AVAILABILITY OF A SYSTEM SUBJECTED TO COMMON CAUSE AND HUMAN ERROR FAILURES: COMPARISON OF REPAIR TI

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AVAILABILITY OF A SYSTEM SUBJECTED TO COMMON CAUSE AND HUMAN

ERROR FAILURES: COMPARISON OF REPAIR TIME DISTRIBUTION

Dr. SANE SREELAKSHMI1 , Dr. S. RUSHMA2

1Lecturer in Mathematics, NSPR Government Degree College for Women, Hindupur, AP.

2Lecturer in Mathematics, Dr YSR Government Degree College, Vedurukuppam, AP.

ABSTRACT: Availability analysis can be used in research and development to evaluate the performance of new systems technologies and components. In the context of three unit series parallel system availability analysis can help to evaluate the reliability and availability of the system, identify the most critical components and sub systems, optimize the system configuration and maintenance strategies and assess the risk of system failures and develop strategies. This paper presents the availability analysis of a three unit series parallel system subjected to common cause and human error failures. A general method for the system steady state availability or limiting availability of the system is developed when failed system repair times are gamma distributed.

KEYWORDS: Reliability, Availability, Three unit series parallel systems, Common cause failures, Human error failures.

1.INTRODUCTION:

The reliability evaluation of two component stand by systems, series and parallel systems, have been studied by manyauthorsunderdifferentassumptions.Inrealitythesystemsundertheconsiderationmaynotbemodeledas seriesorparallelsystems.Forexample,ifweconsiderahumansystem,theheartandthekidneysshouldfunction properly for the human being to survive (assuming the remaining parts of the body are all operative). However this cannot bemodeled as a simple series or parallel system comprising two subsystems with one or two components in each sub-system While the heart does appear as a single component in the first sub-system , the second sub - system representing the kidneys will have to be represented by two components , each one correspondingtooneofthetwokidneys.Thisisnecessitatedbythefactthattheproperfunctioningofanyoneof the two kidneys ensures the survival of the human being. The system may therefore by modeled as a series –parallelsystemasshowninfig.1.

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Here the components A, B and C are statistically independent, i.e., the failure of one component does not affectthefailureorotherwiseoftheothertwocomponents.Theotherexamplesofseries–parallelsystemsare

1. ACPUconnectedtotwoparallelI/Oports.

2. AMultiplexerconnectedtoapairofterminals.

3. AStabilizerpoweredbytwoalternativesources.

In this paper weconsider a three unit series parallel system and we study the availability analysis of the systemissubjectedtocommoncauseandhumanerrorfailures.

2. ASSUMPTIONS AND NOTATIONS:

weconsiderthreemodelsofa threeunitseries-parallel system consisting of the units A, BandCwhich isinfunctioning stateonly, when A is functioning and either BorCareinfunctioningstate,andweassume thatthesystemissubjected tocommoncauseandhumanerrorfailures.

ASSUMPTIONS

Atanytimeepoch‘t’thesystemcanbeinoneofthefollowingeightstates.

0  Thestateofthesystemwithallthethreecomponentsfunctioningstate.

1 ThestateofthesystemwiththecomponentAandoneofthecomponentsBorCis infunctioningstate.

2  ThefailedstateofthesystemcorrespondingtothefailureofAfromtheState‘0’

3  ThefailedstateofthesystemduetothefailureofthecomponentAfromstate1.

4  Thefailurestateofthesystemcorrespondingtothefailureofboththecomponents BandCwhileAisfunctioning.

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5  Thefailedstateofthesystemduetohumanerrorfromstate‘0’.

6  Thefailedstateofthesystemduetocommoncause.

7  ThefailedstateofthesystemduetocommoncauseoffailureofAandB

NOTATIONS :

t =Time

S = Laplacetransformvariable

 = ConstantfailurerateoftheunitsAandB.

C1 = Constantcommoncausefailurerateofthesystemfromthestate‘0’

h = Constanthumanerrorfailurerateofthesystem

C 2 = Constantcommoncausefailurerateofthesystemfromstate1.

A = ConstantfailurerateoftheunitA

AB = CommonfailurerateofAandB

 = Constantrepairrateofaunit

1 = Constantrepairrateofthesystemfromthestate4.

2 =Constantrepairrateofthesystemfromthefailedstate3.

AB =CommonrepairrateofAandB.

PK(x,t)=Probabilitydensity(withrespecttorepairtime)thatthefailedsystem isin stateKandhasanelapsedrepairtimeofxforK=5,6.

k(x),qk(t)=Repairrateandprobabilitydensityfunctionof repairtimeirrespectively whenthefailedsystem isinstatekandhasanelapsedrepairtimeofx fork=5,6.

 = TheshapeparameteroftheGammaprobabilitydensityfunction.

3.ANALYSIS

Withtheabovenotationsthetransitiondiagramofthesystemisgivenby Fig 2: Transition diagram 2

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Fig 2: Transition diagram

Thesystemofintegrodifferentialequationsassociatedwiththismodelaregivenby dt

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Theinitialconditionsaregivenby P0(0)=1andpj(0)=0 for j=1,2,3,4,7 Pk (x,0)=0 for K=5,6.

