Reliability Equivalence Analysis of a Parallel-Series System Subject to Degradation Facility

Transcription

1 Sciece Joural of Applied Mateatics ad Statistics 5; 3(3): 6-64 Publised olie Jue 6 5 (ttp:// doi:.648/j.sjas ISSN: (Prit); ISSN: (Olie) Reliability Equivalece Aalysis of a Parallel-Series Syste Subject to Degradatio Facility M. A. El-Dacese Departet of Mateatics Faculty of Sciece Tata Uiversity Tata Egypt Eail address: eldacese@yaoo.co To cite tis article: M. A. El-Dacese. Reliability Equivalece Aalysis of a Parallel-Series Syste Subject to Degradatio Facility. Sciece Joural of Applied Mateatics ad Statistics. Vol. 3 No. 3 5 pp doi:.648/j.sjas Abstract: Te perforace of a reliability syste ca be iproved by differet etods e.g. te reliability of oe or ore copoets ca be iproved ot or cold udat copoets ca be added to te syste. Soeties tese easures ca be equivalet as tey will ave te sae effect o soe perforace easure of te syste. Tis paper discusses te reliability equivaleces of a parallel series syste. Te syste coside ere cosists of subsystes coected i parallel wit subsyste i cosistig of i idepedet ad idetical copoets i series for. Tree differet etods are used to iprove te syste reliability: (i) te uctio etod (ii) te ot duplicatio etod ad (iii) te cold duplicatio etod. Eac copoet of te syste as four states ad two types of partial failure rates. I tis study te lifeties of te syste copoets are expoetially distributed. A uerical exaple is itroduced to illustrate ow te idea of tis wor ca be applied. Keywords: Partial Failure Rate Reliability Equivalece Factors Parallel-Series Syste. Itroductio I reliability teory oe way to iprove te perforace of a syste is to use te udacy etod. Tere are two ai suc etods:. ot duplicatio etod: i tis case it is assued tat soe of te syste copoets are duplicated i parallel.. Cold duplicatio etod: i tis case it is assued tat soe of te syste copoets are duplicated i parallel via a perfect switc. Ufortuately for ay differet reasos suc as space liitatio ig cost etc it is ot always possible to iprove a syste by duplicatig soe or all of its copoets. For exaple satellites ad space aircrafts ave liited space wic does't allow copoet duplicatio. Also soe icrocips are so expasive tat aufacturers caot afford to duplicate te. I suc cases were duplicatio is ot possible te egieer turs to aoter well-ow etod i reliability teory te so-called uctio etod. I tis etod it is assued tat te failure rates of soe of te syste copoets are uced by a factor ρ < ρ <. Now oce te uctio etod is adopted te ai proble facig te egieer is to decide to wat degree te failure rate sould be decreased i order to iprove te syste. To solve tis proble oe ca ae equivalece betwee te uctio etod ad te duplicatio etod based o soe reliability easures. I oter words te desig of te syste iproved by te uctio etod sould be equivalet to te desig of te syste iproved by oe of te duplicatio etods. Te copariso of te desigs produces te socalled reliability equivalece factors by Sara et al. (8). Te cocept of te reliability equivalece factors was itroduced i te report Rade (989-) ad applied to various reliability systes by Rade ( ). Rade (993a 993b) applied tis cocept for te two-copoet parallel ad series systes wit idepedet ad idetical copoets wose lifeties follow te expoetial distributio. Xia ad Zag (7) coside equivalece factors i Gaa distributio. El-Dacese ad Kalifa (8) obtaied te reliability equivalece factors of seriesparallel systes i te Weibull distributio. Mustafa ad El- Faee () foud te reliability equivalece factors of a geeral parallel syste wit ixture of life tie distributios. Also Sawy et al. (3) coside te reliability equivalece for te expoetiated expoetial distributio. I te previous etioed studies te azard ad te reliability fuctios are decreases or icreases troug te idexed scale paraeter. I reliability geeral frae aalysis tere exists oter lifetie distributios for wic te azard ad

