The Evolution of The Evolution of Traffic Matrix Traffic Matrix

Techniques and Techniques and Applications: Applications:

Past, Present and Past, Present and FutureFutureFan TongliangFan Tongliang

2008120100520081201005College of Communication EngineeCollege of Communication Enginee

ringring

OutlineOutline

Problem StatementProblem Statement Summary: traffic measurementSummary: traffic measurement How have traffic matrix estimation tecHow have traffic matrix estimation tec

hniques evolved?hniques evolved? Applications of traffic matricesApplications of traffic matrices ReferenceReference

Internet EvolutionInternet Evolution

Grows over time…Grows over time…

2

Internet EvolutionInternet Evolution

Say, network Say, network doubles in sizedoubles in size

Key: Key:

Where to add Where to add capacity?capacity?

2

Moore’s-law Moore’s-law like scaling like scaling sufficientsufficient??

If so, good scaling!If so, good scaling!

Uniformly Uniformly scale all scale all capacities?capacities?

Internet EvolutionInternet Evolution

2

Scale some Scale some links faster?links faster?

Moore’s-law Moore’s-law like scaling like scaling insufficient?insufficient?

Internet EvolutionInternet Evolution

2

Congested Congested hot-spotshot-spots

If so, poor scaling!!If so, poor scaling!!

Scale some Scale some links faster?links faster?

Internet EvolutionInternet Evolution

2

How does the worst congestion grow?How does the worst congestion grow? Ideal: O(n)Ideal: O(n)

How much of this is due to…How much of this is due to… Topology?Topology?

““Power-law” structurePower-law” structure Routing algorithm?Routing algorithm?

BGP-Policy routingBGP-Policy routing Traffic demand matrix?Traffic demand matrix?

Uniform vs. non-uniformUniform vs. non-uniform What can be done?What can be done?

Redesign the network?Redesign the network?

Internet EvolutionInternet Evolution

2

Why Measurements ?Why Measurements ?

Optimizing the Internet performance of well-Optimizing the Internet performance of well-connected Internet end-pointsconnected Internet end-points

◊Wide-area bottlenecksWide-area bottlenecks◊Identify and characterize bottlIdentify and characterize bottlenecksenecks

◊Multihoming route controlMultihoming route control◊Quantify benefits and compare Quantify benefits and compare against alternativesagainst alternatives

◊Will these techniques work in the Will these techniques work in the future?future?

◊Current Current best best performinperforminggBGP pathBGP path◊SmartSmartselectioselectionn

1

Why Measurements are Why Measurements are DifficultDifficult

To effectively measure the global Internet, wide cooperation is neeTo effectively measure the global Internet, wide cooperation is needed. However, ISPs are reluctant to coordinate their effortsded. However, ISPs are reluctant to coordinate their efforts

Statistics collection is viewed as a luxury (OC48mon = $100,000) - onStatistics collection is viewed as a luxury (OC48mon = $100,000) - only large ISPs can afford statistics collection and analysis - demand ily large ISPs can afford statistics collection and analysis - demand is still dormants still dormant

Best Effort service, low profit margins for ISPs make operational suBest Effort service, low profit margins for ISPs make operational support difficult – data collection is low prioritypport difficult – data collection is low priority

Traffic volume, high trunk capacity, diversity of protocols, technolTraffic volume, high trunk capacity, diversity of protocols, technologies and applications make traffic monitoring and analysis a challogies and applications make traffic monitoring and analysis a challenging endeavorenging endeavor

Results get obsolete very rapidly: Internet is under very active develResults get obsolete very rapidly: Internet is under very active development – traffic, technology and topology change very fastopment – traffic, technology and topology change very fast

Tremendous growth of Internet – it is difficult to scale measuremeTremendous growth of Internet – it is difficult to scale measurementsnts

OverprovisioningOverprovisioning is a widely practiced solution to network congest is a widely practiced solution to network congestionion

