Modeling teletraffic arrivals by a Poisson cluster process

Queueing Systems

Volumn 54, Issue 2, 2006, Pages 121-140

  Faÿ, Gilles ^a   González Arévalo, Bárbara ^b   Mikosch, Thomas ^c   Samorodnitsky, Gennady ^d  

^a UNIVERSITÉ LILLE 1   (France)

^b ROOSEVELT UNIVERSITY   (United States)

^c UNIVERSITY OF COPENHAGEN   (Denmark)

^d Cornell University   (United States)

Author keywords

Long range dependence; Palm distribution; Poisson cluster model; Regular variation; Teletraffic

Indexed keywords

STOCHASTIC SYSTEMS;

CLUSTER MODELS; LONG RANGE DEPENDENCE; PALM DISTRIBUTION; REGULAR VARIATIONS; TELETRAFFIC;

POISSON DISTRIBUTION;

EID: 33749351362     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1007/s11134-006-9348-z     Document Type: Article

References (40)

1. G. Alsmeyer, On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1992) 1229-1247.
2. F. Baccelli and P. Brémaud, Elements of Queueing Theory. Palm Martingale Calculus and Stochastic Recurrences, 2nd edition (Springer, New York, 2003).
3. J.M. Bardet, G. Lang, E. Moulines, and P. Soulier, Wavelet estimator of long-range dependent processes. Stat. Inference Stoch. Process. 3 (2000) 85-99.
4. 19th "Rencontres Franco-Belges de Statisticiens" (Marseille, 1998).
5. M.S. Bartlett, The spectral analysis of point processes. J. Roy. Statist. Soc., Series B 25 (1963) 264-296.
6. P. Billingsley, Convergence of Probability Measures. (Wiley, New York, 1968).
7. N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation (Cambridge University Press, Cambridge (UK), 1987).
8. P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods, (2nd edition) (Springer, New York, 1991).
9. D.B.H. Cline and T. Hsing, Large deviation probabilities for sums of random variables with heavy or subexponential tails. Technical Report, Texas A& M University (1998).
10. M. Crovella and A. Bestavros, Self-similarity in world wide web traffic: evidence and possible causes. In Proceedings of the 1996 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems (1996) vol. 24, pp. 160-169.
11. M. Crovella, A. Bestavros, and M.S. Taqqu, Heavy-tailed probability distributions in the world wide web. A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions, R., Adler, R. Feldman, and M.S., Taqqu (eds.) (Birkhäuser, Boston, 1998) pp. 3-26.
12. D. Daley, Asymptotic properties of stationary point processes with generalized clusters. Z. Wahrscheinlichkeitstheorie verw. Geb. 21 (1972) 65-76.
13. D. Daley and R. Vesilo, Long range dependence of point processes, with queueing examples. Stochastic Processes and Their Applications 70 (1997) 265-282.
14. D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes (Springer, New York, 1988).
15. P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 1997).
16. P. Embrechts and E. Omey, On subordinated distributions and random record processes. Proc. Camb. Phil. Soc. 93 (1983) 339-353.
17. G. Faÿ, F. Roueff, and P. Soulier, Estimation of the memory parameter of the infinite source Poisson process. Preprint (2005). Available at http://arxiv.org/abs/math/0509371.
18. A. Gut, Stopped Random Walks (Springer, New York, 1988).
19. N.R. Hansen, Markov Controlled Excursions, Local Alignment and Structure. PhD thesis, Institute of Mathematics, University of Copenhagen (2003).
20. D. Heath, S. Resnick, and G. Samorodnitsky, Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Oper. Res. 23 (1998) 145-165.
21. N. Hohn and D. Veitch, Inverting sampled traffic. ACM/SIGCOMM Internet Measurement Conference, (Miami, USA, 2003) pp. 222-233.
22. N. Hohn, D. Veitch, and P. Abry, Cluster processes: a natural language for network traffic. IEEE Trans. Signal Process. 51 (2003), 2229-2244.
23. O. Kallenberg, Foundations of Modern Probability, 2nd edition (Springer, New York, 2001).
24. W.E. Leland, M.S. Taqqu, W. Willinger, and D.V. Wilson, On the self-similar nature of Ethernet traffic. ACM/SIGCOMM Computer Communications Review (1993) 183-193.
25. P.A.W. Lewis, A branching Poisson process model for the analysis of computer failure patterns. J. Roy. Statist. Soc., Series B 26 (1964) 398-456.
26. T. Mikosch and G. Samorodnitsky, The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 (2000) 1025-1064.
27. T. Mikosch and G. Samorodnitsky, Scaling limits for cumulative input processes. Math. Oper. Research (2006) To appear.
28. T. Mikosch, S. Resnick, H. Rootzén, and A. Stegeman, Is network traffic approximated by stable Levy motion or fractional Brownian motion? Ann. Appl. Probab. 12 (2002) 23-68.
29. E. Moulines, F. Roueff, and M. Taqqu, A wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series. Preprint (2005). Available at http://arxiv.org/abs/math.ST/0601070.
30. A.V. Nagaev, Limit theorems for large deviations where Cramér's conditions are violated (in Russian). Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk 6 (1969a) 17-22.
31. A.V. Nagaev, Integral limit theorems for large deviations when Cramér's condition is not fulfilled I,II. Theory Probab. Appl. 14 (1969b) 51-64, 193-208.
32. S.V. Nagaev, Large deviations of sums of independent random variables. Ann. Probab. 7 (1979) 745-789.
33. V.V. Petrov, Limit Theorems of Probability Theory (Oxford University Press, Oxford, 1995).
34. S.I. Resnick, Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18 (1986) 66-138.
35. S.I. Resnick, Extreme Values, Regular Variation, and Point Processes. (Springer, New York, 1987).
36. G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance (Chapman & Hall, New York, 1994).
37. A.J. Stam, Regular variation of the tail of a subordinated probability distribution. Adv. Appl. Probab. 5 (1973) 287-307.
38. M. Westcott, On existence and mixing results for cluster point processes. J. Royal Stat. Soc. Series B 33 (1971) 290-300.
39. W. Willinger, M.S. Taqqu, R. Sherman, and D. Wilson, Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level. Proceedings of the ACM/ SIGCOMM'95, Cambridge, MA. Computer Communications Review 25 (1995) 100-113.
40. G.W. Wornell and A.V. Oppenheim, Estimation of fractal signals from noisy measurements using wavelets. IEEE Transactions on Signal Processing 40 (1992) 611-623.