2 Sciece Joural of Applied Mateatics ad Statistics 5; 3(3): reliability fuctios are ot affected by te scale paraeter ad aily affected by te sape paraeter. Burr type X distributio iitially proposed by Burr (94) ad ivestigated a geeralizatio of te Rayleig distributio by Mudolar ad Srivastava (993). Tis distributios effectively odeled i geeral lifetie data ad coside by Migdadi ad Al-Bata (4). Abdelfatta ad El-Faee (4) applied geeralize reliability equivalece tecique to apply it to a syste of ixture of idepedet ad oidetical lifeties wit delay tie. Te structure of tis paper is orgaized as follows. Sectio presets te reliability of copoet subject to four states ad two types of partial failure rates. Sectio 3 itroduces te illustratio of te parallel-series syste ad te syste reliability. Sectio 4 presets te reliability of te systes iproved accordig to tree differet etods tat ca be applied to iprove te perforace of te ial syste. I Sectio 5 two types of reliability equivalece factors of te syste are discussed. A uerical exaple is itroduced i Sectio 6 to illustrate ow te idea of tis wor ca be applied. Fially Sectio 7 is devoted to te coclusio.. Reliability of Copoet Cosiderig te copoet wit two types of partial failure rates. Te Marov etod is used to develop geeralized expressios for copoet state probabilities; copoet reliability. Accordig to te odel assuptios we te lifetie of copoet is assued to be expoetially distributed te te state of te copoet at tie t {() } is a oogeeous cotiuous-tie Marov cai wit state space Ω = {3 }. Te set of worig/ degraded states is give by = {3 } ad te set of failure states is give by F= {}. Te iitial coditios for tis proble are: = [ ] [ ] P() P () P () P() P () 3 = () Te differetial equatios of te (worig /degraded) state probabilities writte i te atrix for are give by: [ dp / dt d P / dt d P/ dt ] = 3 -( + ) ( ) ( ) ( ) - - [ P t P t P t ] 3 were Pl ( t ) l = 3 probability tat te copoet is i (degraded /worig) state at tie t P ( ) t probability tat te copoet is i dow state at tie t te failure rate of a copoet we it goes fro up state to degraded state of type te failure rate of a copoet we it goes fro up state to degraded state of type. () Usig te iitial coditio fro equatio () ad obtai te values of requi state probabilities Pl ( t ) l = 3 fro equatio () wic are: ad P = exp[ ( + ) t] 3 P = exp[ t] exp[ ( + ) t] = l l l Te copoet reliability fuctio at tie t is: R = P + P + P 3 = exp[ t] + exp[ t] exp[ ( + ) t] (3) 3. Parallel Series Syste We suppose tat te syste cosists of subsystes coected i parallel ad eac subsyste cosists of i copoets coected i series for i =. Te syste operates successfully we at least oe of its subsystes is up ad eac subsyste wors successfully we all copoets are up (see Figure ). We cosider tat te copoets of eac subsyste are idepedet ad idetical. Te failure rates of eac copoet are costat. Let be te reliability fuctio of te copoet j ( j = i ) i subsyste i (i = ) ad let be te reliability fuctio of te subsyste i. ece te reliability fuctio of te ial syste is give by: Figure. Parallel-series syste structure. R = ( Ri) (4) Assuig tat te syste copoets are idepedet ad idetical avig te failure rates ad tis iplies tat R = (exp[ t] + exp[ t] exp[ ( + ) t]) i (5) i Usig (4) ad (5) te reliability fuctio of te ial syste will tae te for: R = ( (exp[ t] + exp[ t] exp[ ( + ) t]) i ) (6)

