The modelling of Ethernet data and of signals that are heavy-tailed with infinite variance. (English) Zbl 1020.60082

Let \(U_i\) be i.i.d. interrenewal times such that \(P(U\geq u)=u^{-\alpha}L_U(u)\), \(\alpha\in(1,2)\); let \(W_i\) be i.i.d. rewards independent of \(\{U_i,\;i=1,2,\dots\}\) with \(E W=0\) and \(\sigma^2=E W^2<\infty\) (FVR, finite variance rewards) or \(P(|W|\geq w)=w^{-\beta}L_W(w)\), \(w>0\) (IVR, infinite variance rewards) (\(L_U\), \(L_W\) are slowly varying functions at infinity). The renewal reward process is defined as \(W(t)=W_n\) if \(t\) belongs to the \(n\)th interrenewal interval, \(W^*(t,M)=\sum_{m=1}^M\int_0^t W^{(m)}(u) du\), where \(W^{(m)}\) are i.i.d. copies of \(W(t)\). The author considers the limit behavior of \(W^*(Tt,M)\) as \(T\to\infty\), \(M\to\infty\). E.g. in the FVR case it is shown that if \(L^*_U\) is a slowly varying function such that \(\forall x>0\), \(L^*_U(u)^{-\alpha}L(u^{1/\alpha}L^*_U(u)x)\to 1\) as \(u\to\infty\) and \[ \lim_{T\to\infty}{M\over T^{\alpha-1}}(L_U^*(MT))^\alpha =\infty, \] then \[ \lim_{T\to\infty}{W^*(Tt,M)\over T^{(3-\alpha)/2}M^{1/2}(L_U(T))^{1/2}}=\sigma_0 B_H(t) \] (in distribution), where \(B_H\) is a standard fractional Brownian motion. In other results the limit processes are the symmetric Lévy motion and a symmetric \(\beta\)-stable process. The processes \(W^{(m)}\) are used to describe a centered load of one workstation \(m\) in the Ethernet local area network at time \(t\). Then \(W^*\) is the aggregated load.


60K15 Markov renewal processes, semi-Markov processes
90B18 Communication networks in operations research
Full Text: DOI


