merca Joural of Operao Reearch, 26, 6, 489-5 hp://www.crp.org/joural/ajor ISSN Ole: 26-8849 ISSN Pr: 26-883 Sadby Redudacy llocao for a Cohere Syem uder I Sgaure Po Proce Repreeao Vaderle da Coa ueo Deparme of Sac, São Paulo Uvery, São Paulo, razl How o ce h paper: da Coa ueo, V. (26) Sadby Redudacy llocao for a Cohere Syem uder I Sgaure Po Proce Repreeao. merca Joural of Operao Reearch, 6, 489-5. hp://dx.do.org/.4236/ajor.26.6645 Receved: Sepember 2, 26 cceped: November 2, 26 Publhed: November 24, 26 brac Wllg o work relably heory a geeral e up, uder ochacally depedece codo, we ed o characerze a o decally pare adby redudacy operao hrough compeaor raform uder a complee formao level, he phyc approach, ha, obervg compoe lfeme. We ed o opmze yem relably uder adby redudacy allocao of compoe, parcularly, uder mmal adby redudacy. o ge reul, we wll ue a cohere yem repreeao hrough a gaure po proce. Copyrgh 26 by auhor ad Scefc Reearch Publhg Ic. h work lceed uder he Creave Commo rbuo Ieraoal Lcee (CC Y 4.). hp://creavecommo.org/lcee/by/4./ Ope cce Keyword Relably, Margale Mehod Relably heory, Sgaure Po Proce, Sadby Redudacy, Cohere Syem. Iroduco I relably heory he ma applcao of redudacy o allocae a reduda pare a yem compoe poo order o opmze yem relably. For ace, ee []-[8], amog oher. here are wo commo ype of redudacy ued relably heory, amely acve redudacy, whch ochacally lead o coder maxmum of radom varable ad adby redudacy, whch ochacally lead o coder covoluo of radom varable. For a k-ou-of- yem, [] coder lkelhood rao orderg ad gve uffce codo o eure ha a ere yem he allocao of a adby pare hould go o he weake compoe whle a parallel yem hould go o he roge. Referece [2] coder he ame problem wh aoher crero of opmaly ad ge he DOI:.4236/ajor.26.6645 November 24, 26
V. da Coa ueo ame reul. I boh above paper, he compoe lfeme are ochacally depede ad he obervao are a yem level. Few paper aaed o he cae where he compoe are ochacally depede. Referece [7] aalyze redudace for a k-ou-of- yem of depede compoe. Referece [6] ude acve redudacy allocao for a k-ou-of- yem of depede compoe whou mulaeou falure. Referece [5] work a parcular form of adby redudacy, called mmal adby redudacy, whch gve he compoe a addoal lfeme a had ju before he falure. For he cae of depede compoe, [5] oberve he yem a compoe level ad ue he revere rule of order 2 (RR2) propery bewee compeaor procee o vegae he problem of where o allocae a pare a k-ou-of- yem. I h paper, we ed o aalyze a o decally pare adby redudacy allocao for a cohere yem of depede compoe whou mulaeou falure, a compoe level, uder a cohere yem gaure po proce repreeao ad prove ha opmal o perform adby redudacy o he weake compoe of a cohere yem order o opmze yem relably. I Seco 2 we characerze a o decally pare adby redudacy hrough compeaor raform for depede compoe. I Seco 3 we reume mahemacal deal of gaure po proce repreeao of a cohere yem ad Seco 4 we vegae he be adby redudacy allocao a depede compoe cohere yem order o opmze yem relably. 2. No Idecally Spare Sadby Operao hrough Compeaor raform We oberve ha each compoe adby redudacy ha wo phae, adby ad operao uder whch hey ca fal. Depedg o compoe falure characerc durg hee phae, adby redudacy clafed o he followg hree ype: ) Ho adby: Each compoe ha he ame falure rae regardle of wheher adby or operao. Sce he falure rae of oe compoe uque ad o affeced by he oher compoe, he ho adby redudacy co of ochacally depede compoe. 2) Warm adby: adby compoe ca fal, bu ha maller falure rae ha he prcpal compoe. Falure characerc of he compoe are affeced by he oher, ad warm adby duce depede compoe falure. 3) Cold adby: Compoe doe o fal whe hey are adby. he compoe have o-zero falure rae oly whe hey are operao. falure of oe prcpal compoe force a adby compoe o ar operao ad o have a o-zero falure rae. hu, falure characerc of oe compoe are affeced by he oher, ad he cold adby redudacy reul muually depede compoe falure. I wha follow, we coder o oberve wo lfeme ad S, whch are fe po- 49
