Reliability Equivalence of a Parallel System with Non-Identical Components

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1 Ieraoa Mahemaca Forum 3 8 o Reaby Equvaece of a Parae Syem wh No-Ideca ompoe M. Moaer ad mmar M. Sarha Deparme of Sac & O.R. oege of Scece Kg Saud Uvery P.O.ox 455 Ryadh 45 Saud raba aarha@yahoo.com brac. Th paper gve he reaby equvaece facor of a parae yem wh depede ad o-deca compoe. I aumed here ha he faure rae of he yem compoe are coa. We ued hree dffere mehod o mprove he yem gve. Two reaby characerc he mea me o faure ad he reaby fuco are ued o perform he yem mproveme. For h purpoe he reaby fuco ad he mea me o faure of he orga ad mproved yem are obaed. The reu gve h paper geeraze he reu gve he eraure by eg. urave umerca exampe preeed o compare he dffere reaby facor obaed. Mahemac Subec afcao: 6F5 Keyword: Expoea drbuo cohere yem mea me o faure.. Iroduco I cae of o repar equvae of dffere deg of he ame yem wh repec o a reaby characerc uch a mea me o faure or urvva fuco eeded. The cocep of reaby equvaece ha bee roduced by Rade []. Rade [-4] ad Sarha [6 7] have apped h cocep o varou yem. Rade [] ha codered hree dffere mehod o mprove he quay of a yem. e uggeed ha he reaby fuco of he yem ca be mproved by: mprovg he quay of oe or evera compoe by decreag her faure rae

2 694 M. Moaer ad. M. Sarha addg a ho compoe o he yem addg a cod reduda compoe o he yem. Sarha [7] ha codered more geera mehod o mprove he quay of a yem. e uggeed he foowg four mehod Improvg he quay of ome compoe by reducg her faure rae by a facor ρ ρ. o dupcao mehod. 3 od dupcao mehod. 4 od dupcao wh mperfec wch mehod. Rade [3 4] ad Sarha [6] ued he urvva fuco a a performace meaure of he yem reaby o compare dffere yem deg. Rade [3 4] ha obaed he reaby equvaece facor for a ge compoe ad for wo depede ad deca compoe ere ad parae yem. Sarha [6] ha obaed he reaby equvaece facor of depede ad o-deca ere yem. Sarha [6] ued he urvva fuco ad mea me o faure a characerc o compare dffere yem deg. e ha derved wo dffere ype of reaby equvaece facor of a bac ere/parae yem. Sarha [5] ha obaed he reaby equvaece facor of a parae yem wh depede ad decay compoe. e aumed ha he faure rae of he compoe o be coa. I h paper we derve he reaby equvaece facor of a parae yem wh depede ad o-decay compoe. We aume ha he ve of he yem compoe are expoeay drbued wh dffere parameer. The urvva fuco ad mea me o faure are ued a performace meaure o compare he deg of orga yem ad ha for he mproved deg. The reu preeed here geeraze he reu gve Sarha [5]. We eed he foowg defo. Defo. [Sarha [6]] reaby equvaece facor of a yem defed a ha facor by whch a characerc of compoe of a yem deg ha o be muped order o reach equay of a characerc of h deg ad a dffere deg. The orgazed a foow. Seco gve he decrpo of he orga yem uded here. The reaby fuco ad mea me o faure of he orga ad

3 Reaby equvaece of a parae yem 695 he mproved yem are preeed Seco 3. o a heorem ha eabhe a comparo amog he mea me o faure of yem ha mproved accordg o he mehod ued gve Seco 3. I Seco 4 we oba he reaby equvaece facor of he yem. The frace of he orga ad mproved yem are obaed Seco 5. Iurave umerca exampe gve Seco 6.. The orga yem The yem codered here co of depede bu o-deca compoe coeced parae. Scheme how he cofgurao of he yem. e T deoe he feme of he compoe. I aumed ha T expoeay drbued wh parameer. Tha he faure rae of he compoe ad reaby fuco { } > ;. R exp.a The yem ca be mproved accordg o oe of he foowg hree dffere mehod:. Reduco mehod: h mehod we reduce he faure rae of e compoe { } by he ame facor ay ρ ρ.. o dupcao mehod: aumed h mehod ha each compoe beog o he e compoe { } dupcaed by ho reduda adby compoe. 3. od dupcao mehod: aumed h mehod ha each compoe beog o he e compoe { } reduda adby compoe. dupcaed by cod The we w mae equvaece of he mproved yem obaed by reduco mehod o: he yem mproved by ho dupcao mehod; he yem mproved by cod dupcao mehod. The fr gve he ho reaby equvaece ad he ecod provde he cod reaby equvaece. We w ue boh he reaby fuco ad mea me o faure o mae he equvaece. For h