[1] DOI: 10.1109/18.650984 · Zbl 0905.94006 · doi:10.1109/18.650984
[2] 2P Abry, P Flandrin, M. S. Taqqu, and D. Veitch (2001 ). Self-similarity and long-range dependence through the wavelet lens . InLong-range dependence: theory and applications(eds P. Doukhan, G. Oppenheim & M. S. Taqqu), Birkhauser, Boston. Preprint. · Zbl 1029.60028
[3] DOI: 10.1109/26.380206 · doi:10.1109/26.380206
[4] 4N. H. Bingham, C. M. Goldie, and J. L. Teugels (1987 ).Regular variation. Cambridge University Press, Cambridge. · Zbl 0617.26001
[5] 5A. Feldmann, A. Gilbert, and W. Willinger (1998 ). Data networks as cascades: investigating the multifractal nature of Internet WAN traffic . InProceedings of the ACM/SIGCOMM’98,28, 42-55. Vancouver, Canada.
[6] Feldmann A., Comput. Comm. Rev. 29 pp 301– (1999)
[7] 7M. W. Garrett, and W. Willinger (1994 ). Analysis, modeling and generation of self-similar VBR video traffic . InProceedings of the ACM Sigcomm ’94, London,UK, 269-280.
[8] 8A. N. Kolmogorov (1941 ). Local structure of turbulence in an incompressible liquid for very large Reynolds numbers .C. R. (Dokl.) Acad. Sci. URSS (N.S.)30, 299-303. Reprinted in Friedlander, G. K. & Topper, L. (1961).Turbulence: classic papers on statistical theory. Interscience, New York.
[9] Levy J., Ann. Sci. Math. Quebec 11 pp 95– (1987)
[10] Levy J. B., Bernoulli 6 pp 23– (2000)
[11] DOI: 10.1109/90.282603 · doi:10.1109/90.282603
[12] 12W. E. Leland, and D. V. Wilson (1991 ). High time-resolution measurement and analysis of LAN traffic: implications for LAN interconnection . InProceedings of the Infocom ’91, Bal Harbour, FL, 1360-1366.
[13] Mandelbrot B. B., Internat. Econom. Rev. 10 pp 82– (1969)
[14] 14P. Mannersalo, and I. Norros (1997 ). Multifractal analysis of real ATM traffic: a first look . Technical report, COST257TD, VTT Information Technology.
[15] 15T. Mikosch, S. Resnick, H. Rootzen, and A. Stegeman (1999 ). Is network traffic approximated by stable Levy motion or fractional Brownian motion? Preprint.
[16] 16K. Park, and W. Willinger (eds) (2000 ).Self-similar network traffice and performance evaluation. Wiley, New York.
[17] Pipiras V., Bernoulli 6 pp 607– (2000)
[18] 18V. Pipiras, M. S. Taqqu, and J. B. Levy (2000 ). Slow, fast and arbitrary growth conditions for renewal reward processes when the renewals and the rewards are heavy-tailed . Preprint. · Zbl 1043.60040
[19] 19S. I. Resnick, and H. Rootzen (1998 ). Self-similar communication models and very heavy tails . ORIE Technical Report 1228, Cornell University.
[20] 20S. I. Resnick, and G. Samorodnitsky (2000 ). Fluid queues, leaky buckets, on-off processes and teletraffic modeling with highly variable and correlated inputs . InSelf-similar network traffic and performance evaluation(eds K. Park & W. Willinger), Wiley Interscience, New York.
[21] 21R. H. Riedi, and J. L. Vehel (1997 ). Multifractal properties of TCP traffic: a numerical study . INRIA Research Report 3129. See also: Levy Vehel, J. & Riedi, R. (1997) Fractional Brownian motion and data traffic modeling. InFractals in engineering, from theory and industrial applications, 185-202. Springer, New York.
[22] 22G. Samorodnitsky, and M. S. Taqqu (1994 ).Stable non-Gaussian processes: stochastic models with infinite variance.Chapman & Hall, New York, London. · Zbl 0925.60027
[23] DOI: 10.1145/359038.359044 · doi:10.1145/359038.359044
[24] 24M. S. Taqqu, and J. Levy (1986 ). Using renewal processes to generate long-range dependence and high variability . InDependence in probability and statistics(eds E. Eberlein & M. S. Taqqu), 73-89, Birkhauser, Boston
[25] Taqqu M. S., Stoch. Models 13 pp 723– (1997)
[26] 26M. S. Taqqu, and V. Teverovsky (1998 ). On estimating the intensity of long-range dependence in finite and infinite variance series . InA practical guide to heavy tails: statistical techniques and applications(eds R. Adler, R. Feldman & M. S. Taqqu), 177-217, Birkhauser, Boston.
[27] 27M. S. Taqqu, V. Teverovsky, and W. Willinger (1995 ). Estimators for long-range dependence: an empirical study .Fractals3, 785-798. Reprinted inFractal geometry and analysis(eds C. J. G. Evertsz, H.O. Peitgens & R. F. Voss) World Scientific Publishing, Singapore, 1996.
[28] Taqqu M. S., Fractals 5 pp 63– (1997)
[29] Taqqu M. S., Comput. Comm. Rev. 27 pp 5– (1997)
[30] Willinger W., Statist. Sci. 10 pp 67– (1995)
[31] 31W. Willinger, M. S. Taqqu, and A. Erramilli (1996 ). A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks . InStochastic networks: theory and applications(eds F. P. Kelly, S. Zachary & I. Ziedins), 339-366, Clarendon Press, Oxford. · Zbl 0855.60086
[32] DOI: 10.1109/90.554723 · doi:10.1109/90.554723
[33] 33W. Willinger, V. Paxson, R. H. Riedi, and M. S. Taqqu (2001 ). Long-range dependence and data network traffic . InLong-range dependence: theory and applications(eds P. Doukhan, G. Oppenheim & M. S. Taqqu), Birkhauser, Boston. Preprint. · Zbl 1109.60317
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