How Measurement ?How Measurement ? Measurement is: data collection, analysis and visualizationMeasurement is: data collection, analysis and visualization Traffic data:Traffic data:

Network Topology and Mapping (connectivity)Network Topology and Mapping (connectivity) Workload (passive or non-intrusive)Workload (passive or non-intrusive) Performance (active)Performance (active) Routing (BGP routing tables)Routing (BGP routing tables)

Active approachActive approach Inject traffic and wait for arrival to the destination or Inject traffic and wait for arrival to the destination or

replyreply Passive approachPassive approach

No traffic injected; Measurements are done over a No traffic injected; Measurements are done over a collection of network monitorscollection of network monitors

Measurement ToolsMeasurement Tools

Can be classified into Can be classified into hardware hardware and and sosoftwareftware measurement tools measurement tools

Hardware: specialized equipmentHardware: specialized equipment Examples: HP 4972 LAN Analyzer, DataGeExamples: HP 4972 LAN Analyzer, DataGe

neral Network Sniffer, others...neral Network Sniffer, others... Software: special software toolsSoftware: special software tools

Examples: tcpdump, xtr, SNMP, others...Examples: tcpdump, xtr, SNMP, others...

Measurement Tools Measurement Tools (Cont’d)(Cont’d)

Measurement tools can also be Measurement tools can also be classified as classified as real-timereal-time or or non-real-timenon-real-time

Real-timeReal-time: collects traffic data as it : collects traffic data as it happens, and may even be able to happens, and may even be able to display traffic info as it happensdisplay traffic info as it happens

Non-real-timeNon-real-time: collected traffic data : collected traffic data may only be a subset (sample) of the may only be a subset (sample) of the total traffic, and is analyzed off-line total traffic, and is analyzed off-line (later)(later)

Measurement ToolsMeasurement Tools Link Based Tools Link Based Tools

CoralReefCoralReef TcpdumpTcpdump

Router Based Tools Router Based Tools SNMP Based SNMP Based

MRTGMRTG NetFlow Based NetFlow Based

FlowScan FlowScan Cflowd Cflowd MADAS MADAS Flowtools Flowtools CISCO NetFlow FlowCollector, NetFlow Data Analyzer CISCO NetFlow FlowCollector, NetFlow Data Analyzer

Detecting Performance Detecting Performance ProblemsProblems

• High utilization or loss statistics for the linkHigh utilization or loss statistics for the link• High delay or low throughput for probesHigh delay or low throughput for probes• Angry customers (complaining via phone?)Angry customers (complaining via phone?)

overload!

Network Operations: Network Operations: Excess TrafficExcess Traffic

Multi-homed customer

Two large flows of traffic

New egress pointfor first flow

Network Operations: Network Operations: Denial-of-Service AttackDenial-of-Service Attack

Web server at its knees…

Install packetfilter

Web server back to life…

Network Operations: Network Operations: Link FailureLink Failure

Link failure

New route overloads a link

Routing change alleviates congestion

What’s a traffic matrix?What’s a traffic matrix?

ingress

egress

Xj

Xj

Yi

PoP (Point of Presence)

Y = A X or Y=RX

Link Measurement Vector

Routing Matrix

“Traffic Matrix”

Example ProblemExample Problem

A B

C D

5 3

4 4

How much traffic flowsbetween origin-destinationpairs?A->CA->DB->CB->D

SNMP byte counts per link

Example: One SolutionExample: One Solution

A B

C D

5 3

4 4

How much traffic flowsbetween?A->D: 4A->C: 1B->C: 3B->D: 0

0

3 41

Example: Another Example: Another SolutionSolution

A B

C D

5 3

4 4

How much traffic flowsbetween?A->D: 2A->C: 3B->C: 1B->D: 2

2

1 23

type of equations: Link1 = XAD + XBD

Link 1

4Mbps 4Mbps

3Mbps5Mbps

Inference: Network Inference: Network TomographyTomography

Sources

Destinations

From link counts to the traffic matrix

11stst Generation Generation ApproachesApproaches

Linear Programming (LP) approach.Linear Programming (LP) approach. O. Goldschmidt - ISMA Workshop 2000O. Goldschmidt - ISMA Workshop 2000