UsingLaplaceTransformations theaboveequationsreducesto

Fromequations12to16

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Fromequations.11

Fromequation17

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Integratingweobtain

logP5(x,s)0

logP5(x,s)

logP5(0,s)

P5(x,s)

P5(0,s)

Againfromequations18weobtain

Fromequation11.wehave

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Fromtheequation21

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Similarlyfromtheequation22weget

Asaspecialcaseweassumethattherepairtimedistributionofthefailedsystemwillfollow Gammadistributionwithshapeparameter . Inthiscaseonehas

(I).Weassumethattheshapeparameter  =1.Inthiscasetherepairrateofthefailedsystemisconstantandits repairtimesareexponentiallydistributed.Hencewehave

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TakingLaplacetransformsweobtain

substitutingtheseresultsintoequation 2.3.23weobtain

Similarly,

where

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N1(S)=2 (S+A2)(S+A3)(S+A4)(S+A5)

N2(S)= A (S+A1)(S+A3)(S+A4)(S+A5)

N3(S)=2A (S+A2)(S+A3)(S+A5)

N4(S)=22 (S+A2)(S+A4)(S+A5)

N5(S)= h (S+A1)(S+A2)(S+A3)(S+A4)(S+A5)

N6(S)=[(S+A1) 1c  +2 2c  ](S+A2)(S+A3)(S+A4)(S+A5)

N7(S)= AB (S+A1)(S+A2)(S+A3)(S+A4)

IfSi,1 i  7aretherealrootsoftheequationD0 (s)=0then fromtheequation28and29 weobtain

And G16eS7t G15eS6t G14eS5t G13eS4t G12eS3t G11eS2t G10eS1t G9 P1(t)

Here thetimedependentavailabilityAv (t)ofthesystemisgivenby Av(t)=P0 (t)+P1 (t)

2 ( 9) 1 (

Limitingavailability(steadystate)Av ofthesystemisgivenby 9) 1 ( ) (

(ii)Weassumethattheshapeparameter 2  

InthiscaserepairtimedistributionisErlangiananditsrepairratesbecometime-dependent.In thiscase onehas x x x q 5 e 2 5 ) 5(    and

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TakingLaplaceTransformsweobtain

Substitutetheseexpressionsin2.3.23 sDR0(s) NR0(s) P0(s)  36 where

9 s DR0(s)

whereB1 toB33 areconstants

Similarly

NR1(S) 0(S) SDR 6)2μ 5)2(Sμ (S P1(S) P0(S)A1 S 2λ P1(S)

NR2(S) 0(S) SDR 6)2μ 5)2(Sμ (S P2(S) P0(S) Aμ S A

NR3(S) 0(S) SDR 6)2μ 5)2(Sμ (S P3(S) P1(S) 2μ S A

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NR4(S) 0(S) SDR 6)2μ 5)2(Sμ (S P4(S) P1(S) 1

NR7(S) 0(S) SDR 6)2μ 5)2(Sμ (S P7(S) P0(S) ABμ S AB

NR5(S) SDR0(S) 6)2μ 5)(S 2μ (S P5(S) q5(S)) P0(S)(1 S

NR6(S) SDR0(S) 5)2μ 6)(S 2μ (S P6(S) P0(S) S q6(S)) (1

where N1(S) A5) A4)(S A3)(S A2)(S 2λλ( NR1(S) 

N2(S) A5) A4)(S A3)(S A1)(S A(Sλ NR2(S) 



  N4(S) A5) A4)(S A2)(S 2(S 2λ NR4(S) 

N3(S) A5) A3)(S A2)(S A(S 2λλ NR3(S)

N5(S) A5) A4)(S A3)(S A2)(S A1)(S h(Sλ NR5(S) 

N6(S) A5) A4)(S A3)(S A2)(S ](S c2 2λλ A1) (Sc1 [λ NR6(S) 

N7(S) A4) A3)(S A2)(S A1)(S AB(Sλ NR7(S) 

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Forrealroots si ,i=1..9oftheequation 0 ) 0(  s DR fromequationsand Weobtain

whereH1 toH20 areconstants

Herethetimedependentavailability AV (t) ofthesystemisgivenby AV(t)= P1(t) P0(t) 

(H3 H12)eS1t (H2 H11) (H1

5 S H16)e (H6 H15)eS4t (H5

(H4

H20)eS9t (H10 H19)eS8t (H9 t 7 S H18)e (H8

(H7

LimitingAvailability(steadystate) AV ofthesystemisgivenby

DISCUSSION

TheAvailabilitycurveisplottedinfigure3.Fromthesegraphweobservethat

1TheAvailabilityofthesystemdecreases withtime;Availabilityofthesystemdecreasesmorerapidly for 2   that 1

2 Asthehumanerrorincreases,theAvailabilityofthesystemdecreases.

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Fig 3: Availability curve

REFERENCES:

1. B.S.Dhillon and O Anulle, common cause failure analysis of a parallel system with warm stand by Miro, Electronics and Reliability,Vol.33,No.9.(1993),pp.1321.

2. B.S.Dhillon and Nianfu Yang Availability of a man mahine system with critical and non critical human error, Micro electronicsandReliability,Vol.33,No.10(1993),pp.1511

3. Anandarajachari. A. , Sastry , M.P. (1994) : System Reliability Analysis in the presence of Lethal and Non Lethal common causeShockFailures.InternationalJournalofManagementandSystems.Vol. 10. No. 7 pp. 1-8.

4. Dhillon, B.S. and Nianfu yang (1993) : Availability of a man- machine System with Critical and Non-critical human error, Micro-ElectronicsandReliability,Vol.33,No.10,pp.1511-1521.