Multifractal analysis of a class of additive processes with correlated non-stationary increments

Electronic Journal of Probability

Volumn 9, Issue , 2004, Pages 508-543

  Barral, Julien ^a   Véhel, Jacques Lévy ^a  

^a INRIA ROCQUENCOURT   (France)

Author keywords

Hausdorff dimension; H lder singularities; Internet Traffic Control Protocol; L vy processes; Multifractal processes; Spectrum of singularities

Indexed keywords

EID: 4944222353     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v9-208     Document Type: Article

References (32)

1. J.-M. Aubry and S. Jaffard, Random wavelet series. Commun. Math. Phys, 227 (2002), 483-514.
2. F. Baccelli and B. Thomas, Window Flow Control in FIFO Networks with Cross Traffic. Inria Tech. Rep. RR-3434 (1998).
3. F. Baccelli and D. Hong, TCP is Max-Plus Linear and what it tells us on its throughput. Inria Tech. Rep. RR-3986 (2000).
4. F. Baccelli and D. Hong, AIMD, Fairness and Fractal Scaling of TCP Traffic. INFO-COM, (June 2002).
5. F. Baccelli and D. Hong, Interaction of TCP Flows as Billiards. Inria Tech. Rep. RR-4437, (2002).
6. J. Barral and J. Levy Vehel, Large deviation spectrum of a class of additive processes with correlated non-stationary increments. In preparation, (2003).
7. J. Barral and J. Levy Vehel, Multifractality of TCP explained. Inria Tech. Rep. (2004).
8. J. Barral and B.B. Mandelbrot, Multifractal products of cylindrical pulses. Probab. Theory Relat. Fields, 124 (2002), 409-430.
9. J. Bertoin, Levy processes. Cambridge University Press, 1996.
10. A. Chaintreau, F. Baccelli and C. Diot, Impact of Network Delay Variation on Multicast Session Performance With TCP-like Congestion Control. Inria Tech. Rep. RR-3987 (2000).
11. M. Crovella and A. Bestavros, “Self-similarity in World Wide Web traffic: Evidence and possible causes,” IEEE/ACM Trans. on Networking, vol. 5, no. 6, pp. 835-846, Dec 1997.
12. R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments. J. Math. Mech., 10 (1961), 493-516.
13. T. D. Dang, S. Molnar and I. Maricza, Queuing Performance Estimation for General Multifractal Traffic. International Journal of Communication Systems, 16 (2), 117-136, 2003.
14. R. Gaigalas and I. Kaj, Convergence of scaled renewal processes and a packet arrival model. Bernoulli, 9(4) (2003), 671-703.
15. C. Houdre and J. Levy Vehel, Large deviation multifractal spectra of certain stochastic processes. Preprint.
16. P. Jacquet, Long term dependences and heavy tails in traffic and queues generated by memoryless ON/OFF sources in series. Inria Tech. Rep. RR-3516 (1998).
17. S. Jaffard, Old friends revisited. The multifractal nature of some classical functions. J. Fourier Anal. Appl., 3 (1997), 1-22.
18. S. Jaffard, The multifractal nature of Levy processes. Probab. Theory Rel. Fields, 114 (1999), 207-227.
19. S. Jaffard, On lacunary wavelet series. Ann. Appl. Probab., 10 (2000), 313-329.
20. J.-P. Kahane, Produits de poids aléatoires et indépendants et applications. in Fractal Geometry and Analysis, J. Belair and S. Dubuc (eds.), 1991, 277-324.
21. M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, 1991.
22. W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, “On the self-similar nature of Ethernet traffic (Extended Version),” IEEE/ACM Trans. on Networking, vol. 2, no. 1, pp. 1-15, Feb 1994.
23. J. Levy Vehel and R. Riedi, Fractional Brownian motion and data traffic modeling: The other end of the spectrum. Fractals in Engineering, J. Levy Vehel, E. Lutton and C. Tricot, Eds., Springer Verlag, 1997.
24. J. Levy Vehel and B. Sikdar, A Multiplicative Multifractal Model for TCP Traffic. Proceedings of IEEE ISCC, 714-719, (2001).
25. B.B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of hight moments and dimension of the carrier. J. fluid. Mech., 62 (1974), 331-358.
26. B.B. Mandelbrot, A class of multinomial multifractal measures with negative (latent) values for the “dimension” f (a). Fractals’ Physical Origins and Properties. Proceedings of the Erice Meeting, 1988. L. Pietronero, Ed., Plenum Press, New York, 1989, pp 3-29.
27. V. Paxson and S. Floyd, Wide area traffic: The failure of Poisson modeling. IEEE/ACM Trans. on Networking, vol. 3, no. 3, pp. 226-244, June 1995.
28. V. Pipiras, M.S. Taqqu and J.B. Levy, Slow, fast and arbitrary growth conditions for renewal reward processes when the renewals and the rewards are heavy tailed. Boston University Preprint, 2002.
29. R. H. Riedi and J. Levy-Vehel, TCP Traffic is multifractal: A numerical study. Inria Tech. Rep. RR-3129 (1997).
30. L. A. Shepp, Covering the line with random intervals. Z. Wahrsch. Verw. Gebeite, 23 (1972), 163-170.
31. W. R. Stevens, TCP/IP illustrated volume 1. Addison Wesley, 1994.
32. W. F. Stout, Almost Sure Convergence. Probability and Mathematical Statistics. Academic Press, New York, 1974.