V. da Coa ueo ve radom varable defed a complee probably pace ( Ω,,P) hrough he famly of ub -algebra ( ) of where σ { { }, { }, > > } S afe Dellachere codo of rgh couy ad compleee. We aume ha P( S ), ha, he lfeme ca be depede bu mulaeou falure are ruled ou. I our geeral e up ad order o mplfy he oao, h paper we aume ha relao uch a,, <,, bewee radom varable ad meaurable e, alway hold wh probably oe, whch mea ha he erm P-a.., uppreed. We recall ha a pove radom varable a -oppg me f, for every, { }. he -oppg me called predcable f a creag equece ( ), of -oppg me, < ex uch ha, a ad a -oppg me oally acceble f P ( S) for all predcable -oppg me S. For a mahemacal ba of ochac procee appled o relably heory ee he book of [9] ad []. Geerally, adby redudacy gve o he compoe a addoal lfeme. I our coex he adby operao of S by defed a he mproveme of S by ( S ) SR SR ad deoed by S, S S ( S ) where ( S) S he e { > S}, SR ad equal o he e { S}. We remark ha, he S lfeme erpreao max, S, whch ha a ull falure rae up o me dffere of a parallel yem lfeme, { } m {, S }. he lfeme SR S ha he falure rae of S before falure. ( { }) Furhermore, relao o σ { },, ad ug he Doob-Meyer decompoo, we coder he predcable compeaor procee ( ) S >, uch ha { } a zero mea uformly egral margale. lo, relao o ( σ { { }, } ), we coder he predcable compeaor procee ( ) >, uch ha { } S a zero mea uformly egral margale. he compeaor proce expreed erm of codoal probably, gve he avalable formao ad geeralze he clacal oo of hazard. Iuvely h correpod o produce wheher he falure goe o occur ow, o he ba of all obervao avalable up o, bu o cludg, he pree. he well kow equvalece bewee drbuo fuco ad compeaor procee follow from [] ad we have l F( ), l G( S). herefore P { > } e ad P{ S > } e. I he cae of depede lfeme, he urvval fuco of he mproved lfeme by SR S { ( ) > } ( ) P S S ( ) { } ( ) P S > P > e d e e e. herefore he -compeaor of SR { S } 49
V. da Coa ueo e d ed e e e d e d. e e e e SR S l e e I h faho ad preervg he depedece cae erpreao, we defe, for depede lfeme, he raformao of ad, -compeaor of { S SR },, wh ad e e e α d, α e e e β d, β We oberve ha < α < ad < β < mplyg a mproveme of he lfeme. a he um of he compeaor ad geg Followg h hkg, a a predcable compeaor uque we are gog o fd a probably meaure uder whch C he a -compeaor of o proceed we coder he compeaor raform e e e e e e e e e d d d.. SR { S } o prove he ma heorem of h eco we are gog o ue he followg Lemma: Lemma 2. Uder h eco aumpo, he followg proce e e e L { } e d e e a oegave -margale wh E L. Proof We coder he -oppg me defed by I uffce o prove ha he proce L { } V f : or. e e e a bouded -margale. Noe ha, for ay -oppg me e d e e e { } e V V d e e e V V we ca wre L V e e d V e e N where N { }. he procedure eay: V < we have O he e { } L V Oherwe, o he e { V } V e e d V d e e e e e e d e. e e 492
V. da Coa ueo he egrad e d e e e d e e V e L e d N V e e e e. e e e d e e e e e e a -predcable proce ad N a -margale, wh E L ad we ge he reul. Secodly, we coder he compeaor raform e e e d d d e e e e e e ad wh he ame argume ued o prove Lemma 2. we ca prove Lemma 2.2: Lemma 2.2 Uder h eco aumpo, he followg proce { } e S d e e L a -margale S e L e S S e e a oegave -margale wh E L. Now, we ca wre he ma heorem: heorem 2.3 Uder h eco aumpo, he followg proce { } { S } L L L e e α β S a oegave local -margale wh E[ L ]. Proof. Ug Lemma 2., Lemma 2.2 ad he Selje dffereao rule we have L L L dl L d L L L. by aumpo ad are couou wh P( S ), we have L L. herefore L L a oegave local -margale wh E L L ad he heorem proved. We are lookg for a probably meaure Q, uch ha, uder Q, C become he a -compeaor of { S SR } wh repec o h modfed probably meaure. Uder cera codo, poble o fd Q. Ideed aume ha he proce L uformly egrable. he follow from Graov heorem, ee [], a well kow reul o po proce margale, ha he dered meaure Q gve by he Rado dq Nkody dervave L. he radom varable L gve by dp L { } { S S } e e S e e e e e e S S ( S ) ( S ) ( e ) ( e ) ( S ) ( S ) e e 493
