École Polytechnique Fédérale de Lausanne

Network Tomography on Correlated Links

Denisa Ghita

Katerina Argyraki

Patrick Thiran

IMC 2010, Melbourne, Australia

Network Tomography

Internet Service Provider

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Network tomography infers links characteristics from path measurements.

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Current Tomographic Methods assume Link Independence

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Current Tomographic Methods assume Link Independence

Links can be correlated!

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Can we use network tomography when links are correlated?

Yes, we can!

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All

Link Correlation Model

links are independent.Some

possibly correlated

independent

Independence among correlation sets!

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How to find the Possibly Correlated Links?

Links in the same local-area network may be correlated!

Links in the same administrative domain may be correlated!

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The Probability that a Link is Faulty

link is faultyP( ) = ?

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Our Main Contribution

P( link faulty) = ?

P( link faulty) = ?

P( link faulty) = ?

P( link faulty) = ?

Theorem that states the necessary and sufficient condition to identify the probability that each link is faulty when links in the network are correlated.

P( link faulty) =…

P( link faulty) =…

P( link faulty) =…

P( link fa

ulty) =…

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Our ConditionEach subset of a correlation set must be covered by a different set of paths!

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A

B

Identifiable

Our Condition

Subset of aCorrelation Set Covered Paths

eAB eBC eBD eBC, eBD

Each subset of a correlation set must be covered by a different set of paths!

C

D

1. Define the subsets of the correlation sets.

2. Find the paths that cover each subset.

3. Are any subsets covered by the same paths?

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Our ConditionA

B

C

D

Identifiable

ESubset of aCorrelation Set

eAB eBC eBD eBC, eBD

Covered Paths

eEB

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The Gist behind the Algorithm

Solvable!3 equations 4 unknowns

P( PAC good ) = P(eAB good) P(eBC good)

P( PAD good ) = P(eAB good) P(eBD good)

P( PED good ) = P(eEB good) P(eBD good)

BC

DE

A

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The Gist behind the Algorithm

P( PAC good ) = P(eAB good) P(eBC good)

P( PAD good ) = P(eAB good) P(eBD good)

P( PED good ) = P(eEB good) P(eBD good)

BC

DE

A

P( PAC , PAD good ) = P(eAB good) P(eBD ,eBC good)

P(eBDgood)P(eBC good)

≠

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The Gist behind the Algorithm

P( PAC good ) = P(eAB good) P(eBC good)

P( PAD good ) = P(eAB good) P(eBD good)

P( PED good ) = P(eEB good) P(eBD good)

BC

DE

A

P( PAC , PAD good ) = P(eAB good) P(eBD ,eBC good)

P( PAD , PED good ) = P(eAB good) P(eEB good) P(eBD good)

Solvable !5 unknowns5 equations

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The Gist behind the Algorithm

P( PAC good ) = P(eAB good) P(eBC good)

P( PAD good ) = P(eAB good) P(eBD good)

P( PED good ) = P(eEB good) P(eBD good)

BC

DE

A

P( PAC , PAD good ) = P(eAB good) P(eBD ,eBC good)

P( PAD , PED good ) = P(eAB good) P(eEB good) P(eBD good)

Solvable !5 unknowns5 equations

Correlation set of 40 links -> 240 unknowns !!!

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The Gist behind the Algorithm

P( PAC good ) = P(eAB good) P(eBC good)

P( PAD good ) = P(eAB good) P(eBD good)

P( PED good ) = P(eEB good) P(eBD good)

BC

DE

A

P( PAC , PAD good ) = P(eAB good) P(eBD ,eBC good)

P( PAD , PED good ) = P(eAB good) P(eEB good) P(eBD good)

Solvable !5 unknowns5 equations

Correlation set of 40 links -> 240 unknowns !!!

Consider only sets of paths that do not cover correlated links !

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The Gist behind the Algorithm

P( PAC good ) = P(eAB good) P(eBC good)

P( PAD good ) = P(eAB good) P(eBD good)

P( PED good ) = P(eEB good) P(eBD good)

BC

DE

A

P( PAC , PAD good ) = P(eAB good) P(eBD ,eBC good)

P( PAD , PED good ) = P(eAB good) P(eEB good) P(eBD good)

Consider only sets of paths that do not cover correlated links !

Solvable!4 unknowns 4 equations

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Simulations – Domain Level Tomography

Actual Topology Measured Topology

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Simulations – Domain Level Tomography

absolute error between the actual probability that a link is faulty, and the probability inferred by the algorithm.

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Simulations – Domain Level Tomography

absolute error between the actual probability that a link is faulty, and the probability inferred by the algorithm.

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Conclusion

• We study network tomography on correlated links.

• We formally prove under which necessary and sufficient condition the probabilities that links are faulty are identifiable.

• Our tomographic algorithm determines accurately the probabilities that links are faulty in a variety of congestion scenarios.

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Thank you!denisa.ghita@epfl.ch