1. Small scale analysis of data traffic models B. D’Auria - Eurandom joint work with S. Resnick - Cornell University

2. 14/03/2006B. D'Auria - EURANDOM Content Data traffic - stylized facts: Heavy-Tails, LRD, self-similarity, Burstiness M/G/∞ input models Small-scale asymptotic results Conclusions

3. 14/03/2006B. D'Auria - EURANDOM Telecommunication data traffic 1 st column Ethernet Traffic 2 nd column Poisson Model Taqqu et al., (1997)

4. 14/03/2006B. D'Auria - EURANDOM Stylized facts Heavy tails for distributions of such quantities as –File sizes –Transmission rates –Transmission durations Long Range Dependence (LRD) Self-similarity (s-s) Burstiness

5. 14/03/2006B. D'Auria - EURANDOM Heavy tails We use the class of regular varying distributions where L(x) is a slowly varying function at ∞ File sizes –Arlitt and Williamson (1996) –Resnick and Rootzén (2000) Transmission durations –Maulik et al. (2002) –Resnick (2003) Number of packets per slot –Leland et al. (1993) –Willinger et al. (1995)

6. 14/03/2006B. D'Auria - EURANDOM Long Range Dependence (LRD) In our context, we define LRD as the non-summability of the covariance function, i.e. a stationary stochastic process is Long Range Dependent if

7. 14/03/2006B. D'Auria - EURANDOM Self-similarity (s-s) A stationary process is strictly self-similar (ss-s) if 0<H<1 is called Hurst parameter.

8. 14/03/2006B. D'Auria - EURANDOM Self-similarity (s-s) A ss-s process has covariance function When H>1/2 self-similarity implies LRD.

9. 14/03/2006B. D'Auria - EURANDOM Let be a stationary process Weak self-similarity Exact 2 nd order self-similarity (es-s) Asymptotic 2 nd order self-similarity (as-s)

10. 14/03/2006B. D'Auria - EURANDOM Burstiness Following Sarvotham et al. (2005), data traffic can be spitted in two parts: α-traffic - large files at very high rate β-traffic - the rest. The α-component is a small fraction of the total workload but is entirely responsible for burstiness The β-component is responsible for the dependence structure At high levels of aggregation traffic appears to be Gaussian

11. 14/03/2006B. D'Auria - EURANDOM α-traffic and β-traffic from Sarvotham et al. (2001)

12. 14/03/2006B. D'Auria - EURANDOM M/G/∞ Input Model Transmitting sources arrive according to a Poisson Process with rate λ The generic transmission k has associated 4 parameters: – the arrival time –the transmission rate –the file size –the transmission length

13. 14/03/2006B. D'Auria - EURANDOM M/G/∞ Input Model

14. 14/03/2006B. D'Auria - EURANDOM Two Models Model RF where Model RL where

15. 14/03/2006B. D'Auria - EURANDOM M/G/∞ Input Model Given the relation by Breiman’s theorem we have the following: Model RF Model RL

16. 14/03/2006B. D'Auria - EURANDOM Poisson Random Measure The counting function, N, of the points is a Poisson Random measure on, given by with mean measure

17. 14/03/2006B. D'Auria - EURANDOM The data process A(δ) Fixed the window size, δ, we consider the discrete time process where represents the total amount of work inputted to the system in the k-th time slot We assume that the arrival rate is function of δ

18. 14/03/2006B. D'Auria - EURANDOM Decomposition of A(0,δ] It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions

19. 14/03/2006B. D'Auria - EURANDOM Decomposition of A(0,δ] It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions

20. 14/03/2006B. D'Auria - EURANDOM Decomposition of A(0,δ] It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions

21. 14/03/2006B. D'Auria - EURANDOM Decomposition of A(0,δ] It is possible to decompose A(δ)=A(0,δ ] in 4 independent parts related to the following 4 regions

22. 14/03/2006B. D'Auria - EURANDOM Random Measure decomposition The restriction of the Poisson measure to the 4 regions give 4 independent Poisson processes and

23. 14/03/2006B. D'Auria - EURANDOM A >0,1 (0,δ ] Contributions from sessions starting in (0,δ ] and terminating before δ. Where P >0,1 (δ) is Poisson distributed with parameter

24. 14/03/2006B. D'Auria - EURANDOM Region Proposition. If with P >0,1 (0) Poisson distributed with parameter and RF

25. 14/03/2006B. D'Auria - EURANDOM We have that implies Having that and using we finally get the result. Proof RF

26. 14/03/2006B. D'Auria - EURANDOM A >0,2 (0,δ ] Contributions from sessions starting in (0,δ ] and terminating after δ. Where P >0,2 (δ) is Poisson distributed with parameter

27. 14/03/2006B. D'Auria - EURANDOM Region Proposition. X >0,2 is infinitely divisible with Lévy measure with density where RF

28. 14/03/2006B. D'Auria - EURANDOM We have that then we prove that Then having convergence of the r.h.s of the following relation we get the result Proof RF

29. 14/03/2006B. D'Auria - EURANDOM A <0,1 (0,δ ] Contributions from sessions starting before 0 and terminating in (0,δ ]. Where P <0,1 (δ) is Poisson distributed with parameter Proposition.

30. 14/03/2006B. D'Auria - EURANDOM Proof

31. 14/03/2006B. D'Auria - EURANDOM A <0,2 (0,δ ] Contributions from sessions starting before 0 and terminating after δ. Where P <0,2 (δ) is Poisson distributed with parameter

32. 14/03/2006B. D'Auria - EURANDOM Region Proposition. where RF

33. 14/03/2006B. D'Auria - EURANDOM and we prove that Proof We have that RF

34. 14/03/2006B. D'Auria - EURANDOM Comparing the models Model RFModel RL

35. 14/03/2006B. D'Auria - EURANDOM Normal contribution for the RL Model RL

36. 14/03/2006B. D'Auria - EURANDOM Dependence structure Proposition. where and RF

37. 14/03/2006B. D'Auria - EURANDOM A(0,δ ] and A(kδ,(k+1)δ ] RF

38. 14/03/2006B. D'Auria - EURANDOM A(0,δ ] and A(kδ,(k+1)δ ] RF

39. 14/03/2006B. D'Auria - EURANDOM A(iδ,(i+1)δ ] with 0 ≤i ≤k RF

40. 14/03/2006B. D'Auria - EURANDOM Dependence structure Proposition. where is infinite divisible with Lévy measure with density and RL

41. 14/03/2006B. D'Auria - EURANDOM Question The over-sampling asymptotically implies perfect correlation. What can we say about correlation structure for finite δ?

42. 14/03/2006B. D'Auria - EURANDOM LRD for δ>0 Proposition. For fixed δ>0, as k→∞, RF

43. 14/03/2006B. D'Auria - EURANDOM LRD for δ>0 RF

44. 14/03/2006B. D'Auria - EURANDOM with LRD for fixed t>0 Proposition. For fixed t>0, as δ→0, RF and

45. 14/03/2006B. D'Auria - EURANDOM with LRD for fixed t>0 Proposition. For fixed t>0, as δ→0, RL

46. 14/03/2006B. D'Auria - EURANDOM Some simulations RF

47. 14/03/2006B. D'Auria - EURANDOM Simulation analysis RF

48. 14/03/2006B. D'Auria - EURANDOM Conclusions The Model RF seems to well represent real data traces as it show gaussianity in the limit. The Model RL instead converges marginally to heavy- tailed infinite divisible limit. That implies difficulty in handling with correlation structure and LRD.