Transcript Slide 1

Neuro-Fuzzy Processing of Packet Dispersion Traces
for Highly Variable Cross-Traffic Estimation
2
Universidad de los Andes, Bogotá, Colombia
University of South Florida, Tampa, FL
CLOSE (qi )  exp  i | qi | ,
 FAR (qi )  1  CLOSE (qi ) for i=1,2,3 with 1  2
CLOSE (q 4 )  exp  4q 4  ,
 FAR (q 4 )  1  CLOSE (q 4 )
i
Mbps
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Xn C
xn 
C
L
xn  d n 
C T
Normalized simple estimator (SE)
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1100
Dn  T
dn 
T
t
t
T
D
A link carries a cross-traffic, characterized by a given coefficient of variation and a
given Hurst parameter, along with a probing traffic consisting L-bits long packets
sent every T seconds. How much correlation is there between the average crosstraffic arrival rate at the nth measurement period, Xn, and the corresponding packet
dispersion measure, Dn?
H = 0.65
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C=2
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ro
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log2(T)
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ro
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log2(T)
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dispersion mean
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close1
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S-norm
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Exact
Poor
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ro
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log2(T)
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ro
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log2(T)
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pdf
As we increase the variability through the coefficient of variation (C = 1,2,4), the
correlation becomes less dependent on the utilization factor, ro. Similarly, as we
increase the variability through the Hurst parameter (H = 0.5, 0.65, 0.8, 0.95), the
correlation becomes less dependent on T as well. Correspondingly, as traffic
variability increases, we can test the link over a wide range of time scales obtaining a
high correlation even with a low utilizations factor.
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Theta1
pdf of Theta3
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Exact
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Theta2
pdf of Theta4
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Exact
Poor
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f E ( x)
 F ( x) 
f E ( x)  f P ( x)
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-1
-2
0
Fitting the histograms of each
input variable conditioned on
an “exact” (fE(.)) or “poor”
(fP(.)) performance of SE
(shown left), we define fuzzy
sets “Far from zero” and
“Close to zero” through the
relationships
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Theta4
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C ( x )  1   F ( x )
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Utilization
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T
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The SNR is still high for another traffic
trace, a 768 kbps MPEG4 version of
“Jurassic Park”, a very different cross
traffic trace than the one used for
training, revealing the good
generalization properties of the system.
Fair
Poor
far4
T-norm = product
S-norm = maximum
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• A different affine estimator, based on (q1, q2), for each class above,
q1
Queue
simulation
q4
q2
w^G
q1
^x
n
mux
^x
n
T
• In particular, dispersion measurements can still be highly correlated with the
cross traffic on a wide range of measurement time scales, even for lowly used
links.
Neuro
Fuzzy
Estimator
q3
0
• The dynamics of packet pair probing techniques are influenced by traffic
variability, beyond what constant rate fluid flow models have suggested.
^
w^F
2
Conclusions
wP
Fuzzy
Inference
System
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0.4
0.2
The final system, a Heuristically-modified Neuro-Fuzzy Estimator (HNFE), adds
two additional elements:
q4
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Utilization
q2
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Good
close4
q3
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The performance for the training
trace, in terms of the SNR (where the
traffic trace is the signal and the
estimation error is the noise) is high,
even under low utilization factors, on
a wide range of measurement time
scales, for different link utilizations
and different measurement periods.
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far3
q4
1 11
2
q4 ( n)    dnk  q3 ( n) 
11 k 0
pdf of Theta2
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S-norm
far2

pdf of Theta1
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T-norm
close3
Selected input variables:
Based on the experimental data (dn, xn), we estimated that information about xn
conveyed by these four selected variables, I(xn ; {qi(n), i=1,2,3,4}), is comparable to
that conveyed by the 12 previous dispersion measurements together,
I(xn;{dn-k, k=0,…,11}).
0.5
25
+1
q2 (n)  dn1
1
0.2
1
( )
1 11
q3 ( n)   dnk
12 k 0
1.5
Time (seconds)
• A queue simulator [qn = max(0, qn-1 + x^n + L/(CT))], to recover SE whenever the
queue length has exceed a given threshold, thr.
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IF one scale says SE is poor
AND the other does not say it is good
THEN
SE is poor
ELSEIF one scale says SE is good
AND the other does not say it is poor
THEN
SE is good
ELSE
SE is fair
END
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dispersion mean
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q1 (n)  dn
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q
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far1
q1
Neuro-Fuzzy Estimator
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Considering the mean and
variance of the previous 12
dispersion measures, D and
D2, we plot the relative error
of SE (left) to establish the
following intuitively plausible
rules:
These rules call for a fuzzy approach to our estimation problem
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C=4
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• If D is far from 0, the simple estimation is exact
• If D is close to 0 and D2 is small, the simple estimation is poor
• If D is close to 0 and D2 is large, the simple estimation is fair.
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Cross-traffic
Simple Estimator
HNFE
Combining the result of each scale
close2
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H = 0.95
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q
T-norm
q3
pdf
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Which leads to the following fuzzy inference system for classifying SE,
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pdf
C=1
1
H = 0.8
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0
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1
qq
0.60
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pdf
H = 0.5
0
q2
dispersion standard deviation
Pr
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far
close
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dispersion standard deviation
L/C
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both variables are close to 0
THEN
SE is poor
ELSEIF
both variables are far from 0
THEN
SE is good
ELSE
SE is fair
END
1000
relative
error
average error
Pt