Bayesian estimation.Bayesian estimation. C. Tebaldi, M. West - J. of American Statistical AssC. Tebaldi, M. West - J. of American Statistical Ass

ociation, June 1998.ociation, June 1998. Expectation Maximization (EM) apExpectation Maximization (EM) ap

proach.proach. J. Cao, D. Davis, S. Vander Weil, B. Yu - J. of AmeriJ. Cao, D. Davis, S. Vander Weil, B. Yu - J. of Ameri

can Statistical Association, 2000.can Statistical Association, 2000.

Linear ProgrammingLinear Programming

Objective:Objective:

Constraints:Constraints:

Statistical ApproachesStatistical Approaches

Bayesian ApproachBayesian Approach

Assumes Assumes P(XP(Xjj)) follows a Poisson distribution follows a Poisson distribution with mean with mean λλ j. j. (independently dist.)(independently dist.)

needs to be estimated. (a prioneeds to be estimated. (a prior is needed)r is needed)

Conditioning on link counts: P(X,Conditioning on link counts: P(X,ΛΛ|Y)|Y)Uses Markov Chain Monte Carlo (MCMC) simuUses Markov Chain Monte Carlo (MCMC) simu

lation method to get posterior distributions.lation method to get posterior distributions. Ultimate goal: compute P(X|Y)Ultimate goal: compute P(X|Y)

Expectation Expectation Maximization (EM)Maximization (EM)

Assumes Xj are ind. dist. Gaussian.Assumes Xj are ind. dist. Gaussian.

Y=AX implies:Y=AX implies:

Requires a prior for initialization.Requires a prior for initialization. Incorporates multiple sets of link measuremIncorporates multiple sets of link measurem

ents.ents. Uses EM algorithm to compute MLE.Uses EM algorithm to compute MLE.

22ndnd generation methods generation methods MOTIVATION: The fundamental problem is that of MOTIVATION: The fundamental problem is that of

an under-constrained, or ill-posed, system. some sort an under-constrained, or ill-posed, system. some sort of side information or assumptions must then be of side information or assumptions must then be added to make the estimation problem well-posed.added to make the estimation problem well-posed. What options do we have for getting more data What options do we have for getting more data

into the problem?into the problem? Approach 1:Approach 1:

MLE estimation methods require a “starting MLE estimation methods require a “starting point” (initial condition/prior/etc)point” (initial condition/prior/etc)

Can we find “intelligent starting points” based on Can we find “intelligent starting points” based on network properties?network properties?

Approach 2:Approach 2: What can we do to increase the rank of the What can we do to increase the rank of the

routing matrix?routing matrix?

DirectionsDirections Lessons learned:Lessons learned:

Model assumptions do not reflect the true nature Model assumptions do not reflect the true nature of traffic. (multimodal behavior)of traffic. (multimodal behavior)

Dependence on priorsDependence on priors Link count is not sufficient (Generally more data Link count is not sufficient (Generally more data

is available to network operators.)is available to network operators.) Proposed Solutions:Proposed Solutions:

Use choice models to incorporate additional infoUse choice models to incorporate additional information.rmation.

Generate a good prior solution: Gravity model.Generate a good prior solution: Gravity model. Information-Theoretic Information-Theoretic Assignment modelAssignment model

Let RLet Rii be total amount of traffic entering the network th be total amount of traffic entering the network that is sourced at POP iat is sourced at POP i

Traffic POP(i->j)Traffic POP(i->j) = R= Ri i ijij What is What is ijij ? ?

the proportion of traffic at ingress node ‘i’ headed to egress the proportion of traffic at ingress node ‘i’ headed to egress node ‘j’node ‘j’

{{ijij for all for all j j } called the “} called the “fanoutfanout”” Problem: estimate the fanouts Problem: estimate the fanouts ijij