V. da Coa ueo where S m { S, }. Remark 2.4. I referece o he fr paragraph of h eco, he above eg we ca defy he meaure L wh warm adby whch cae he compoe adby ca fal before he compoe operao. I he cae of cold adby redudacy, doe o fal before S, we ca coder S < ad we have S S e e L S S. e e I he cae where ad S are decally drbued, we have ad he compeaor raform gve by e 2 2e 2e 2 e 2 d d whch ca be ued o defe a adby redudacy hrough compeaor raform whe he adby compoe ad he compoe operao are ochacally depede bu decally drbued a [6]. 3. Reul Sgaure Po Proce { } Due mporace we pree hee reul h eco whch appear [2]. I our geeral eup, we coder he vecor (, 2,, ) of compoe lfeme whch are fe ad pove radom varable defed a complee probably pace ( Ω,,P) wh P ( j) for all j,, j C {,, }, he dex e of compoe. herefore, he lfeme ca be depede bu mulaeou falure are ruled ou. he evoluo of compoe me defe a marked po proce gve hrough he falure me ad he correpodg mark. We deoe < ( 2) < < he ordered lfeme, 2,, a hey appear me ad by X j: j he correpodg mark. a coveo we e ( 2) ad X X 2 e where e a fcou mark o C he dex e of he compoe. he equece (, X ) defe a marked po proce. he mahemacal decrpo of our obervao, he complee formao level,, where gve by a famly of ub σ algebra of, deoed by ( ) σ, X j,, j C, < { > }, afe he Dellachere codo of rgh couy ad compleee. Iuvely, a each me he oberver kow f he eve {, X j} have eher occurred or o ad f had, he kow exacly he value ad he mark X. We coder, coveely, he lfeme, j defed by he falure eve X j wh her ub-drbuo fuco, uable adardzed {, } j ( ) F P X, j., 494
V. da Coa ueo he behavor of he po proce P ( I ), a he formao flow cououly me gve by he followg heorem: heorem 3. Le, 2,, be he compoe lfeme of a cohere yem wh lfeme. he, P ( ). k j {, } {, } k k j K J { ( k) } { ( k) } Proof. From he oal probably rule we have P ( I ) P( { } { k I ) E k k I { ( k) } { ( k) } ad ( k ) are -oppg me ad well kow ha he eve ( k ) k where I : I,, { } { { } k k } { } k { ( k) } we coclude ha I - meaurable. herefore P ( I ) E k I { } { } k k k { ( k) } { ( k) }. he above decompoo allow u o defe he gaure proce a compoe level. Defo 3.2 he vecor, k, j { ( k ), j } defed a he marked po gaure proce of he yem ΦΦ. Remark 3.3 We oe ha he above repreeao ca be e wo way. We would prefer he oe whch preerve he compoe dex becaue, by example, we could alk abou he relably mporace of compoe j for he yem relably a he k-h falure. lo, a P ( j) for all, j, he colleco {{, }, k, j k j } form a paro of Ω ad {, }. herefore k k j P ( > ) k j k j { ( k), j} ( k), j K, J k j {, } k j ( K), J k j ( k), j K, J> { } { } { } { } { } Remark 3.4 Ug Remark 3.3 we ca calculae he yem relably a P ( > ) E P ( > ) E k j { ( k), j} { ( K), J> } P > k j ({ ( k), j} { ( K), J }) If he compoe lfeme are couou, depede ad decally drbued we have, ( k) k ( ( k) ) P> P P >, 495