4 696 M. Moaer ad. M. Sarha purpoe we gve he reaby fuco ad he mea me o faure of he orga yem. The yem reaby fuco R R exp{ }.b The foowg reu eeded. Oe ca verfy ha where exp{ }. { } exp Ug. ad. we ca wre he reaby fuco R a he foowg form R.3 From.3 ad accordg o he we ow reao bewee he reaby fuco ad he mea me o faure oe ca ge he yem mea me o faure a The mproved yem I h eco we pree he mproved yem whch ca be derved accordg o he hree mehod meoed above. 3. Reduco mehod.

5 Reaby equvaece of a parae yem 697 I aumed h mehod ha he yem ca be mproved by reducg he faure rae of he e compoe { } by he facor ρ ρ. e R ρ be he reaby fuco of he yem mproved accordg o he reduco mehod. Oe ca derve R ρ a he foowg form ere { } \ exp{ ρ } exp{ } R ρ 3. he compemeary e of. ppyg. o exp{ ρ } ad exp{ } ad where ad Therefore exp{ } we ge ρ ρ 3. exp{ } 3.3 { } { } exp 3.4 : exp{ ρ } exp{ } ρ Thu R ρ ca be wre a he foowg form R ρ ρ ρ ρ. 3.5

6 698 M. Moaer ad. M. Sarha e ρ be he mea me o faure of he yem mproved by mprovg he e compoe accordg o he reduco mehod. From 3.5 we may wre d d ρ ρ d ρ u d : ρ ρ d : d : : ρ ρ Therefore ρ become : ρ ρ : : :. ρ o dupcao mehod. I aumed h mehod ha he yem ca be mproved by mprovg he e compoe { } accordg o ho dupcao mehod. The compoe

7 Reaby equvaece of a parae yem 699 ad o be mproved by he ho dupcao mehod f dupcaed wh aoher deca compoe ha coeced parae wh a deca compoe. e R be he reaby fuco of he yem mproved accordg o he ho dupcao mehod by mprovg he e compoe. Oe ca oba R a foow [ R ] [ R ] R 3.7 where R e e { } \. The fuco R ca be wre a where R R e { } exp{ } exp{ } ad exp 3.8. Smar o 3.3 we ca expre exp { } a he foowg form exp{ } 3.9 where are gve 3.4. From 3.3 ad 3.9 we have exp{ } exp{ } Thu R ca be wre a R exp{ } exp{ }. 3. e be he mea me o faure of he yem mproved by mprovg he e compoe accordg o he ho dupcao mehod. From 3. we have

8 7 M. Moaer ad. M. Sarha [ ] d e d e d e d e Sovg he egra above gve : : : : od dupcao mehod. I aumed h mehod ha he yem ca be mproved by mprovg he e compoe { } accordg o cod dupcao mehod. The compoe ad o be mproved by he cod dupcao mehod f dupcaed wh aoher deca compoe va a perfec wch.. e R be he reaby fuco of he yem mproved accordg o he cod dupcao mehod by mprovg he e compoe. The fuco R ca be obaed a foow R R R 3. where e R ee o ad a 983. The fuco R ca be wre a { } { } R exp exp 3.3 where. The expre ca be wre a he foowg form

