Failure Probability Bounds Failure Probability Bounds of Complex of Complex

Telecommunication System Telecommunication System by Use of LPby Use of LP

SupervisorSupervisor Dr. Alexan Simonyan Dr. Alexan Simonyan

RefereeReferee Sargis ZeytunyanSargis Zeytunyan StudentStudent Yelena Vardanyan Yelena Vardanyan

American University of ArmeniaAmerican University of Armenia

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OutlineOutline IntroductionIntroduction Chapter 1: Chapter 1: Reliability and Failure Probability Analysis for 9 StationsReliability and Failure Probability Analysis for 9 Stations

Gamma DistributionGamma Distribution Weibull DistributionWeibull Distribution Exponential DistributionExponential Distribution

Chapter 2: Chapter 2: Theoretical BackgroundTheoretical Background: :

Failure Probability Bounds of the Whole System by the use of LPFailure Probability Bounds of the Whole System by the use of LP LP’s Size and Decomposition ApproachLP’s Size and Decomposition Approach LP Formulation LP Formulation Advantages of LP bound’s method Advantages of LP bound’s method LP Formulation for Conditional ProbabilityLP Formulation for Conditional Probability

Chapter 3: Chapter 3: LP formulation of the telecommunication systemLP formulation of the telecommunication system LP formulation for sub-componentLP formulation for sub-component LP formulation for Conditional ProbabilityLP formulation for Conditional Probability General LP for the sub-componentGeneral LP for the sub-component LP formulation for entire systemLP formulation for entire system

Conclusions and Recommendations for Future WorkConclusions and Recommendations for Future Work ReferencesReferences

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• Reliability Reliability is the probability that system will not fail under some specified set of circumstances.

IntroductionIntroductionThe main goal of this research is to give general picture of the complex telecommunication system: which percent of time the system is available with its 100% working condition and which percent of time the system is not available (failure probability bounds). This work is done, based on the results of T. Ghazaryan’s thesis: the failure time distribution of all stations with their estimated parameters. The mentioned thesis is done in terms of power supply, one from the series of problems which can cause outages, based on the real-life data.

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Reliability and Failure Reliability and Failure Probability Analysis for 9 Probability Analysis for 9 StationsStations

Site Code Failure time (distribution)

Yer_001 Gamma Yer_002 Weibull Yer_006 Weibull Yer_010 Gamma Yer_012 Exponential Yer_016 Exponential Yer_018 Weibull Yer_019 Gamma Yer_020 Gamma

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Gamma DistributionGamma Distribution

Failure Gamma function

Estimated Parameters Site Code

k λ

Yer_001 1,4303 0,0032 Yer_010 14,6759 0,0145 Yer_019 0,7382 0,0103 Yer_020 2,4099 0,0021

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Reliability and Failure Reliability and Failure Probability Analysis: Probability Analysis: Gamma DistributionGamma Distribution

dueuk

tTPtRt uk

f

0

11)()(

)()(1)( tTPtRtF

dueuk

t uk

0

1 )(tF

for t > 0for t > 0

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Weibull DistributionWeibull Distribution

Failure Weibull function

Estimated Parameters Site Code

α λ

Yer_002 0,5213 0,0892 Yer_006 0,5966 0,0463 Yer_018 0,5377 0,0798

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Reliability and Failure Reliability and Failure Probability Analysis: Probability Analysis: Waibull DistributionWaibull Distribution

etTPtR f

t )()(

)(

)*(1)( tetF

for t > 0for t > 0

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Exponential DistributionExponential Distribution

Site Code Failure Exponential function

Estimated Parameters

λ Yer_012 0,0157 Yer_016 0,0058

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Reliability and Failure Reliability and Failure Probability Analysis: Probability Analysis: Exponential DistributionExponential Distribution

etTPtR ft

)()(

etFt

1)(

for t > 0for t > 0

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Calculated Reliabilities and Calculated Reliabilities and Failure ProbabilitiesFailure Probabilities

Site Code Failure time (distribution)

Reliability (1 hour)

%

Failure probability

(1 hour) %

Yer_001 Gamma 0.9968 0.0032 Yer_002 Weibull 0.752 248.0 Yer_006 Weibull 0.854 0.146 Yer_010 Gamma 0.9857 0.0143 Yer_012 Exponential 0.98 0.02 Yer_016 Exponential 0.994 0.006 Yer_018 Weibull 0.7752 0.2248 Yer_019 Gamma 0.9899 0.0101 Yer_020 Gamma 0.9979 0.0021

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Failure Probability Bounds of the Whole Failure Probability Bounds of the Whole

System by use ofSystem by use of LPLP

The system failure probability bounds The system failure probability bounds was old enough announced in 1965 was old enough announced in 1965 and first was explored by Hailperin. and first was explored by Hailperin. Then Kounias and Marin in 1976 Then Kounias and Marin in 1976 used the method to look at the used the method to look at the accuracy of some theoretical accuracy of some theoretical bounds.bounds.

