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

VarianceLRD Poisson

HT 2 T

)(krk

X

Multiscale Tree Models

Model relationship between dyadic scales

Additive and Multiplicative Models

),0( 2..

scale

dii

scaleW ),(..

scalescale

dii

scale ppA Gaussian non-Gaussian (asymp. Lognormal)

Queuing

);)((sup TctKQ TT

t

t

TtjjT XtK

1

)(

Multiscale Queuing

•Exploit tree for queuing

Restriction to Dyadic Scales

Only dyadic scales:

tDt

j

jDt

QQ

ctKQ j

,

2, )2)((sup

Approximate queuing formulas:

)2)0((1)()2)0((sup2,02

bcKPbQPbcKP j

jD

j

jjj

Critical dyadic time scale(CDTSQ)

Multiscale queuing formula(MSQ)

Multiscale Queuing Formula: Intuition

MSQEPEPbQP

bcKE

jj

jjD

jj j

)(1)(1)(

}2{Let

,0

2

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

)(sup)2(sup2

bTcKPbcKP TT

j

jj

•Equality if critical time scale is a dyadic scale•fractional Gaussian noise: equality at b=const. j2

Convergence of MSQ

)2(1)(2

0)( bcKPbMSQ j

Nj

Nj

•For infinite terms is MSQ(b)=1?•Result: There exists N such that

2)()()( )21)(()()( NNN bMSQbMSQbMSQ

Tree depth

Non-Stationarity of Models

Commonparent

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?

0t

Non-Stationarity

Stationary traffic:

;tX Non-stationary model: tY

;)(1

t

Ttii

YT YtK;)(

1

t

Ttii

XT XtK

Theorem: If the autocorrelation of X is positive and non-increasing,

))(())0(())0(( tKVarKVarKVar YT

YT

XT

Implication: The model captures the variance of traffic bestat the edge (t=0) of the tree => best location to study queuing

Asymptotic Queuing

tat timeinput queue as with size queue YQYt

)()/1(lim0

1

1 bQPLPLt

Lt

Lavg

)log(/n slower tha increases )(

if )(log))(/1(lim)(log))(/1(lim

2 TTKVar

bQPbfbPbf

XT

X

bavg

b

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

Probingdelay 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 multiplicativemultiscale 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