V. da Coa ueo recoverg he clacal reul a [3]. Remark 3.5 he marked N (, j) { X, j} a j -meaurable, egrable, ad ( ) ( ),. -ub-margale, ha, {, } (, ) E N j N j for all Follow ha, from Doob-Meyer decompoo, here ex a uque proce, (, j), ( ), ha M (, j) N (, j) (, j) -margale. We aume ha (, j) predcable N, j, uch -compeaor of ( ) j, called he a zero mea uformly egrable are aboluely couou -compeaor procee ad ha, j are oally acceble -oppg me. he -compeaor of { } where he yem lfeme e he followg heorem: heorem 3.6 Le, 2,,, be he compoe lfeme of a cohere yem wh lfeme. he, uder he above hypohe ad oao, he -ub-margale P ( ), ha he -compeaor Proof. We coder he proce { ( k), j} k j ( ) d k, j. w,. { } k, j ( k), j { } ( w ) I lef couou ad -predcable. herefore { } k, j d M k, j a a -margale. a fe um of { ( k), j} k j ( ) d M k, j -margale a { } ( ) k, j ( k), j k j k j { } -margale ( ) d N k, j d k, j. -margale. he compeaor uque we ge he reul. 4. Sadby Redudacy a Cohere Syem of Depede Compoe We are cocered wh he problem of where o allocae a pare compoe ug adby redudacy a cohere yem order o opmze yem relably mproveme. We le Φ ( ) be he lfeme of a cohere yem wh compoe lfeme ( ), P ( ), for all j,, j uder he hypo-,,, 2 j he ad oao of Seco 3. Furhermore, le Φ (,,, S,,, ) be he yem lfeme reulg from a adby redudacy operao of compoe hrough a pare wh lfeme S, o decally drbued a. I parcular we cou h yem falure hrough N { a coug proce wh } -compeaor,. o compare he yem lfeme reulg from redudacy operao we are gog o compare he compoe po procee compeaor hrough cumula- 496
V. da Coa ueo ve hazard order a [4] Defo 4. Coder wo po procee, lfeme vecor defed a complee probably pace N correpodg o he compoe Ω,, P ad N S, relao o he compoe lfeme vecor S pobly defed o a dffere probably pace, wh correpodg couou compeaor procee [, [ o ;,,, [, [ m o S S ;,,, m m m whch are, max, m ad m, ( m) for all < < <, < < < ad,, we ay ha S maller ha he cumulave hazard order, deoed by S. ch lo, we are gog o ue he followg reul from [5]. heorem 4.2 Coder wo po procee, N correpodg o he compoe lfeme vecor defed a complee probably pace ( Ω,, P ) ad N S, correpodg o he compoe lfeme vecor S pobly defed o a dffere probably pace. If S maller ha cumulave hazard order, S, ch he P almo urely, couou. If for all { } ψ EP ψ N EP N S for all decreag real ad rgh couou fuco wh lef had lm ψψ, whch mple N N. S 4.. Mmal Sadby Redudacy a Cohere Syem of Depede Compoe I h fr ubeco we reume he reul from [5] edg o pree a geeralzao of he ma heorem from a k-ou-of- yem o cohere yem. Iuvely, a mmal adby redudacy gve o he compoe a addoal lfeme a had ju before he falure. I a radom evrome where he compoe affeced by he behavor of oher compoe, [5] fd a compeaor approach for mmal adby redudacy coderg he Graov heorem argume where he compoe compeaor N raformed hrough proce of wh α ad j α d, α for j. he reul : uder he meaure Q defed by he Rado Nkod dervave dq dp, he compoe compeaor raform of N,. Oberve ha d l ( ), 497
V. da Coa ueo ad, he aboluely couou cae, where l P ( ) recover, he depedece cae, he clacal expreo >, [], we ca ( > ) P S F F l F. Recoverg our eg, le Φ (,,, S,,, ), he yem lfeme reulg from a mmal adby redudacy operao of he lfeme, of compoe. We cou h yem falure hrough N { a coug proce wh } -compeaor,. Φ be he lfeme of a cohere yem wh com- heorem 4.. Le be le poe lfeme ( ), 2,,, P ( j), for all j,, j. Uder a mmal adby redudacy operao, he hypohe ad oao of Seco 3, f, j j < j, he N N, < j. Proof From heorem 3.6 we have o compare yem compeaor expecao value o he form { } ( ) { } ( ) k, j k, { ( k), j} k j k k j ( ) k, j k, k, j. for where he oao (( k), j ) mea he rerco of erval ] ] ad j 2., k k. Clearly, uffce o prove for { } ( ) ( ) k, { k, j} k k j 2 ( ) { ( k),} { ( k),2} k k { ( k), j} k j 3 ( ) 2 ( ( )) ( ) ( ) k, l k, k, j ( ) ( ) k, k,2 l k,2 k, j ( ) ( ) l k, l k,2 k, k,2. he fal reul follow from heorem 4.2 4.2. Sadby Redudacy a Cohere Syem of Depede Compoe j, o he I wha follow we coder a uque pare wh lfeme S, a Seco 2, wh compeaor procee ( ), uch ha { } S a zero mea uformly egral margale, o be allocaed bewee he compoe, order o opmze yem relably: heorem 4.2. Le be le Φ ( ) be he lfeme of a cohere yem wh compoe lfeme ( ), P ( ), for all j,, j. U-, 2,, j der adby redudacy ad he hypohe ad oao of Seco 3, f, j j < j, he N N, < j. Proof. Follow, from Seco 2, ha he adby redudacy hrough compeaor raform of he compoe by a pare wh compeaor SR S e d ed l e e. e e 498
