queuing analysis of tree-based lrd traffic models
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
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
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