Traffic Matrix Estimation in Non-Stationary Environments

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Traffic Matrix Estimation in Non-Stationary
Environments

• Presented by
• R. L. Cruz
• Department of Electrical Computer Engineering
• University of California, San Diego
• Joint work with
• Antonio Nucci
• Nina Taft
• Christophe Diot
• NISS Affiliates Technology Day on Internet
  Tomography
• March 28, 2003

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The Traffic Matrix Estimation Problem

• Formulated in Y. Vardi, Network Tomography
  Estimating Source-Destination Traffic From Link
  Data, JASA, March 1995, Vol. 91, No. 433, Theory
  Methods

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The Traffic Matrix Estimation Problem
Xj
Yi
ingress
egress
Xj
PoP (Point of Presence)
Y A X
Traffic Matrix
Link Measurement Vector
Routing Matrix
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The Traffic Matrix Estimation Problem

• Importance of Problem capacity planning, routing
  protocol configuration, load balancing policies,
  failover strategies, etc.
• Difficulties in Practice
• missing data
• synchronization of measurements (SNMP)
• Non-Stationarity (our focus here)
• long convergence time needed to obtain estimates

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What is Non-Stationary?

• Traffic Itself is Non-Stationary

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What is Non-Stationary?

• Also, Routing is Non-Stationary
• e.g. Due to Link Failures
• Essence of Our Approach
• Purposely reconfigure routing in order to help
  estimate traffic matrix
• More information leads to more accurate estimates
• Effectively increases rank of A
• We have developed algorithms to reconfigure the
  routing for this purpose (beyond the scope of
  this talk)

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Outline of Remainder of Talk

• Describe the Stationary Method
• Stationary traffic, non-stationary routing
• Stationary traffic assumption is reasonable if we
  always measure traffic at the same time of day
  (e.g. peak period of a work day)
• Briefly Describe the Non-Stationary Method
• Both non-stationary traffic and non-stationary
  routing
• More complex but allows estimates to be obtained
  much faster

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Network and Measurement Model

• Network with L links, N nodes, PN(N-1) OD pair
  flows
• K measurement intervals, 1 k K
• Y(k) is the link count vector at time k (L x 1)
• A(k) is the routing matrix at time k (L x P)
• X(k) is the O-D pair traffic vector at time k (P
  x 1)
• X(k) (x1(k) , x2(k) , xP(k))T

Y(k) A(k) X(k)
Y(k) and A(k) can be truncated to reflect missing
and redundant data
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Traffic Model Stationary Case

• X(k) is the O-D pair traffic vector at time k
  (P x 1)
• X(k) (x1(k) , x2(k) , xP(k))T

X(k) X W(k)

• W(k) Traffic Fluctuation Vector
• Zero mean, covariance matrix B
• B diag(X)

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Matrix Notation
Linear system of equations
where
LK
LKP
LKKP
KP
P
Choose Routing Configurations such that
Rank(A) P
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Traffic matrix Estimation-Stationary Case
Y AX CW

• Initial Estimate Use Psuedo-Inverse of A-
  does not require statistics of W (covariance B)
• Gauss-Markov Theorem Assume B is known
• - Unbiased, minimum variance estimate-
  Coincides with Maximum Likelihood Estimate
• if W is Gaussian

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Traffic matrix Estimation-Stationary Case
Y AX CW

• Minimum Estimation Error (assumes B is
  known)

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Traffic matrix Estimation-Stationary Case

• Recall we assume B cov(W) satisfies B
  diag(X)
• Set

• Recursion for Estimates

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Traffic matrix Estimation-Stationary Case

• Our estimate is a solution to the equation

• Open questions for fixed point equation
• Existence of Solution?- Uniqueness?
• Is solution an un-biased estimate?

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Numerical Example-Stationary case

• N10 nodes, L24 links and P90 connections.
• Three set of OD pairs with mean x equal to
• 500 Mbps, 2 Gbps and 4 Gbps.
• Gaussian Traffic Fluctuations

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Stationary case b1 Samples/Snapshot1
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Stationary case b1 Samples/Snapshot1
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Stationary case b1 Samples/Snapshot15
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Stationary case b1 Samples/Snapshot15
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Stationary case b1.4 Samples/Snapshot1
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Stationary case b1.4 Samples/Snapshot1
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Stationary case b1.4 Samples/Snapshot15
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Stationary case b1.4 Samples/Snapshot15
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Stationary and Non-Stationary traffic

• 20 snapshots / 4 samples per snapshot / 5 min per
  sample
• Stationary Approach 20 min per day (same time)
  / 20 days
• Non-Stationary Approach aggregate all the
  samples in
• one window time large 400 min (7 hours)

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Traffic Model Non-Stationary Case

• Each OD pair is cyclo-stationary
• Each OD pair is modeled as
• Fourier series expansion

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Mean estimation Results-Non Stationary case

• Three set of OD pairs
• where are linear independent Gaussian
  variables with

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Non Stationary case b1 Link Count
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Non Stationary case b1 Mean estimation