Journal of Simulation

, Volume 7, Issue 2, pp 83–89 | Cite as

Efficient online generation of the correlation structure of the fGn process



Detailed observations of communications networks have revealed singular statistical properties of the measurements, such as self-similarity, long-range dependence and heavy tails, which cannot be overlooked in modelling Internet traffic. The use of stochastic processes consistent with these properties has opened new research fields in network performance analysis and particularly in simulation studies, where the efficient synthetic generation of samples is one of the main topics. In this paper, we describe an efficient and online generator of the correlation structure of the fractional Gaussian noise process.


fGn process M/G/∞ process heavy tails efficiency online generation 


  1. Abry P and Veitch D (1998). Wavelet analysis of long-range dependent traffic. IEEE Transactions on Information Theory 44 (1): 2–15.CrossRefGoogle Scholar
  2. Abry P, Borgnat P, Ricciato F, Scherrer A and Veitch D (2010). Revisiting an old friend: On the observability of the relation between long range dependence and heavy tail. Telecommunications Systems 43 (3–4): 147–165.CrossRefGoogle Scholar
  3. Albert R, Jeong H and Barabási AL (1999). Diameter of the World Wide Web. Nature 401: 130–131.CrossRefGoogle Scholar
  4. Arlitt MF and Williamson CL (1997). Internet web serves: Workload characterization and performance implications. IEEE/ACM Transactions on Networking 5 (5): 631–645.CrossRefGoogle Scholar
  5. Beran J, Sherman R, Taqqu MS and Willinger W (1995). Long-range dependence in variable-bit-rate video traffic. IEEE Transactions on Communications 43 (2–4): 1566–1579.CrossRefGoogle Scholar
  6. Brichet F, Roberts J, Simonian A and Veitch D (1996). Heavy traffic analysis of a storage model with long-range dependent on/off sources. Queueing Systems 23 (1–4): 197–215.CrossRefGoogle Scholar
  7. Conti M, Gregori E and Larsson A (1996). Study of the impact of MPEG-1 correlations on video sources statistical multiplexing. IEEE Journal on Selected Areas in Communications 14 (7): 1455–1471.CrossRefGoogle Scholar
  8. Cox D and Isham V (1980). Point Processes. Chapman and Hall: London, UK.Google Scholar
  9. Crovella ME and Bestavros A (1997). Self-similarity in World Wide Wed traffic: Evidence and possible causes. IEEE/ACM Transactions on Networking 5 (6): 835–846.CrossRefGoogle Scholar
  10. Davies RB and Harte DS (1987). Test for Hurst effect. Biometrika 74 (1): 95–102.CrossRefGoogle Scholar
  11. Duffield N (1987). Queueing at large resources driven by long-tailed M/G/∞ processes. Queueing Systems 28 (1–3): 245–266.Google Scholar
  12. Eliazar I (2007). The M/G/∞ system revisited: Finiteness, summability, long-range dependence and reverse engineering. Queueing Systems 55 (1): 71–82.CrossRefGoogle Scholar
  13. Eliazar I and Klafter J (2009). A unified and universal explanation for Lévy laws and 1/f noises. Proceedings of the National Academy of Sciences 106 (30): 12251–12254.CrossRefGoogle Scholar
  14. 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.CrossRefGoogle Scholar
  15. Faloutsos M, Faloutsos P and Faloutsos C (1999). On power-law relationships of the Internet topology. Proceedings SIGCOMM’99. ACM: Cambridge, MA, pp 251–262.Google Scholar
  16. Graf HP (1983). Long-range correlations and estimation of the self-similarity parameter PhD Thesis, University of Zurich.Google Scholar
  17. Hosking JRM (1984). Modeling persistence in hydrological time series using fractional differencing. Water Resources Research 20 (12): 1898–1908.CrossRefGoogle Scholar
  18. Hurst HE (1951). Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116: 770–799.Google Scholar
  19. Jiang M, Nikolic M, Hardy S and Trajkovic L (2001). Impact of self-similarity on wireless data network performance. Proceedings ICC’01. IEEE: Helsinki, Finland, pp 477–481.Google Scholar
  20. Kaplan L and Kuo C (1994). Extending self-similarity for fractional Brownian motion. IEEE Transactions Signal Processing 42 (12): 3526–3530.CrossRefGoogle Scholar
  21. Krunz M and Makowski A (1998). Modeling video traffic using M/G/∞ input processes: A compromise between Markovian and LRD models. IEEE Journal on Selected Areas in Communications 16 (5): 733–748.CrossRefGoogle Scholar
  22. Lau WC, Erramilli A, Wang JL and Willinger W (1995). Self-similar traffic generation: The random midpoint displacement algorithm and its properties. Proceedings ICC’95. IEEE: Seattle, WA, pp 466–472.Google Scholar
  23. L'Ecuyer P and Touzin R (2000). Fast combined multiple recursive generators with multipliers of the form a=+−2q+−2r. Proceedings of the 2000 Winter Simulation Conference. ACM: Orlando, FL, pp 683–689.CrossRefGoogle Scholar
  24. Ledesma S and Liu D (2000). Synthesis of fractional Gaussian noise using linear approximation for generating self-similar network traffic. ACM Computer Communication Review 30 (2): 4–17.CrossRefGoogle Scholar
