1. Link Dimensioning for Fractional Brownian Input Chen Jiongze PhD student, Electronic Engineering Department, City University of Hong Kong Supported by Grant [CityU 124709] Moshe Zukerman Electronic Engineering Department,, City University of Hong Kong Ron G. Addie Department of Mathematics and Computing, University of Southern Queensland, Australia

2. Outline: Background A New Analytical Result of an FBM Queue Simulation Link Dimensioning Discussion & Conclusion

3. Outline: Background A New Analytical Result of an FBM Queue Simulation Link Dimensioning Discussion & Conclusion

4. Fractional Brownian Motion (fBm) process of parameter H, M t H are as follows: Gaussian process N(0,t 2H ) Covariance function: For H > ½ the process exhibits long range dependence … traffic Queue

6. … traffic Queue Long Range Dependence Gaussian By Central limit theorem Its statistics match those of real traffic (for example, auto-covariance function) - Gaussian process & LRD A small number of parameters - Hurst parameter (H), variance Amenable to analysis Is fBm a good model? YES

7. Outline: Background A New Analytical Result of an FBM Queue Simulation Link Dimensioning Discussion & Conclusion

8. A New Analytical Result of an fBm Queue … traffic Queue Queuing Model fBm traffic Hurst parameter (H) variance (σ 1 2 ) drift / mean rate of traffic (λ) Single server Queue with ∞ buffers service rate (τ) steady state queue size (Q) mean net input (μ = λ - τ)

9. Analytical results of (fBm) Queue No exact results for P(Q>x) for H ≠ 0.5 Existing asymptotes : By Norros [9] [9] I. Norros, “A storage model with self-similar input,” Queueing Syst., vol. 16, no. 3-4, pp. 387–396, Sep. 1994.

10. Analytical results of (fBm) Queue Existing asymptotes (cont.): By Husler and Piterbarg [14] [14] J. H¨usler and V. Piterbarg, “Extremes of a certain class of Gaussian processes,” Stochastic Processes and their Applications, vol. 83, no. 2, pp. 257 – 271, Oct. 1999.

11. Approximation of [14] is more accurate for large x but with no way provided to calculate Our approximation: Analytical results of (fBm) Queue

12. Our approximation VS asymptote of [14]: Advantages: a distribution accurate for full range of u/x provides ways to derive c Disadvantages: Less accurate for large x (negligible) Analytical results of (fBm) Queue [14] J. H¨usler and V. Piterbarg, “Extremes of a certain class of Gaussian processes,” Stochastic Processes and their Applications, vol. 83, no. 2, pp. 257 – 271, Oct. 1999.

13. Outline: Background A New Analytical Result of an FBM Queue Simulation Link Dimensioning Discussion & Conclusion

14. Simulation Basic algorithm (Lindley’s equation):

15. n012… U n (Mb)1.234– 0.35510.743… m(t) (Mb)-0.5 … Q n (Mb)0max(0, 1.234 – 0.5) =0.734 max (0, 0.734 – 0.3551 – 0.5) =0 … 1 ms Length of U n = 2 22 for different Δt, it is time- consuming to generate U n for different time unit)

16. An efficient approach Instead of generating a new sequence of numbers, we change the “units” of work (y-axis).

17. 1ms -> Δt ms variance of the fBn sequence (U n ): V Rescale the Y-asix

18. An efficient approach Instead of generating a new sequence of numbers, we change the “units” of work (y-axis). 1 unit = S instead of 1 where Rescale m and P(Q>x) m = μΔt/S units, so P(Q>x) is changed to P(Q>x/S) Only need one fBn sequence

19. Simulation Results Validate simulation

20. Simulation Results

24. Outline: Background A New Analytical Result of an FBM Queue Simulation Link Dimensioning Discussion & Conclusion

25. Link Dimensioning We can drive dimensioning formula by Incomplete Gamma function: Gamma function:

26. Finally Link Dimensioning where C is the capacity, so.

27. Link Dimensioning

32. Outline: Background Analytical results of a fractional Brownian motion (fBm) Queue Existing approximations Our approximation Simulation An efficient approach to simulation fBm queue Results Link Dimensioning Discussion & Conclusion

33. Discussion fBm model is not universally appropriate to Internet traffic negative arrivals ( μ = λ – τ) Further work re-interpret fBm model to alleviate such problem A wider range of parameters

34. Conclusion In this presentation, we considered a queue fed by fBm input derived new results for queueing performance and link dimensioning described an efficient approach for simulation presented agreement between the analytical and the simulation results comparison between our formula and existing asymptotes numerical results for link dimensioning for a range of examples

35. References:

38. Q & A

39. Background Self-similar (Long Range Dependency) “Aggregating streams of traffic typically intensifies the self similarity (“burstiness”) instead of smoothing it.”[1] Very different from conventional telephone traffic model (for example, Poisson or Poisson-related models) Using Hurst parameter (H) as a measure of “burstiness” [1] 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. Networking, vol. 2, no. 1, pp. 1–15, Feb. 1994.

40. Background Self-similar (Long Range Dependence) “Aggregating streams of traffic typically intensifies the self similarity (“burstyiness”) instead of smoothing it.”[1] Very different from conventional telephone traffic model (for example, Poisson or Poisson-related models) Using Hurst parameter (H) as a measure of “burstiness” Gaussian (normal) distribution When umber of source increases [1] 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. Networking, vol. 2, no. 1, pp. 1–15, Feb. 1994. [6] M. Zukerman, T. D. Neame, and R. G. Addie, “Internet traffic modeling and future technology implications,” in Proc. IEEE INFOCOM 2003,vol. 1, Apr. 2003, pp. 587–596. process of Real trafficGaussian process [2] Central limit theorem Especially for core and metropolitan Internet links, etc.

41. Analytical results of (fBm) Queue A single server queue fed by an fBm input process with - Hurst parameter (H) - variance (σ 1 2 ) - drift / mean rate of traffic (λ) - service rate (τ) - mean net input (μ = λ - τ) - steady state queue size (Q) Complementary distribution of Q, denoted as P(Q>x), for H = 0.5: [16] [16] J. M. Harrison, Brownian motion and stochastic flow systems. New York: John Wiley and Sons, 1985.