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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

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

Xj

Yi

ingress

egress

Xj

PoP (Point of Presence)

Y = A X

“Traffic Matrix”

Link Measurement Vector

Routing Matrix

- 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

- Traffic Itself 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)

- Purposely reconfigure routing in order to help estimate traffic matrix

- 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

- Network with L links, N nodes, P=N(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

- 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)

Linear system of equations:

where:

[LK]

[LK][P]

[LK][KP]

[P]

[KP]

Choose Routing Configurations such that

Rank(A) = P

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

Y = AX + CW

- Minimum Estimation Error: (assumes B is known)

- Recall we assume B = cov(W) satisfiesB = diag(X)
- Set

- Recursion for Estimates:

- Our estimate is a solution to the equation:

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

N=10 nodes, L=24 links and P=90 connections.

Three set of OD pairs with mean x equal to:

500 Mbps, 2 Gbps and 4 Gbps.

Gaussian Traffic Fluctuations:

- 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)

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

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