Choice ModelsChoice Models

POP 1

POP 2

POP 4

POP 313

14

12

Gravity modelGravity model Router-to-router gravity model: Router-to-router gravity model:

[Zhang,Roughan, et. al. Sigco[Zhang,Roughan, et. al. Sigcomm04]mm04]

Use this to as a smart initial condition for optimUse this to as a smart initial condition for optimizationization

Solve min ||X – XSolve min ||X – Xgg|| s.t. || AX – Y|| is minimized|| s.t. || AX – Y|| is minimized Use a least squares type solutionUse a least squares type solution

kk

dst

jdst

isrc

jig RX

RXRXRRX

)(

)()(),(

Gravity-based OD Flow Gravity-based OD Flow ModelModel

What does the gravity model say about What does the gravity model say about OD flows?OD flows?

Assume Assume nodesnodes are independent are independent The gravity model is a The gravity model is a spatial modelspatial model

among OD flowsamong OD flows Gravity model is calibrated using SNMP Gravity model is calibrated using SNMP

from access and peering links from access and peering links entering/exiting router nodesentering/exiting router nodes this is not the same SNMP data as the inter-this is not the same SNMP data as the inter-

router links used in estimationrouter links used in estimation

Route Change MethodRoute Change Method Idea: change the link weights - the new shorteIdea: change the link weights - the new shorte

st paths computed will lead to new routes betst paths computed will lead to new routes between some OD pairs ween some OD pairs [Soule, Nucci, Cruz, et. al. Sig[Soule, Nucci, Cruz, et. al. Sigmetrics04]metrics04] Each routing induces a different Y=A(r)*X where Each routing induces a different Y=A(r)*X where

A(r) is the routing matrix for weight setting case A(r) is the routing matrix for weight setting case ‘r’.‘r’.

Hope: by combining all the linear constraints into Hope: by combining all the linear constraints into one big system, we increase the rank of A from thone big system, we increase the rank of A from the original system. It works!e original system. It works!

Caveat: the SNMP link counts from different rCaveat: the SNMP link counts from different routing configurations need to be collected ovouting configurations need to be collected over many hours or even days -> so we are in ther many hours or even days -> so we are in the non-stationary regime of OD traffic flows.e non-stationary regime of OD traffic flows.

An Information-An Information-Theoretic ApproachTheoretic Approach

Maximum EntropyMaximum Entropy Entropy is a measure of uncertaintyEntropy is a measure of uncertainty

More information = less entropyMore information = less entropy To include measurements, maximize entropy To include measurements, maximize entropy

subject to the constraints imposed by the datsubject to the constraints imposed by the dataa

Impose the fewest assumptions on the resultsImpose the fewest assumptions on the results Instantiation: Maximize “relative entropInstantiation: Maximize “relative entrop

y”y” Minimum Mutual InformationMinimum Mutual Information

Assignment modelAssignment model We may see our problem as follows. We may see our problem as follows.

and =1and =1

The ultimate value of OD pair can be The ultimate value of OD pair can be described by: described by:

and =1and =1

z iz ix l yz

izl

z zt ztt

x w xt

ztw

33rdrd generation models generation models

Carriers set a 10% average error rate Carriers set a 10% average error rate as general target.as general target.

22ndnd generation methods achieving generation methods achieving average errors in the range of 15-average errors in the range of 15-20%, roughly. 20%, roughly.

Can we further reduce errors? Can we further reduce errors? What other kinds of What other kinds of

information/measurements can be information/measurements can be brought into the picture?brought into the picture?

Two-step Statistical ApproachTwo-step Statistical Approach

First stepFirst step: Mlogit and Linear Choice Models: Mlogit and Linear Choice Models

Step 2: Step 2: Expectation Maximization AlgorithmExpectation Maximization Algorithm The division of the TM estimationprocess into The division of the TM estimationprocess into

two steps offers great flexibility for combinintwo steps offers great flexibility for combiningand evaluating different strategies that could gand evaluating different strategies that could be applied to solve theinference problem.be applied to solve theinference problem.

