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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
2.10.2007 2
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
2.10.2007 3
• 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.
2.10.2007 5
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
2.10.2007 6
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
2.10.2007 7
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
2.10.2007 8
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
2.10.2007 9
Reliability and Failure Reliability and Failure Probability Analysis: Probability Analysis: Waibull DistributionWaibull Distribution
etTPtR f
t )()(
)(
)*(1)( tetF
for t > 0for t > 0
2.10.2007 10
Exponential DistributionExponential Distribution
Site Code Failure Exponential function
Estimated Parameters
λ Yer_012 0,0157 Yer_016 0,0058
2.10.2007 11
Reliability and Failure Reliability and Failure Probability Analysis: Probability Analysis: Exponential DistributionExponential Distribution
etTPtR ft
)()(
etFt
1)(
for t > 0for t > 0
2.10.2007 12
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
2.10.2007 13
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.
2.10.2007 14
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
2.10.2007 15
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
2.10.2007 16
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
2.10.2007 17
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
2.10.2007 18
Conditional ProbabilityConditional Probability
P(AB) = P(B|A)*P(A) P(AB) = P(B|A)*P(A)
)(
)()|(
AP
ABPABP
2.10.2007 19
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
2.10.2007 20
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
2.10.2007 23
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
2.10.2007 25
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.
2.10.2007 27
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
2.10.2007 29
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.
2.10.2007 30
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.
2.10.2007 31
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.
2.10.2007 32
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