0.4
IF
2000
Cross traffic
Probing traffic
q 
0.6
At each scale, (q1, q2) and (q3, q4)
Normalized variables
C
0.8
SNR
queue length in packets
4000
Simulation experiment with the Bellcore traffic trace BCpAug89 on a T1 link, with
24-byte probing packets sent every second.
t
0.8
0.8
We have two inputs at the scale of the measurement period T, q1 and q2, and two
inputs at the scale of 12T, q3 and q4, so we can classify SE according to each scale
and then combine their results according to the following fuzzy classifiers:
time in seconds
Xt
1
time in seconds
• This HNFE exploits the high variability property of modern traffic by using an
extremely low rate probing traffic and multiple time scale analysis, achieving
high accuracy, low computational cost, very low transmission overhead and high
robustness against varying network conditions.
Packet Dispersion and High Variability
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0
-1
2
For the training trace and with the training conditions, the proposed system reduced
the estimation error, even when the probing packets do not belong to the same
occupation period.
4
1
Cross-Traffic
simple estimation
3
Results
i
4
0.2
Simple cross-traffic estimation
• As an example of how the previous finding can be exploited for estimation
purposes, we design a heuristically-modified neuro-fuzzy estimator (HNFE) for
real time high resolution traffic estimation that, instead of looking for a long run
average, tracks the cross traffic rate signal.
i
4

For a FIFO link of capacity C bps that does not
become idle between the nth and n+1st probing
packets, the nth dispersion measurement Dn, will
give an exact value of the cross-traffic arrival rate
during the nth measurement period, Xn.
q 
• However, here we show evidence that, with highly variable traffic, it is possible
to have very short probing packets at a very low rate and still get an important
correlation between dispersion and traffic, over a long range of measurement
time scales, even when the utilization factor is low.
Dn  C  L
Xn 
T
Universidad Distrital, Bogotá, Colombia
Leading closely to the following closed form expressions
A Simple Estimator
• In the context of constant-rate fluid-flow traffic, it has been shown that there is a
minimum probing traffic rate at which the dispersion exhibits correlation with
cross traffic, so packet length and input gap must be adjusted to reach that
minimum.
3
Traffic rate (Mbps)
Main Ideas
Miguel A.

1
Néstor M.
2
Labrador
q 
Marco A.
1
Peña ,
SNR
1,2,3
Alzate ,
Simple
estimator
+1
• We do not need to exhaust available bandwidth with probing traffic in order to
obtain good cross-traffic estimates, as long as the measurement time scale falls
within the high variability range.
• By exploiting the properties above, our HNFE demonstrates that it is possible to
obtain a good cross-traffic tracking signal (not just a long run average) with a
negligible probing load.
Training:
1. Initialize 1, 3, and 4 by fitting the conditional histograms.
2. Compute the nine linear parameters of the affine estimators by a least square
procedure.
3. Compute the optimal exponents {i, i=1,3,4} through a quasi-Newton line search
algorithm.
4. Iterate steps 2 and 3 until convergence.
5. Look for the optimal queue threshold through bracketing.
Through its reduced structural complexity, HNFE achieves
•
•
•
•
Low computational cost,
Good generalization (low overfiting),
good interpretability (rules were chosen, not learned)
fast learning speed.
References
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IEEE Network Magazine, Vol. 17, No. 6, pp. 27--35, Nov/Dec. 2003.
2. N. Hu and P. Steenkiste, “Evaluation and Characterization of Available Bandwidth Probing Techniques,” IEEE JSAC, Vol. 21,
No. 6, pp. 879-894, Aug., 2003.
3. M. Jain and C. Dovrolis, “End-to-End Available Bandwidth Measure Methodology, Dynamics and Relation with TCP
throughput,” IEEE/ACM Transactions on Networking, Vol. 11, No. 4, August 2003, pp. 537-549.
4. V. Ribeiro, R. Riedi, R. Baraniuk, J. Navratil and L. Cottrell, “PathChirp: Efficient Available Bandwidth Estimation for
Network Paths,” Proceedings of Passive and Active Measurements (PAM) Workshop, La Jolla, CA, USA, Apr. 2003.
5. J. Strauss, D. Katabi, F. Kaashoek, and B. Prabhakar, “Spruce: A Lightweight End-to-End Tool for Measuring Available
Bandwidth,” Proceedings of Internet Measurement Conference (IMC) 2003, Miami, Florida, October 2003.
6. R. Ribeiro, M. Coates, R. Riedi, S. Sarvotham, B. Hendricks, and R. Baraniuk, “MultiFractal Cross-Traffic Estimation,”
Proceedings of ITC Specialist Seminar on IP Traffic Measurement, Monterey California, September 18-20 2000
7. Lawrence Berkeley National Laboratory. The Internet Traffic Archives, BC – Ethernet traces of LAN and WAN traffic,
http://ita.ee.lbl.gov/html/contrib/BC.html
8. Video Traces Research Group, Arizona State University,
http://trace.eas.asu.edu/TRACE/pics/FrameTrace/mp4/Verbose_Jurassic.dat.
Passive and Active Measurement Conference, PAM 2007, Louvain-la-neuve, Belgium, April 5-6 2007