  25. Ledesma S, Liu D and Hernández D (2007). Two approximation methods to synthesize the power spectrum of fractional Gaussian noise. Computational Statistics and Data Analysis 52 (2): 1047–1062.CrossRefGoogle Scholar
  26. Leland WE, Taqqu MS, Willinger W and Wilson DV (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Transactions on Networking 2 (1): 1–15.CrossRefGoogle Scholar
  27. Le-Ngoc T and Subramanian SN (2000). A Pareto-modulated Poisson process (PMPP) model for long-range dependent traffic. Computer Communications 23 (2): 123–132.CrossRefGoogle Scholar
  28. Likhanov N, Tsybakov B and Georganas ND (1995). Analysis of an ATM buffer with self-similar (fractal) input traffic. Proceedings INFOCOM’95. IEEE: Boston, MA, pp 985–992.Google Scholar
  29. López JC, López C, Suárez A, Fernández A and Rodríguez RF (2000). On the use of self-similar processes in network simulation. ACM Transactions on Modeling and Computer Simulation 10 (2): 125–151.CrossRefGoogle Scholar
  30. Ma S and Ji C (2001). Modeling heterogeneous network traffic in wavelet domain. IEEE/ACM Transactions on Networking 9 (5): 634–649.CrossRefGoogle Scholar
  31. Mandelbrot BB (1971). A fast fractional Gaussian noise generator. Water Resources Research 7 (3): 543–553.CrossRefGoogle Scholar
  32. Mandelbrot BB and Van Ness JW (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (4): 422–437.CrossRefGoogle Scholar
  33. Maulik K and Resnick S (2003). Small and large time scale analysis of a network traffic model. Queueing Systems 43 (3): 221–250.CrossRefGoogle Scholar
  34. Norros I (1995). On the use of fractional Brownian motion in the theory of connectionless networks. IEEE Journal on Selected Areas in Communications 13 (6): 953–962.CrossRefGoogle Scholar
  35. Novak M (2004). Thinking in Patterns: Fractals and Related Phenomena in Nature. World Science: Singapore.CrossRefGoogle Scholar
  36. Parulekar M (1997). Buffer engineering for M/G/∞ input process PhD Thesis, University of Maryland.Google Scholar
  37. Paxson V (1994). Empirically-derived analytic models of wide-area TCP connections. IEEE/ACM Transactions on Networking 2 (4): 316–336.CrossRefGoogle Scholar
  38. Paxson V (1997). Fast, approximate synthesis of fractional Gaussian noise for generating self-similar traffic. Computer Communication Review 27 (5): 5–18.CrossRefGoogle Scholar
  39. Paxson V and Floyd S (1995). Wide-area traffic: The failure of Poisson modeling. IEEE/ACM Transactions on Networking 3 (3): 226–244.CrossRefGoogle Scholar
  40. Purczynsky J and Wlodarski P (2006). On fast generation of fractional Gaussian noise. Computational Statistics and Data Analysis 50 (10): 2537–2551.CrossRefGoogle Scholar
  41. Resnick S (2007). Heavy-Tail Phenomena. Springer: Berlin, Germany.Google Scholar
  42. Resnick S and Rootzen H (2000). Self-similar communication models and very heavy tails. Annals of Applied Probability 10 (3): 753–778.CrossRefGoogle Scholar
  43. Samorodnitsky G and Taqqu MS (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall: New York, NY.Google Scholar
  44. Sousa ME, Suárez A, López JC, López C and Fernández M (2008). On improving the efficiency of a M/G/∞ generator of correlated traces. Operations Research Letters 36 (2): 184–188.CrossRefGoogle Scholar
  45. Sousa ME, et al (2007). Application of the Whittle estimator to the modeling of traffic based on the M/G/∞ process. IEEE Communications Letters 11 (10): 817–819.CrossRefGoogle Scholar
  46. Sousa ME, Suárez A, Rodríguez RF and López C (2010). Flexible adjustment of the short-term correlation of LDR M/G/∞-based processes. Electronic Notes on Theoretical Computer Science 161: 131–145.CrossRefGoogle Scholar
  47. Suárez A, et al (2002). A new heavy-tailed discrete distribution for LRD M/G/∞ sample generation. Performance Evaluation 47 (2/3): 197–219.CrossRefGoogle Scholar
  48. Taqqu MS and Teverovsky V (1998). On estimating the intensity of long-range dependence in finite and infinite variance time series. In: Adler RJ, Feldman RE and Taqqu MS (eds). A Practical Guide to Heavy Tails. Birkhauser: Boston, MA, pp 177–218.Google Scholar
  49. Taqqu MS, Willinger W and Sherman R (1997). Proof of a fundamental result in self-similar traffic modeling. Computer Communication Review 27 (2): 5–23.CrossRefGoogle Scholar
  50. Whittle P (1953). Estimation and information in stationary time series. Arkiv Matematick 2 (23): 423–434.CrossRefGoogle Scholar
  51. Willinger W, Taqqu MS, Sherman R and Wilson DV (1997). Self-similarity through high variability: Statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Transactions on Networking 5 (1): 71–86.CrossRefGoogle Scholar

Copyright information

© Operational Research Society 2013

Authors and Affiliations

  1. 1.University of VigoVigoSpain

Personalised recommendations