9 Reaby equvaece of a parae yem 7 a a a 3.4 Repacg wh 3.3 we ge { } exp. From 3.3 ad 3.4 we have { } a a exp Thu R ca be wre a { } { } a R exp exp a. 3.5 e be he mea me o faure of he yem mproved by mprovg he e compoe accordg o he cod dupcao mehod. From 3.5 we have [ ] d e d e a d e d e a 3.6 u [ ] d e Γ Subug from egra o 3.6 gve [ ] Γ a

10 7 M. Moaer ad. M. Sarha a Γ [ ] Reaby equvaece facor I h eco we derve wo reaby equvaece facor: he urvva reaby equvaece facor SREF; he mea reaby equvaece facor MREF. The foowg are he defo of hee wo facor. D Defo 4. SREF [Sarha ] The ho cod SREF ay ρ D defed a ha facor by whch he faure rae of he e compoe houd be reduced order o mprove he yem reaby o be a ha reaby of he yem mproved by aumg ho cod dupcao of he e compoe. D Defo 4. MREF [Sarha ] The ho cod MREF ay ζ D defed a ha facor by whch he faure rae of he e compoe houd be reduced order o mprove he yem o be a ha of he yem mproved by aumg ho cod dupcao of he e compoe. D aed o he defo 4. he ho cod SREF ζ D ca be derved by ovg he foowg equao wh repec o D ρ ζ : D ρ D 4. Ug 4. whe D 3.5 ad 3. we ge he foowg yem of wo oear equao ρ ρ exp{ } exp{ }

11 Reaby equvaece of a parae yem 73 To ge he ho SREF we have o ove he yem of he o-ear equao ρ 4. ad 4.3. eem h yem ha o aayca ouo herefore we have o ue umerca echque o ge. For h purpoe we ued he Mahad pacage. ρ Smary ug 4. whe D 3.5 ad 3.5 we ge he foowg yem of wo o-ear equao ρ ρ exp{ } exp{ } a 4.4 a. 4.5 Thu o oba he cod SREF we have o ove he yem of he o-ear ρ equao 4.4 ad 4.5. eem h yem ha o aayca ouo herefore we have o ue umerca echque o ge. For h purpoe we ued he Mahad pacage. ρ Foowg he defo 4. ad ug equao ad 3.7 he ho D cod MREF ρ ζ D ca be derved by ovg he foowg equao: D ρ where ad. 4.6 gve by 3. ad 3.7 repecvey. Sovg D equao 4.6 we ge he ζ D a

12 74 M. Moaer ad. M. Sarha D D ζ The -frace We dcu he -frace of he orga ad mproved yem. The foowg defo gve he -frace. Defo 5. For a gve he -frace of a yem wh he reaby fuco R he ouo of he foowg equao wh repec o w.r.. R 5. where. Ug reao.3 ad 5. we ca ge he -frace of he orga yem by ovg he foowg o-ear equao w.r.. exp. 5. Smary we ca ge he -frace of he yem mproved accordg o he ho cod dupcao mehod we have o ove mar equao o he equao 5. whe he reaby fuco of he mproved yem repaced he fuco R. Tha he -frace of he yem mproved by mprovg he e compoe accordg o he DM he ouo of he foowg equao { } { } exp exp

13 Reaby equvaece of a parae yem ad he -frace of he yem mproved by mprovg he e compoe accordg o he DM he ouo of he foowg equao { } { } exp exp a a. 5.4 The above equao do o have aayc ouo. Therefore umerca echque mehod houd be ued o ge he frace. 6. Iurave exampe I h eco we aume a parae yem wh hree depede compoe. The feme of compoe 3 expoea wh parameer where. 5.5 ad 3.. The mea me o faure of h yem 9.9. We mproved h yem accordg o he mehod meoed he eco 3. Tabe how he mea me o faure of he yem mproved by he dupcao mehod wh dffere pobe e. Fgure ad gve ρ aga ρ for dffere e. I eem from fgure ha reducg he faure rae of a ge compoe wh maer faure rae gve a beer yem he ee of havg hgher mea me o faure. From fgure oe ca cocude ha reducg he faure rae of wo compoe for whch he um of her faure rae maer ha ha of ay oher wo compoe produce a modfed yem wh hgher mea me o faure. Reducg he faure rae of he compoe wh hghe faure rae gve ghy mproveme pecay whe ρ >..