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Advantages of LP Bounds Advantages of LP Bounds MethodMethod

• Any type of information can be usedAny type of information can be used– Marginal component failure probabilities Marginal component failure probabilities – Joint component failure probabilities Joint component failure probabilities

• The method guarantees the narrowest The method guarantees the narrowest possible boundspossible bounds

• The method is applicable to general The method is applicable to general systemssystems

• Easy identification of critical components Easy identification of critical components and cut sets within a systemand cut sets within a system

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LP’s Size and Decomposition ApproachLP’s Size and Decomposition Approach

The approach is the following:The approach is the following:• Decompose the system into a Decompose the system into a

number of subsystemsnumber of subsystems• Consider each subsystem and Consider each subsystem and

perform analyses separatelyperform analyses separately• Consider subsystems as components Consider subsystems as components

for the whole systemfor the whole system

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LP FormulationLP Formulation

The general formulation of LP is the The general formulation of LP is the following:following:

Minimize/maximize Psys. = CTp

subject to A1p = B1

A2p B2 (1)

A3p B3

12

1

n

i jp

pj 0 , j = 1,2,….n

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LP FormulationLP Formulation

• n equality or inequality constraints n equality or inequality constraints results from knowledge of uni-results from knowledge of uni-component probabilities, component probabilities,

equality or inequality constraints equality or inequality constraints results from knowledge of bi-results from knowledge of bi-component probabilities,component probabilities,

!2)!*2(

!2

n

nCn

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Conditional ProbabilityConditional Probability

P(AB) = P(B|A)*P(A) P(AB) = P(B|A)*P(A)

)(

)()|(

AP

ABPABP

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LP Formulation for Conditional LP Formulation for Conditional ProbabilityProbability

Ar r

ABr r

p

pBAP )|(

Minimize/maximize (CTAB - CT

A )*p

subject to A1p = B1 (5)

A2p B2

A3p B3

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LP Formulation for LP Formulation for Telecommunication SystemTelecommunication System

• The number of unknown variables The number of unknown variables would be 2would be 299=512,=512,

• 9 equality constraints result from 9 equality constraints result from the knowledge of the marginal (uni-) the knowledge of the marginal (uni-) component failure probabilities, component failure probabilities,

• CC992 2 = 36 equality constraints result = 36 equality constraints result

from the knowledge of the joint (bi-) from the knowledge of the joint (bi-) component failure probabilitiescomponent failure probabilities

• Probability axiomsProbability axioms

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LP Formulation for Super-LP Formulation for Super-ComponentComponent

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LP Formulation for Conditional LP Formulation for Conditional ProbabilityProbability

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Calculated Conditional Calculated Conditional ProbabilitiesProbabilities

Lower Bound of joint Failure probability

Upper Bound of joint Failure probability

P(AB) 0 0,00001024

P(AC) 0 0,00001024

P(AD) 0 0,00001024

P(AE) 0 0,00001024

P(BC) 0 0,036208

P(BD) 0 0,0035464

P(BE) 0 0,00496

P(CD) 0 0,0020878

P(CE) 0 0,00292

P(DE) 0 0,000204

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LP for the super-componentLP for the super-component

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Calculated Bounds for Calculated Bounds for Super-ComponentSuper-Component

The failure probability of sub-The failure probability of sub-component in terms of defined component in terms of defined system event:system event:

(AUBUCUDUE)(AUBUCUDUE) ЄЄ (0.3815328; (0.3815328; 0.4315) interval.0.4315) interval.

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LP Formulation for Entire LP Formulation for Entire SystemSystem

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Calculated Conditional Calculated Conditional ProbabilitiesProbabilities

Lower Bound of joint Failure probability

Upper Bound of joint Failure probability

P(AB) 0 0,000036

P(AC) 0 0,000036

P(AD) 0 0,0000126

P(AE) 0 0,000036

P(BC) 0 0,00227048

P(BD) 0 0,00047208

P(BE) 0 0,05053504

P(CD) 0 0,00002121

P(CE) 0 0,00010201

P(DE) 0 0,00000441

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LP Formulation for Entire LP Formulation for Entire SystemSystem

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Calculated Bounds for Calculated Bounds for Entire SystemEntire System

The failure probability of entire The failure probability of entire telecommunication system in terms telecommunication system in terms of defined system event:of defined system event:

(A(A11UBUB11UCUC11UDUD11UEUE11) ) ЄЄ (0.571007; (0.571007; 0.6745) interval.0.6745) interval.

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ConclusionConclusion

This means in general, the working This means in general, the working condition of the whole condition of the whole telecommunication system varies from telecommunication system varies from 100 % working condition 57-67 % in time100 % working condition 57-67 % in time

OrOr

The working condition of the whole The working condition of the whole telecommunication system varies from telecommunication system varies from 100% working condition 34-40 minutes 100% working condition 34-40 minutes in one hour.in one hour.

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RECOMMENDATIONS for RECOMMENDATIONS for FUTURE WORKFUTURE WORK

• To get failure probability of the To get failure probability of the entire system by use of Simulationentire system by use of Simulation

• To do sensitivity analysis and find To do sensitivity analysis and find out the weakest component (station) out the weakest component (station) in this system.in this system.

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ReferencesReferences

• A.D.Kiureghian, Junho Song, “A.D.Kiureghian, Junho Song, “Multi-scale Reliability Multi-scale Reliability Analysis and Updating of Complex Systems by Use of Analysis and Updating of Complex Systems by Use of Linear Programming”Linear Programming” 2005 2005

• Arnljot Hoyland, Marvin Rausand, “Arnljot Hoyland, Marvin Rausand, “System Reliability System Reliability

TheoryTheory” 1994” 1994

• Richard A. Johnson, “Richard A. Johnson, “Miller & Freunds Probability & Miller & Freunds Probability & Statistics for EngineersStatistics for Engineers” 1994” 1994

• Tigran Ghazaryan, thesis work “Tigran Ghazaryan, thesis work “Availability, Reliability Availability, Reliability and Maintainability of the power supply system of the and Maintainability of the power supply system of the Telecommunication CompanyTelecommunication Company ”, Yerevan, 2006.”, Yerevan, 2006.

• Sheldon M. Ross “Sheldon M. Ross “Introduction to probability and Introduction to probability and statistics for engineers and scientistsstatistics for engineers and scientists” 1987” 1987

• E. E. Lewis “E. E. Lewis “Reliability engineeringReliability engineering” 1996” 1996

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Thank YouThank You