3 6 M. A. El-Dacese: Reliability Equivalece Aalysis of a Parallel-Series Syste Subject to Degradatio Facility Usig equatio (6) te ea tie to syste failure ( ) ca be derived i te followig for: dt (7) 4. Te Iproved Systes Te reliability of te syste ca be iproved accordig to oe of te followig two differet etods: - Reductio etod. - Stadby udacy etod: (a) ot stadby udacy called ot duplicatio etod (b) Cold stadby udacy called cold duplicatio etod. I te followig sectios we will derive te reliability fuctios ad te ea tie to failures of te systes iproved accordig to te etods etioed above. 4.. Te Reductio Metod I tis etod it is assued tat te reliability of i i idetical copoets of te subsyste i.. is iproved by icreasig te reliability fuctio troug ultiplyig te failure rates by factors ρ ad s respectively < ρ s <. Terefore usig (3) te reliability of eac of te i copoets of te subsyste i.. is give by: R( ρ s) = exp[ ( ρ) t] + exp[ ( s) t] exp[ ( ρ + s) t] (8) Tis iplies te reliability of te syste iproved by te uctio etod is give by: R = [ ( R) ( R ) ] i i i ( ρ s) t t i i ρ = [ (exp[ ] + exp[ ] exp[ ( + ) t]) (exp[ ( ) t] + exp[ ( s ) t] exp[ ( ρ + s) t]) i ] (9) Usig equatio (9) te ea tie to syste failure ca be derived i te followig for: dt () 4.. ot Duplicatio Metod I tis etod it is assued tat soe of te syste copoets are duplicated i parallel. If i copoets are ot duplicatio te reliability for eac of te i copoets is give by: R = ( R) R () Tis iplies te reliability of te syste iproved by te ot duplicatio etod is give by: i i i R = [ ( R) ( R ) ] t t exp[ ( = [ (exp[ ] + exp[ ] + ) t]) i ( exp[ t ] exp[ t] + exp[ ( + ) t]) i ] () Usig equatio () te ea tie to syste failure ca be derived i te followig for: dt (3) 4.3. Cold Duplicatio Metod I tis etod soe of te syste copoets are duplicated i parallel via a perfect switc. Followig Rade (989-) te reliability fuctio of eac copoet iproved by a cold via perfect switc ca be give by: R = ( + l(/ R)) R (4) c Tis iplies te reliability of te syste iproved by te cold duplicatio etod is give by: i i i R = [ ( R) ( R ) ] C t t exp[ ( = [ (exp[ ] + exp[ ] + ) t]) i ( + l(/(exp[ t ] + exp[ t] exp[ ( + ) t]))) i ] (5) c Usig equatio (5) te ea tie to syste failure C ca be derived i te followig for:

4 Sciece Joural of Applied Mateatics ad Statistics 5; 3(3): ( ) C = RC t dt (6) 5. Reliability Equivalece Factors A reliability equivalece factor is a factor by wic a caracteristic of copoets of a syste desig as to be ultiplied i order to reac equality of a caracteristic of tis desig ad a differet desig regarded as a stadard (Migdadi ad Al-Bata (4)). As etio above te reliability equivalece factor is defied as te factor by wic te failure rates of soe of te syste s copoets sould be uced i order to reac equality of te reliability of aoter better syste. I tis sectio te reliability equivalece factors of te iproved systes are derived. Te reliability equivalece D D factor(s) deoted by ρ ( α ) s ( α ) D = (C) for ot (cold) duplicatio is defied as tat factor(s) ρ s by wic te failure rates for te set of syste copoets sould be uced or equivaletly te reliability fuctio icreased so tat oe could obtai a desig of te syste wit a reliability fuctio of a desig obtaied fro te ial syste. For te ot duplicatio ρ ca be obtaied by solvig te set of te two equatios were R = exp[ t] + exp[ t] exp[ ( + ) t] R( ρ s) = exp[ ( ρ) t] + exp[ ( s) t] exp[ ( ρ + s) t] R = ( R) R R tl RtRt For exaple give tat =. =.3 ρ=.3 ad s=.5 oe ay be tepted to calculate te fuctios of reliability of te syste versus te tie are sow i Figure. R = α ad R = α (7) For te cold duplicatio ρ ca be obtaied by solvig te set of te two equatios R = α ad RC = α (8) 6. Illustrative Exaple I tis exaple we cosider tat te