1077544606
Project Euclid: euclid.bj/1077544606
[45] Resnick, S. I. (1997). Heavy tail modeling and teletraffic data. The Annals of Statistics 25 1805–1869.
Mathematical Reviews (MathSciNet):
MR1474072
Zentralblatt MATH:
0942.62097
Digital Object Identifier: doi:10.1214/aos/1069362376
Project Euclid: euclid.aos/1069362376
[46] Roughan, M. and Kalmanek, C. R. (2003). Pragmatic modeling of broadband access traffic. Computer Communications 26 (8) 804–816.
[47] Roughan, M. (2005). Simplifying the synthesis of Internet traffic matrices. Computer Communication Review 35 93–96.
[48] Roughan, M., Greenberg, A., Kalmanek, C., Rumsewicz, M., Yates, J. and Zhang, Y. (2003). Experience in measuring Internet backbone traffic variability: Models, metrics, measurements, and meaning. Proc. ITC 18 379–388. Berlin, Germany.
[49] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, London.
Mathematical Reviews (MathSciNet):
MR1280932
Zentralblatt MATH:
0925.60027
[50] Sommers, J., Barford, P., Duffield, N. and Ron, A. (2007). Efficient network-wide SLA compliance monitoring. Computer Communication Review (Proc. of ACM/Sigcomm’07, Kyoto, Japan) 39 (4), to appear.
[51] Tang, A., Wang, J., Low, S. H., and Chiang, M. (2005). Equilibrium of heterogeneous congestion control: Existence and uniqueness. Proc. IEEE Infocom 2005.
[52] Taqqu, M. S., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Computer Communication Review 27 5–23.
[53] Vetterli, M. and Kovacevic, J. (1995). Wavelets and Subband Coding. Prentice Hall, Englewood Cliffs, NJ.
[54] Willinger, W., Taqqu, M. S. and Erramilli, A. (1996). A bibliographical guide to self-similar traffic and performance modeling for modern highspeed networks. In Stochastic Networks: Theory and Applications (F. P. Kelly, S. Zachary and I. Ziedins, eds.) 339–366. Oxford University Press, Oxford, UK.
[55] Willinger, W., Taqqu, M. S., Leland, W. E. and Wilson, D. V. (1995). Self-similarity in high-speed packet traffic: Analysis and modeling of Ethernet traffic measurements. Statistical Science 10 (1) 67–85.
Mathematical Reviews (MathSciNet):
MR1113418
Digital Object Identifier: doi:10.1111/j.1467-9965.1991.tb00004.x
[56] Willinger, W., Taqqu, M. S., Sherman, R. and Wilson, D. V. (1997). Self-similarity through high-variability: Statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Transactions in Networking 5 (1) 71–86.
[57] Willinger, W., Paxson, V. and Taqqu, M. S. (1998). Self-similarity and heavy tails: Structural modeling of network traffic. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications (R. Adler, R. Feldman and M. S. Taqqu, eds.) 27–53. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1652283
[58] Willinger, W., Alderson, D. and Li, L. (2004). A pragmatic approach to dealing with high-variability in network measurements. Proc. 2004 ACM/Sigcomm Internet Measurement Conference (IMC’04) 88–100.
[59] Zhang, Y., Roughan, M., Lund, C. and Donoho, D. (2005). Estimating point-to-point and point-to-multipoint traffic matrices: An information-theoretic approach. IEEE/ACM Transactions on Networking 13 947–960.
Institute of Mathematical Statistics Collections