TomogravityTomogravity

Two step modeling.Two step modeling. Gravity Model:Gravity Model: Initial solution obtained us Initial solution obtained us

ing edge link load data and ISP routing poling edge link load data and ISP routing policy.icy.

Tomographic Estimation:Tomographic Estimation: Initial solution i Initial solution is refined by applying quadratic programms refined by applying quadratic programming to minimize distance to initial solution ing to minimize distance to initial solution subject to tomographic constraints (link csubject to tomographic constraints (link counts).ounts).

Genetic-Assignment Genetic-Assignment algorithmalgorithm

the key link the key link CC== RR RRTT

““troublesome” OD pairs troublesome” OD pairs QQ== R RTTRR

problemproblem1

( )P

ij zt iz i ij t

r w l y y

ijij

ii jj

c

c c

ijij

ii jj

q

q q

PCA MethodPCA Method Using the Using the measuredmeasured time series of all the O time series of all the O

D flows - do PCA analysisD flows - do PCA analysis output of PCA: eigenflows – a new time seroutput of PCA: eigenflows – a new time ser

iesies cyclical ones, bursty ones, and noisy onescyclical ones, bursty ones, and noisy ones

Each OD flow can be represented by a weiEach OD flow can be represented by a weighted sum of a small number (<10) eigenflghted sum of a small number (<10) eigenflowsows

PCA SolutionPCA Solution

Rather than estimate the traffic matrix,Rather than estimate the traffic matrix, estimate the eigenflows (elements of t estimate the eigenflows (elements of the low-dim representation)he low-dim representation) this is well posed.this is well posed.

Rebuild the traffic matrix using the apRebuild the traffic matrix using the appropriate weighted sum of the eigenflopropriate weighted sum of the eigenflows.ws.

Issues in 3Issues in 3rdrd gen methods gen methods Model Recalibration: need to keep models up Model Recalibration: need to keep models up

to date as traffic evolvesto date as traffic evolves For models based on 24-hours of measurements: For models based on 24-hours of measurements:

need scheme for detecting change and deciding need scheme for detecting change and deciding when to launch a new measurement collection when to launch a new measurement collection episode.episode.

For model with 1-flow at a time, no change For model with 1-flow at a time, no change detection needed; the model is essentially self detection needed; the model is essentially self updating on an ongoing basis.updating on an ongoing basis.

OverheadsOverheads a tradeoff is induced: measurement overhead versus a tradeoff is induced: measurement overhead versus

gain in error reductiongain in error reduction

Areas of ApplicationAreas of Application

Route selection Route selection how to choose link weights for shortest how to choose link weights for shortest

path routingpath routing Evaluating the impact of policy Evaluating the impact of policy

changes on trafficchanges on traffic Anomaly detectionAnomaly detection

Application Area #1:Application Area #1:Selecting Link Weights for Selecting Link Weights for

RoutingRouting Link weights selection algorithms use a Link weights selection algorithms use a

traffic matrix as input. Goal: balance traffic traffic matrix as input. Goal: balance traffic across links well. across links well. suppose the input TM has errors?suppose the input TM has errors? how does this affect our ability to choose routes?how does this affect our ability to choose routes?

Want a set of routes to last many days Want a set of routes to last many days without requiring changes. But the TM is a without requiring changes. But the TM is a dynamic fluctuating thing. dynamic fluctuating thing. Can a single set of weights be good for along Can a single set of weights be good for along

time, i.e., over a variety of TMs?time, i.e., over a variety of TMs?

