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

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)

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

Multiscale Tree Models Model relationship between dyadic scales

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

Queuing

Multiscale Queuing Exploit tree for queuing

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

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

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

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

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.

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

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?

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

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

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

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

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

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

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

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

Chirp Cross-Traffic Inference

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

Conclusion Efficient chirp probing scheme for cross- traffic estimation