1. Queuing Analysis of Tree-Based LRD Traffic Models Vinay J. Ribeiro R. Riedi, M. Crouse, R. Baraniuk

2. Research Topics LRD traffic queuing Internet path modeling: probing for cross- traffic estimation Open-loop vs. closed-loop traffic modeling (AT&T labs) Sub-second scaling of Internet backbone traffic (Sprint Labs)

3. Long-range dependence (LRD) Process X is LRD if Scale (T) 1 ms 2 ms 4 ms Variance LRD Poisson

4. Multiscale Tree Models Model relationship between dyadic scales

5. Additive and Multiplicative Models Gaussian non-Gaussian (asymp. Lognormal)

6. Queuing

7. Multiscale Queuing Exploit tree for queuing

8. Restriction to Dyadic Scales Only dyadic scales: Approximate queuing formulas: Critical dyadic time scale (CDTSQ) Multiscale queuing formula (MSQ)

9. Multiscale Queuing Formula: Intuition independence Assumption: dyadic scales far enough apart to allow independence

10. Simulation: Accuracy of Formula Berkeley Traffic Additive model Multiplicative model Queue size b log P(Q>b)

11. Issues Restriction to dyadic scales Convergence of MSQ Non-stationarity of models

12. How good is the dyadic restriction? Compare CDTSQ to well known critical time scale approximation Equality if critical time scale is a dyadic scale fractional Gaussian noise: equality at b=const.

13. Convergence of MSQ For infinite terms is MSQ(b)=1? Result: There exists N such that Tree depth

14. Non-Stationarity of Models Common parent No common parent Tree models are non-stationary Queue distribution changes with time Formulas for edge of tree (t=0) How is queue at t=0 related to the queue at other times t? How is does the models’ queuing compare with that of the stationary modeled traffic?

15. Non-Stationarity Stationary traffic:Non-stationary model: Theorem: If the autocorrelation of X is positive and non-increasing, Implication: The model captures the variance of traffic best at the edge (t=0) of the tree => best location to study queuing

16. Asymptotic Queuing Conjecture: Note: The conjecture is true for fGn (Sheng Ma et al)

17. Conclusions Developed queuing formulas for multiscale traffic models Studied the impact of using only dyadic scales, tree depth and non-stationarity of the models Ongoing work: accuracy of formulas for non-asymptotic buffer sizes

18. End-to-End Path Modeling Goal: Estimate volume of cross-traffic Abstract the network dynamics into a single bottleneck queue driven by `effective’ crosstraffic

19. Probing delay spread of packet pair correlates with cross-traffic volume

20. Probing Uncertainty Principle Small T for accuracy –But probe traffic disturbs cross-traffic (overflow buffer!) Larger T leads to uncertainties –queue could empty between probes To the rescue: model-based inference

21. Multifractal Cross-Traffic Inference Model bursty cross-traffic using the multiplicative multiscale model

22. Efficient Probing: Packet Chirps Tree inspires geometric chirp probe MLE estimates of cross-traffic at multiple scales

23. Chirp Cross-Traffic Inference

24. ns-2 Simulation Inference improves with increased utilization Low utilization (39%)High utilization (65%)

25. Conclusion Efficient chirp probing scheme for cross- traffic estimation