Queuing Analysis of TreeBased LRD Traffic Models Vinay

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 crosstraffic estimation • Open-loop vs. closed-loop traffic modeling (AT&T labs) • Sub-second scaling of Internet backbone traffic (Sprint Labs)

Long-range dependence (LRD) Scale (T) 4 ms 2 ms 1 ms • Process X is LRD if 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 log P(Q>b) Berkeley Traffic Multiplicative model Additive model Queue size 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 Tree depth • For infinite terms is MSQ(b)=1? • Result: There exists N such that

Non-Stationarity of Models • 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? No common Common parent 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 f. Gn (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 Abstract the network dynamics into a single bottleneck queue driven by `effective’ crosstraffic • Goal: Estimate volume of cross-traffic

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 crosstraffic estimation