Empirical testing of the infinite source poisson data traffic model

Stochastic Models

Volumn 19, Issue 2, 2003, Pages 151-200

  Guerin, C A ^a   Nyberg, H ^b   Perrin, O ^c   Resnick, S ^d   Rootzen H ^e   Starica C ^e  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b ERICSSON RESEARCH   (Sweden)

^c UNIVERSITY OF TOULOUSE CAPITOLE   (France)

^d Cornell University   (United States)

^e CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Data transmission modelling; Heavy tails; Internet traffic; Pareto tails; Regular variation; Scaling; Self similarity

Indexed keywords

APPROXIMATION THEORY; DATA COMMUNICATION SYSTEMS; INTERNET; PARETO PRINCIPLE; POISSON DISTRIBUTION; QUEUEING THEORY; RANDOM PROCESSES; TELECOMMUNICATION TRAFFIC; VARIATIONAL TECHNIQUES;

DATA TRAFFIC MODEL; INFINITE SOURCE POISSON MODEL; INTERNET TRAFFIC; QUEUE APPROXIMATION;

DATA STRUCTURES;

EID: 0037592876     PISSN: 15326349     EISSN: None     Source Type: Journal    
DOI: 10.1081/STM-120020386     Document Type: Article

References (88)

1. Abry, P. Ondelettes et Turbulences-Multirésolutions, Algorithmes de Décompositions, Invariance d'Echelle et Signaux de Pression. Diderot, Editeur des Sciences et des Arts, Paris, 1997.
2. Abry, P.; Delbecke, L.; Flandrin, P. Waveletbased estimator for the self-similarity parameter of α-stable processes. IEEE-ICASSP 99 p III 1581-1584, Phoenix, Ar, USA, March 1999.
3. Abry, P.; Flandrin, P.; Taqqu, M.S.; Veitch, D. Wavelets for the analysis, estimation and synthesis of scaling data. In Self-similar network analysis and performance evolution; Park, K., Willinger, W., Eds.; Wiley: New York, 2000; 30-88.
4. Abry, P.; Veitch, D. Wavelet analysis of long-range-dependent traffic. IEEE Trans. Inform. Theory 1998, 44, 2-15.
5. Adler, R.J. An introduction to continuity, extrema, and related topics for general Gaussian processes. I.M.S. Lecture Notes-Monogr. Series 1990, 12.
6. Adler, R.J.; Feldman, R.; Taqqu, M.S. A practical guide to heavy tails: statistical techniques and applications; Brikhauser: Boston, 1998.
7. Arlitt, M.; Williamson, C.L. Web server workload characterization: the search for invariants (extended version). Proceedings of the ACM Sigmetrics Conference, Philadelphia, PA, 1996. Available from {mfa126,carey}@cs.usaska.ca.
8. Andersen, A.T.; Nielsen, B.F. On the statistical implications of certain random permutations in Markovian Arrival Processes (MAP)s and second order self-similar processes. Perform. Eval. 2000, 41, 67-82.
9. Asmussen, S. Applied Probability and Queues; Wiley: Chichester, UK, 1987.
10. Barford, P.; Bestavros, A.; Bradely, A.; Crovella, M. E. Changes in web client access patterns: characteristics and caching implications. World Wide Web, Special Issue on Characterization and Perform. Eval. 1999, 2, 15-28.
11. Beirlant, J.; Vynckier, P.; Teugels, J. Tail index estimation, Pareto quantile plots, and regression diagnostics. J. Am. Stat. Assoc. 1996, 91, 1659-1667.
12. Beran, J. Statistics for long-memory processes; Chapman & Hall: New York, 1994.
13. Billingsley, P. Convergence of probability measures, 2nd Ed.; Wiley: New York, 1999.
14. Cambanis, S.; Houdré, C. On the continuous wavelet transform of second-order random processes. IEEE Trans. Inform. Theory 1995, 41, 628-642.
15. Cohen, J.; Resnick, S.; Samorodnitsky, G. Sample correlations of infinite variance time series models: and empirical and theoretical study. J. Appl. Math. Stochast. Anal. 1998, 1, 255-282.