parallel-series syste is cosisted of two subsystes coected i parallel ( = ) ad cosider tat te first subsyste as tree copoets i series ( 3) ad te secod oe as two copoets i series ( ). Our ai is to iprove te reliability of tis parallel-series syste by iprovig te perforace of soe copoets istead of icreasig te uber of tese copoets. Te fuctios of reliability of te syste (for = =) are defied as follows: R ( R) ( R) ( R) 3 5 = + R t R t R t R t R t ( ) = ( ( )) ( ρ s) ( ) + ( ) ( ρ s) ( ) ( R) ( R ) R = ( R) R + R R ( R) ( R ) 3 R t R t R t R t R t C( ) = (( ( )) + c( ) ( ( )) c( )) ( R Rc R Rc) (( R) Rc 3 ( ρ s) Figure. Te R C(t) (solid lie) R (t) (dased lie) R (t) (dased dotted lie) R (t) (dotted lie). Also calculated te ea tie to syste failure ad te results are sow i Table. Table. Te ea tie to failure of te ial ad iproved systes. C Value Table represet te α-fractiles te ot duplicatio ρ ad cold duplicatio ρ correspodig to R (t) ad R C (t) for = = we R (t) for = = ca be derived as follows: R = ( R) + ( R ) ( R) ( R ) 3 3 ( ρ s) ( ρ s) ad i tese calculatio te level is cose to be differet values of α-fractiles. +. ( R) Rc)( R Rc R Rc)

5 64 M. A. El-Dacese: Reliability Equivalece Aalysis of a Parallel-Series Syste Subject to Degradatio Facility Table. Te α-fractiles ρ ρ. α ρ ( α) s( α ) ρ ( α) s( α ) Coclusio I tis paper we discussed te reliability equivalece of a parallel-series syste wit idepedet ad idetical copoets. It is assued tat te eac copoet of te syste avig two types of partial failure rates. Tree ways aely te uctio ot duplicatio ad cold duplicatio etods are used to iprove te syste reliability. A reliability equivalece factor was derived. A uerical exaple is used to illustrate ow te results obtaied ca be applied. Refereces [] Abdelfatta M. ad El-Faee Adel A.( 4) Reliability equivalece factors of a syste wit ixture of idepedet ad o-idetical lifeties wit delay tie Joural of te Egyptia Mateatical Society 96. [] Burr I. W. (94) Cuulative frequecy fuctios Te Aals of Mateatical Statistics 3() 5-. [3] El-Dacese M. A. ad Kalifa M. M.( 8) Reliability equivalece factors of a series-parallel systes i Weibull distributio. Iteratioal Joural of Reliability ad Applicatios 9() [4] Migdadi. S. ad Al-Bata M. S.( 4) Testig Reliability Equivalece Factors of a Series- Parallel Systes i Burr Type X Distributio Britis Joural of Mateatics & Coputer Sciece 4(8) [5] Mudolar G.S. Srivastava D.K. (993) Expoetiated Weibull faily for aalyzig battub failure rate data IEEE Trasactios o Reliability 4()99-3. [6] Mustafa A. ad El-Faee A. A.( ) Reliability equivalece factors of a geeral parallel syste wit ixture of lifeties. Applied Mateatical Scieces 6(76) [7] Rade L. Reliability Equivalece Studies i Statistical Quality Cotrol ad Reliability 989- Mateatical Statistics Calers Uiversity of Tecology. [8] Rade L. Reliability Systes of 3-state Copoets Studies i Statistical Quality Cotrol ad Reliability 99-3 Mateatical Statistics Calers Uiversity of Tecology. [9] Rade L. Perforace Measures for Reliability Systes wit a Cold Stadby wit a Rado Switc Studies i Statistical Quality Cotrol ad Reliability 99 Calers Uiversity of Tecology. [] Rade L. (993a) Reliability Equivalece Microelectroics & Reliability 33(3) [] Rade L. (993b) Reliability Survival Equivalece Microelectroics & Reliability 33(6) [] Sara A. M. Tadj L. Al-edairi A. ad Mustafa A. (8) Equivalece Factors of a Parallel-Series Syste Applied Scieces 9-3. [3] Sawy A. I. Abdelader Y.. ad Al-Oally M. I.( 3) Reliability equivalece factors i expoetiated expoetial distributio. Wulfeia Joural (3) [4] Xia Y. ad Zag G.( 7) Reliability equivalece factors i Gaa distributio. Applied Mateatics ad Coputatio 87()