Application Area #1: Application Area #1: Some FindingsSome Findings

[Roughan et. al. IMC03][Roughan et. al. IMC03] yes there is some sensitivity, but it’s not too badyes there is some sensitivity, but it’s not too bad

except: “optimal” routing (MPLS) is more sensitive except: “optimal” routing (MPLS) is more sensitive than than near-optimal algorithms (OSPF)near-optimal algorithms (OSPF)

can find a routing that is robust to daily can find a routing that is robust to daily fluctuationsfluctuations

[Applegate/Cohen Sigcomm03][Applegate/Cohen Sigcomm03] theoretical result, using oblivious routing...theoretical result, using oblivious routing... showed that can find a single routing that works showed that can find a single routing that works

well well under a wide variety of cases of traffic matricesunder a wide variety of cases of traffic matrices

Application Area #2:Application Area #2:Impact of Routing Policy Impact of Routing Policy

ChangeChange Using a TM, can get broad view of policy Using a TM, can get broad view of policy

changeschanges Questions:Questions:

what kinds of fluctuations do we see in the TM what kinds of fluctuations do we see in the TM due to changes in internal routing (IGP) ? due to changes in internal routing (IGP) ? [Agarwal, et. al. Sigmetrics04][Agarwal, et. al. Sigmetrics04]

what kinds of fluctuations do we see in the TM what kinds of fluctuations do we see in the TM due to changes in inter-domain routing (BGP)? due to changes in inter-domain routing (BGP)? [Teixeira, et. al. PAM05][Teixeira, et. al. PAM05]

Answer: Not often, but when they happen, Answer: Not often, but when they happen, they are big (affect a lot of traffic).they are big (affect a lot of traffic).

Application Area #3:Application Area #3:Anomaly DetectionAnomaly Detection

A set of traffic matrices over time can be A set of traffic matrices over time can be used to describe “normal” traffic. used to describe “normal” traffic. We now have lots of models for OD flows.We now have lots of models for OD flows. Can we then identify abnormalities?Can we then identify abnormalities?

Subspace Method Subspace Method [Lakhina, et. al. SIGCOMM04][Lakhina, et. al. SIGCOMM04] Builds on the PCA idea - projects traffic flows Builds on the PCA idea - projects traffic flows

onto low-dimensional representation and onto low-dimensional representation and extracts outliersextracts outliers

There is much more that can be done There is much more that can be done here ...here ...

Application Area #3:Application Area #3:Anomaly DetectionAnomaly Detection

Advantages of using TMs for security: Advantages of using TMs for security: have network-wide perspectivehave network-wide perspective

If see attack on a set of links, maybe this all If see attack on a set of links, maybe this all belongs to one OD flow, i.e., the same belongs to one OD flow, i.e., the same attackattack permits easy identification of point of entrypermits easy identification of point of entry

If one attacker attacked multiple victims, If one attacker attacked multiple victims, anomalies show up in one row of a TManomalies show up in one row of a TM

If multiple zombies attack a single victim, If multiple zombies attack a single victim, anomalies show up in a column of the TManomalies show up in a column of the TM

Traffic Matrix: Operational Traffic Matrix: Operational UsesUses

Short-term congestion and performance problemsShort-term congestion and performance problems Problem: predicting link loads after a routing changeProblem: predicting link loads after a routing change Map the traffic matrix onto the new set of routesMap the traffic matrix onto the new set of routes

Long-term congestion and performance problemsLong-term congestion and performance problems Problem: predicting link loads after topology changesProblem: predicting link loads after topology changes Map traffic matrix onto the routes on new topologyMap traffic matrix onto the routes on new topology

Reliability despite equipment failuresReliability despite equipment failures Problem: allocating spare capacity for failoverProblem: allocating spare capacity for failover Find link weights such that no failure causes overloadFind link weights such that no failure causes overload

Traffic Matrix: Traffic Traffic Matrix: Traffic Engineering ExampleEngineering Example

Problem Problem Predict influence of weight changes on traffic flowPredict influence of weight changes on traffic flow Minimize objective function (say, of link utilization)Minimize objective function (say, of link utilization)

InputsInputs Network topology: capacitated, directed graphNetwork topology: capacitated, directed graph Routing configuration: integer weight for each linkRouting configuration: integer weight for each link Traffic matrix: offered load for each pair of nodesTraffic matrix: offered load for each pair of nodes