16. Constantine, A.G.; Hall, P. Characterizing surface smoothness via estimation of effective fractal dimension. J. R. Stat. Soc. B 1994, 56, 97-113.
17. Crovella, M.; Bestavros, A. Explaining world wide web traffic self-similarity. Preprint available as TR-95-015 from {crovella,best}cs.bu.edu, 1995.
18. Crovella, M.; Bestavros, A. Self-similarity in world wide web traffic: evidence and possible causes. Proceedings of the 1996 ACM SIGMETRICS International Conference on Measurement and Modelling of Computer System, 1996.
19. Crovella, M.; Bestravros, A.; Taqqu, M. Heavy tailed distributions in the world wide web. A practical guide to heavy tails: statistical techniques and applications; Birkhauser: Boston, 1998.
20. Crovella, M.; Kim, G.; Park, K. On the relationship between file sizes, transport protocols, and self-similar network traffic. Proceedings of the Fourth International Conference on Network Protocols (ICNP '96), 1996; 171-180.
21. Crovella, M.; Park, K.; Kim, G. On the effect of traffic self-similarity on network performance. Proceedings of the SPIE International Conference on Performance and Control of Networked Systems, 1997.
22. Cunha, C.R.; Crovella, M.; Bestavros, A. Characteristics of www client-based traces. Preprint BU-CS-95-010, Boston University, 1995.
23. Daubechies, I. Ten lectures on wavelets; SIAM: Philadelphia, PA, 1992.
24. Davis, R.; Resnick, S.I. Limit theory for bilinear processes with heavy tailed noise. Ann. Appl. Probab. 1996, 6, 1191-1210.
25. Durrett, R.; Resnick, S. Weak convergence with random indices. J. Stochast. Proc. Appl. 1977, 5, 213-220.
26. Embrechts, P.; Klüppelberg, C.; Mikosh, T. Modelling extremal events for insurance and finance; Springer: Heidelberg, 1997.
27. Erramilli, A.; Narayan, O.; Willinger, W. Experimental queueing analysis with long-range dependent packet traffic. IEEE/ACM Trans. Network Comput. 1996, 4, 209-223.
28. Feldman, A.; Gilbert, A.C.; Willinger, W. Data networks as cascades: Investigating the multifractal nature of Internet WAN traffic. Proc. of the ACM/SIGCOMM '98, Vancouver, B.C., 1998; 25-38.
29. Flandrin, P. Wavelets analysis and synthesis of fractional Brownian motion. IEEE Trans. Inform. Theory 1992, 38, 910-917.
30. Feigin, P.; Resnick, S. Pitfalls of fitting autoregressive models for heavy tailed time series. Extremes 1999, 1, 391-422.
31. Gilbert, A.C.; Willinger, W.; Feldmann, A. Scaling analysis of conservative cascades, with applications to network traffic. IEEE Trans. Inform. Theory 1999, 45 3, 971-991.
32. Gogl, H. Data Acquisition and Data Analysis in ATM based Networks. Report 1998, Technical University of Munich, available at http://wwwjessen.informatik.tu-muenchen.de/forschungs/leistungsanalyse/ATM_Proje ct/atm_index.html
33. Gogl, H. Traffic statistics and application profiles in ATM networks. Report 1998, Technical Uiversity of Munich, available at http://wwwjessen.informatik.tu-muenchen.de/forschung/leistungsanalyse/ATM_Projec t/atm_index.html.
34. Gladyshev, E.G. A new limit theorem for processes with Gaussian increments. Theory Probab. Appl. 1961, 6, 52-61.
35. Guérin, C.A. Wavelet analysis and covariance structure of some classes of non-stationary processes. J. Fourier Anal. Appl. 2000, 4 6, 403-425.
36. Hall, P.; Wood, A. On the performance of box-counting estimators of fractal dimension. Biometrika 1993, 80, 246-252.
37. Hall, P.; Wood, A.; Feuerverger, A. Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings. J. Timer Ser. Anal. 1994, 15 no 6, 587-606.
