Link Dimensioning for Fractional Brownian Input

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

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, MtH are as follows: • Gaussian process N(0,t2H) • Covariance function: • For H > ½ the process exhibits long range dependence … traffic Queue

6. Gaussian By Central limit theorem Long Range Dependence Is fBm a good model? YES • 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 … traffic Queue

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 Queuing Model fBm traffic Single server Queue with ∞ buffers Hurst parameter (H) variance (σ12) drift / mean rate of traffic (λ) service rate (τ) steady state queue size (Q) … mean net input (μ = λ - τ) traffic Queue

9. Analytical results of (fBm) Queue No exact results for P(Q>x) for H ≠ 0.5 Existing asymptotes: • By Norros [9]

10. Analytical results of (fBm) Queue Existing asymptotes (cont.): • By Husler and Piterbarg [14]

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

12. Analytical results of (fBm) Queue • 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)

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

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

15. Length of Un = 222 for different Δt, it is time-consuming to generate Un for different time unit) 1 ms

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

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

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

21. Simulation Results

22. Simulation Results

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

27. Link Dimensioning

28. Link Dimensioning

29. Link Dimensioning

30. Link Dimensioning

31. 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:

36. References:

37. 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”

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 Central limit theorem • process of Real traffic Gaussian process [2] 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 (σ12) - 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]