OutputsOutputs Shortest path(s) for each node pairShortest path(s) for each node pair Volume of traffic on each link in the graphVolume of traffic on each link in the graph Value of the objective functionValue of the objective function

SummarySummary Traffic Matrix estimation methods today Traffic Matrix estimation methods today

are quite good.are quite good. Who knows, maybe sampling will obviate Who knows, maybe sampling will obviate

the need for inference...the need for inference... Research has produced interesting Research has produced interesting

models for Origin-Destination flowsmodels for Origin-Destination flows Applications using traffic matrices are Applications using traffic matrices are

gaining momentum, but there’s lots gaining momentum, but there’s lots more to domore to do

Giving the community tools to generate Giving the community tools to generate synthetic traffic matrices would improve synthetic traffic matrices would improve evaluation studies.evaluation studies.

ReferenceReference [1]Y. Vardi, Network tomography: estimating source-destination traffic intensi[1]Y. Vardi, Network tomography: estimating source-destination traffic intensi

ties from link data, Journal of the American Statistical Association, Vol. 91, No.ties from link data, Journal of the American Statistical Association, Vol. 91, No. 433, 1996, pp. 365-377. 433, 1996, pp. 365-377.

[2] C. Tebaldi, M. West, Bayesian Inference of Network Traffic Using Link Cou[2] C. Tebaldi, M. West, Bayesian Inference of Network Traffic Using Link Count Data, Journal of the American Statistical Association, Vol. 93, No. 442, 1998, nt Data, Journal of the American Statistical Association, Vol. 93, No. 442, 1998, pp. 557-576.pp. 557-576.

[3] J. Cao, D. Davis, S. Vander Weil and B. Yu, Time-Varying Network Tomogra[3] J. Cao, D. Davis, S. Vander Weil and B. Yu, Time-Varying Network Tomography: Router Link Data, Journal of the American Statistical Association, Vol. 95, phy: Router Link Data, Journal of the American Statistical Association, Vol. 95, No. 452, 2000, pp.1063-1075.No. 452, 2000, pp.1063-1075.

[4] A. Medina, N. Taft, K. Salamatian, S. Bhattacharyya and C. Diot, Traffic Mat[4] A. Medina, N. Taft, K. Salamatian, S. Bhattacharyya and C. Diot, Traffic Matrix Estimation: Existing Techniques Compared and New Directions, ACM Sigcrix Estimation: Existing Techniques Compared and New Directions, ACM Sigcomm, Pitsburgh, PA, August 2002, pp. 161-174.omm, Pitsburgh, PA, August 2002, pp. 161-174.

[5] Y. Zhang, M. Roughan, N. Duffield and A. Greenberg, Fast Accurate Compu[5] Y. Zhang, M. Roughan, N. Duffield and A. Greenberg, Fast Accurate Computation of Large-Scale IP Traffic Matrices from Link Loads, Proceedings of ACM tation of Large-Scale IP Traffic Matrices from Link Loads, Proceedings of ACM Sigmetrics San Diego, CA, June 2003, pp. 2061-217.Sigmetrics San Diego, CA, June 2003, pp. 2061-217.

[6] Y. Zhang, M. Roughan, C. Lund and D. Donoho, An Information Theoretic [6] Y. Zhang, M. Roughan, C. Lund and D. Donoho, An Information Theoretic Approach to Traffic Matrix Estimation, Proceedings of ACM Sigcomm, KarlsruApproach to Traffic Matrix Estimation, Proceedings of ACM Sigcomm, Karlsruhe, Germany, August 2003, pp. 301-312.he, Germany, August 2003, pp. 301-312.

[7] H. Tang, TL. Fan and GF. Zhao. An Assignment Model on Traffic Matrix Est[7] H. Tang, TL. Fan and GF. Zhao. An Assignment Model on Traffic Matrix Estimation. LNCS 4222imation. LNCS 4222 ，， Part II, 2006, pp.295-304. Part II, 2006, pp.295-304.