38. Michael Harrison, J. Brownian Motion and Stochastic Flow Systems; Wiley: New York, 1985.
39. Heath, D.; Resnick, S.; Samorodnitsky, G. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res. 1998, 23, 145-165.
40. Heath, D.; Resnick, S.; Samorodnitsky, G. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Ann. Appl. Probab. 1999, 9, 352-375.
41. Hill, B.M. A simple general approach to inference about the tail of a distribution. Ann. Stat. 1975, 3, 1163-1174.
42. Hunt, F. Error analysis and convergence of capacity dimension algorithms. SIAM J. Appl. Math. 1990, 50, 307-321.
43. Ibragimov, L.A.; Rozanov, Y.A. Gaussian random processes; Springer Verlag: Berlin, 1978.
44. Istas, J. Wavelets coefficients of a Gaussian process and applications. Ann. Inst. Henri Poincaré 1992, 28, 537-556.
45. Istas, J.; Lang, G. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré 1997, 33, 407-436.
46. Istas, J.; Laredo, C. Estimation de la fonction de covariance d'un processus gaussien stationnaire par méthodes d'échelles. C. R. Acad. Sci., Tome 1993, 316, 495-498, Série I.
47. Jelenković P.R.; Lazar, A.A. Subexponential asymptotics of a Markov-modulated G/G/1 queue. Technical report, Columbia University, 1995.
48. Jelenković P.R.; Lazar, A.A. Multiplexing on/off sources with subexponential on-periods. Technical Report CTR. 457-96-23, Columbia University, 1996.
49. Jelenković, P.; Lazar, A. Asymtotic results for multiplexing subexponential on-off processes. Adv. Appl. Probab. 1999, 31, 394-421.
50. Kaj, I. Convergence of scaled renewal processes to fractional Brownian motion. Technical report, available at http://www.math.uu.se/~ikaj/papers.html, 1999.
51. Kent, J.T.; Wood, A.T.A. Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. R. Stat. Soc. B 1997, 59 no 3, 679-699.
52. Klüppelberg, C.; Mikosch, T. Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1995, 1, 125-148.
53. Konstantopoulos, T.; S.-J., Lin Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 1998, 28, 215-243.
54. Kratz, M.; Resnick, S. The qq-estimator and heavy tails. Stochast. Models 1996, 12, 699-724.
55. Kurtz, T. Limit theorems for workload input models. In Stochastic Networks: Theory and Applications; Zachary, S., Kelly, F.P., Ziedins, I., Eds.; Clarendon Press: Oxford, 1996.
56. Lamperti, J.W. Semi-stable stochastic processes. Trans. Am. Math. Soc. 1962, 62-78.
57. Lang, G. Estimation de la régularité et de la longue dépendence de processus gaussiens. Thése de l'université Toulouse III, 1994.
58. Leland, W.E.; Taqqu, M.S.; Willinger, W.; Wilson, D.V. On the self-similar nature of Ethernet traffic. ACM/SIGCOMM Computer Communications Review, 1993. Proceedings of the ACM/SIGCOMM93. To appear.
59. Leland, W.E.; Taqqu, M.S.; Willinger, W.; Wilson, D.V. On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 1994, 2, 1-15.
60. Leland, W.E.; Taqqu, M.S.; Willinger, W.; Wilson, D.V. Statistical analysis of high time resolution Ethernet Lan traffic measurements. In. PCmpScSt25, 146-155, 1993
61. Maulik, K.; Resnick, S.; Rootzen, H. Asymptotic independence and a network traffic model. J. Appl. Probab. 2002, 4, 671-699.
62. Maulik, K.; Resnick, S. Small and large time scale analysis of a network traffic model. Technical report, available at www.orie.cornell.edu/trlist/trlist.html, 2001.
63. Miokosch, T.; Resnick, S.; Rootzén, H.; Stegeman, A. Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Preprint 1999, available as #1999:32 at http://www.md.chalmers.se/Stat/Research/Preprints.