14 76 M. Moaer ad. M. Sarha The urvva ad mea reaby equvaece facor of h yem are compued for dffere pobe e ad ed repecvey abe - 5. Tabe. The mea me o faure of he mproved yem. {} {} {3} {} {3} {3} {3} Fgure. The behavor of aga ρ whe. ρ

15 Reaby equvaece of a parae yem 77 Fgure. The behavor of aga ρ whe. ρ

16 78 M. Moaer ad. M. Sarha Tabe : The ho SREF ρ for dffere e ad. {} {} {} {3} {} {3} { 3} {3} {} {} {} {3} {} {3} { 3} {3} {3} {} {} {3} {} {3} { 3} {3}

17 Reaby equvaece of a parae yem 79 Tabe 3: The cod SREF ρ for dffere e ad. {} {} {} {3} {} {3} { 3} {3} { } {} {} {3} {} {3} { 3} {3} { 3} {} {} {3} {} {3} { 3} {3} Tabe 4: The -frace of he orga yem ad yem mproved by DM. Orga {} {3} { 3} {} { } { 3} {3} Tabe 5: The -frace of he orga yem ad yem mproved by DM. Orga {} {3} { 3} {} { } { 3} {3}

18 7 M. Moaer ad. M. Sarha From he above abe we ca cocude ha Improvg he e {} compoe accordg o he ho dupcao mehod creae:. The yem mea me o faure from 9.9 o 9.9 ee abe.. The.-frace of he orga yem from 4.95 o 4.93 ee abe 7. The ame reu ca be reached by dog he foowg. Reducg he faure rae of he ame compoe by he facor ρ.94 ee abe. { }{}.. Reducg he faure rae of he e {} compoe by he facor ρ ee abe.. { }{} 3 Improvg he e {} compoe accordg o he cod dupcao mehod creae: 3. The yem mea me o faure from 9.9 o 9.57 ee abe. 3. The.-frace of he orga yem from 4.95 o ee abe 8. 4 The ame reu ca be reached by dog he foowg 4. Reducg he faure rae of he ame compoe by he facor ρ ee abe 3. { }{}. 4. Reducg he faure rae of he e {} compoe by he facor ρ ee abe 3.. { }{}. 5 I he ame maer oe ca read he ree of he reu abe ad ocuo I h paper we derved wo reaby equvaece facor of a parae yem cog of depede ad o-deca compoe. We aumed ha he faure rae of he yem compoe are coa. We dcued hree dffere mehod o mprove he yem. We derved boh he reaby fuco ad he mea me o faure of each mproved yem. We uraed he probem o a umerca

19 Reaby equvaece of a parae yem 7 exampe o expa how oe ca uze he heoreca reu obaed. The probem uded h paper ca be exeded o may cae uch a: whe he compoe are o depede; he faure rae of he compoe are o coa wh he depedecy aumpo; o-coa faure rae ad o-depedecy aumpo. cowedgeme. Th paper uppored by he Reearch eer a oege of Scece Kg Saud Uvery uder he umber Sa/6/3. Referece []. Rade Reaby equvaece ude aca quay coro ad reaby. Mahemaca Sac. hamer Uvery of Techoogy 989. []. Rade Reaby yem of 3-ae compoe ude aca quay coro ad reaby. Mahemaca Sac. hamer Uvery of Techoogy 99. [3]. Rade Performace meaure for reaby yem wh a cod adby wh radom wch ude aca quay coro ad reaby. Mahemaca Sac. hamer Uvery of Techoogy 99. [4]. Rade Reaby equvaece. Mcroeecroc ad Reaby [5].M. Sarha Reaby equvaece facor of a parae yem Reaby Egeerg & Syem Safey [6].M. Sarha Reaby equvaece wh a bac ere /parae yem pped Mahemac ad ompuao

20 7 M. Moaer ad. M. Sarha [7].M. Sarha Reaby equvaece of depede ad o-deca compoe ere yem Reaby Egeerg & Syem Safey Receved: February 7 8