ReferenceReference [8]A. Nucci, R. Cruz, N. Taft, and C. Diot, Design of IGP link weight changes for e[8]A. Nucci, R. Cruz, N. Taft, and C. Diot, Design of IGP link weight changes for e

stimation of traffic matrices, Proceedings of IEEE INFOCOM 2004, Vol. 4, Hong stimation of traffic matrices, Proceedings of IEEE INFOCOM 2004, Vol. 4, Hong Kong, 2004, pp. 2341- 2351.Kong, 2004, pp. 2341- 2351.

[9] Y. Zhang, Z. Ge, Finding critical traffic matrices, Proceedings of the Internati[9] Y. Zhang, Z. Ge, Finding critical traffic matrices, Proceedings of the International Conference on Dependable Systems and Networks, 2005, pp. 188-197.onal Conference on Dependable Systems and Networks, 2005, pp. 188-197.

[10] A. Nucci, R. Cruz, N. Taft and C. Diot, How to Identify and Estimate the Larg[10] A. Nucci, R. Cruz, N. Taft and C. Diot, How to Identify and Estimate the Largest Traffic Matrix Elements in a Dynamic Environment, ACM SIGMETRICS Perfoest Traffic Matrix Elements in a Dynamic Environment, ACM SIGMETRICS Performance Evaluation Review, Vol. 32, No. 1, 2004, pp. 73-84rmance Evaluation Review, Vol. 32, No. 1, 2004, pp. 73-84

[11] A.Medina, K. Salamatian, N. Taft, I. Matta and C. Diot, A two-step statistical [11] A.Medina, K. Salamatian, N. Taft, I. Matta and C. Diot, A two-step statistical approach for inferring network traffic demands, Technical Report, BUCS-2004-0approach for inferring network traffic demands, Technical Report, BUCS-2004-011, Computer Science, Boston University, 2004.11, Computer Science, Boston University, 2004.

[12] M. Gh. Negoita, M. Giuclea and H. Dediu, GA to optimize approximate soluti[12] M. Gh. Negoita, M. Giuclea and H. Dediu, GA to optimize approximate solutions of fuzzy relational equations for fuzzy systems or controllers, Proceedings of ons of fuzzy relational equations for fuzzy systems or controllers, Proceedings of the 2nd New Zealand Two-Stream International Conference on Artificial Neural the 2nd New Zealand Two-Stream International Conference on Artificial Neural Networks and Expert SystemsNetworks and Expert Systems,, Dunedin, New Zealand, Nov. 1995, pp. 124 – 127 Dunedin, New Zealand, Nov. 1995, pp. 124 – 127

[13] TL. Fan, H. Tang and YB. Liu, How to Identify Critical Links and Relation bet[13] TL. Fan, H. Tang and YB. Liu, How to Identify Critical Links and Relation between OD Pairs, Dynamics of Continuous, Discrete & Impulsive Systems, Series B:ween OD Pairs, Dynamics of Continuous, Discrete & Impulsive Systems, Series B: Applications & Algorithms, Vol. 14 (S3), Supplementary, August, 2007, ISSN1492- Applications & Algorithms, Vol. 14 (S3), Supplementary, August, 2007, ISSN1492-8760, Part 1, pp.74-79.8760, Part 1, pp.74-79.

[14] A. Feldmann, A. Greenberg, C. Lund, N. Reingold, J. Rexford, and F. True, D[14] A. Feldmann, A. Greenberg, C. Lund, N. Reingold, J. Rexford, and F. True, Deriving traffic demands for operational IP networks: Methodology and experienceriving traffic demands for operational IP networks: Methodology and experience, IEEE/ACM Transactions on Networking, Vol.9, No.3, June 2001, pp. 265–279.e, IEEE/ACM Transactions on Networking, Vol.9, No.3, June 2001, pp. 265–279.

That’s it!That’s it! For more detail, see For more detail, see

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