64. Nolan, J.P. Maximum likelihood estimation and diagnostics for stable processes. Levy processes; Birkhaüser Boston: Boston, MA, 2001.
65. Prabhu, N.U. Stochastic Storage Processes: Queues, Insurance risk, Dams and Data Communication; Springer: New York, 1998.
66. Reiss, R.D.; Thomas, M. Statistical analysis of extreme values; Birghäuser: Basel, 1997.
67. Resnick, S.I. Extreme Values, Regular Variation and Point Processes; Springer: New York, 1987.
68. Resnick, S.I. Heavy tail modelling and teletraffic data. Ann. Stat. 1997, 25, 1805-1869.
69. Resnick, S.; Samorodnitsky, G. Performance decay in a single server exponential queuing model with long range dependence. Oper. Res. 1997, 45, 235-243.
70. Resnick, S.; Samorodnitsky, G.; Xue, F. How misleading can sample acf's of stable ma's be? (Very!). Ann. Appl. Probab. 1999, 9 3, 797-817.
71. Resnick, C. Smoothing the Hill estimator. Adv. Appl. Probab. 1997, 29, 271-293.
72. Resnick, S.I.; Rootzén, H. Self-similar communication models and very heavy tails. Ann. Appl. Probab. 2000, 10, 753-778.
73. Resnick, S.; van den Berg, E. A test for nonlinearity of time series with infinite variance. Extremes 2000, 3 4, 145-172.
74. Resnick, S.; van den Berg, E. Weak Convergence of high-speed network traffic models. J. Appl. Probab. 2000, 37 2, 575-597.
75. Riedi, R.H. Multifractal processes. Long range dependence: theory and applications; Birkhauser: Boston, 2001.
76. Riedi, R.H.; Willinger, W. Toward an improved understanding of network traffic dynamics. Self-similar Network Traffic and Performance Evaluation; Wiley: New York, 2000.
77. Samorodnitsky, G.; Taqqu, M. Stable non-Gaussian random processes; Chapman and Hall: New York, 1994.
78. Tajvidi N. Confidence Intervals and Accuracy Estimation for Heavy tailed Generalized Pareto Distribution. Part of the PhD thesis, 1996, Chalmers, available at http://www.mai.liu.se/~nataj/
79. Taqqu, M.; Willinger, W.; Sherman, R. Proof of a fundamental result in self-similar traffic modelling. Comp. Commun. Rev. 1997, 27.
80. Mandelbrot, B.B.; Taqqu, M.S. Robust R/S analysis of long-run serial correlation. Proceedings of the 42nd Session of the International Statistical Institute Manila, Vol.48, Book 2, pp. 69-104, Bulletin of the ISI, 1979.
81. Tewfik, A.H.; Kim, M. Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans. Inform. Theory 1992, 38, 904-909.
82. Veitch, D.; Abry, P. Estimation conjointe en ondelettes des paramèters du phénomène de dépendence longue. Proc. 16ième Colloque GRETSI, Grenoble, France, 1997; 1451-1454.
83. Veitch, D.; Abry, P. A wavelet based joint estimator for the parameters of Ird. IEEE Trans. Inform. Theory. 1998, special issue "Multiscale Statistical Signal Analysis and its Application".
84. Whitt, W. Some useful functions for functional limit theorems. Math. Oper. Res. 1980, 5, 67-85.
85. Willinger, W.; Taqqu, M.S.; Leland, M.; Wilson, D. Self-similarity in high-speed packet traffic: analysis and modelling of ethernet traffic measurements. Stat. Sci. 1995, 10, 67-85.
86. Willinger, W.; Taqqu, M.S.; Leland, M.; Wilson, D. Self-similarity through high variability: statistical analysis of ethernet lan traffic at the source level. Comput. Commun. Rev. 1995, 25, 100-113, Proceedings of the ACM/SIGCOMM'95, Cambridge, MA.
87. Willinger, W.; Taqqu, M.S.; Leland, M.; Wilson, D. Wilson, D. Self-similarity through high variability: statistical analysis of ethernet lan traffic at the source level (extended version). IEEE/ACM Trans. Networking 1996. To appear.
88. d. J. Comput. Graphical Stat. 1994, 3, 409-432.