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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

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

The Traffic Matrix Estimation Problem

Xj

Yi

ingress

egress

Xj

PoP (Point of Presence)

Y = A X

“Traffic Matrix”

Link Measurement Vector

Routing Matrix

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

What is Non-Stationary?

- Traffic Itself is Non-Stationary

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)

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

Network and Measurement Model

- 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

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)

Matrix Notation

Linear system of equations:

where:

[LK]

[LK][P]

[LK][KP]

[P]

[KP]

Choose Routing Configurations such that

Rank(A) = P

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

Traffic matrix Estimation-Stationary Case

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

- Recursion for Estimates:

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?

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:

Numerical Example-Stationary caseStationary 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)

Traffic Model: Non-Stationary Case

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

Mean estimation Results-Non Stationary case

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

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