V. da Coa ueo Clearly, uffce o prove for ad j 2. (( k), ) { } ( ) k, { k, j} k k j 2 (( k),2) { } (( k) ) { } (( k) ) k, k,2 k k 2 { } (( k), j) k, j k j 3 k, l e e k, j,, 2 l e e ( ( )) ( ) ( ) ( ) l k, l k,2 k, k,2. he fal reul follow from heorem 4.2. by hypohe, ( j), < j we are coderg compoe weaker ha compoe j he ee ha he hazard proce for falure of compoe larger ha he hazard proce for falure of compoe j, alo mple ha ochacally le ha j. herefore, uder heorem 4.2. we uderad ha, a compoe level, opmal o perform acve redudacy allocao o he weake compoe of a cohere yem of couou depede compoe wh o mulaeou falure. We ca, alo coder wo pare wh lfeme S ad S 2, { S } wh -compeaor ad { S 2 } wh -compeaor ( 2), o be allocaed bewee he compoe, order o opmze yem relably. he followg corollary ca be ealy proved ug he ame argume of heorem 4.2.. Corollary 4.2. Le be le Φ ( ) be he lfeme of a cohere yem wh compoe lfeme ( ), P ( ), for all j,, j. Uder, 2,, j adby redudacy ad he hypohe ad oao of Seco 3, f j j j 2, he N N, < j, where, < ad Φ ( S ) 5. Cocluo,,,,,,. effce mehod o opmze he relably of a cohere yem o add redudacy compoe o he yem. herefore very gfca o kow abou he allocao whch be opmze yem relably. I he la decade, may reearcher devoed hemelve o h opc, geeral aalyzg k-ou-of- yem ad followg a aural ad clacal approach: coderg ha he compoe lfeme were ochacally depede ad o obervg he yem a level hrough σ { { } > } Few paper aemp o he cae where he compoe are ochacally depede whou mulaeou falure. [5] ad [6] coder ochacally depede compoe lfeme ad oberve he complee formao a compoe level σ { { },, } > 499
V. da Coa ueo geg reul for k-ou-of- yem. Wh rece reul gaure heory ad exeo o a gaure po proce, we geeralze reul from k-ou-of- o cohere yem, parcularly for mmal adby redudacy ad adby redudacy. I alo mpora o oe he characerzao of adby operao reul wh o decally pare. he dcuo abou h ew approach ad he clacal oe ca be e comparg reul of P ( > ) wh P ( > ). We coclude ha, a compoe level, opmal o perform acve redudacy allocao o he weake compoe of a cohere yem of couou depede compoe wh o mulaeou falure whe ug he hazard rae orderg bewee he compoe lfeme. ckowledgeme h work wa parally uppored by São Paulo Reearch Foudao (FPESP), gra 25/2249-. Referece [] olad, P.J., El Neweh, E. ad Procha, F. (992) Sochac Order for Redudacy llocao Sere ad Parallel Syem. dvaced ppled Probably, 4, 6-7. hp://do.org/.7/s86782426 [2] Sgh, H. ad Mra, N. (994) O Redudacy llocao Syem. Joural of ppled Probably, 3, 4-4. hp://do.org/.7/s29299526 [3] Kuo, W., Praad, V.R. lma, F. ad Hwag, L. (2) Opmal Relably Deg. Cambrdge Uvery Pre, Cambrdge [4] Praad, V.R., Kuo, W. ad Km, K.M. (999) Opmal llocao of Idecal, Mul Fucoal Spear a Sere Syem. IEEE raaco o Relably, 48, 8-26. hp://do.org/.9/24.784269 [5] ueo, V.C. (25) Mmal Sadby Redudacy llocao a k-ou-of-:f Syem of Depede Compoe. Europea Joural of Operaoal Reearch, 65, 786-793. hp://do.org/.6/j.ejor.23..4 [6] ueo, V.C. ad Carmo, I.M. (27) cve Redudacy llocao for a k-ou-of-:f Syem of Depede Compoe. Europea Joural of Operaoal Reearch, 76, 4-5. hp://do.org/.6/j.ejor.25.9.2 [7] elzuce, F. Marez-Puera, H. ad Ruz, J.M. (23) O llocao of Reduda Compoe for Syem wh Depede Compoe. Europea Joural of Operaoal Reearch, 23, 573-58. hp://do.org/.6/j.ejor.23.5.4 [8] Zhao, P., Cha, P.S. ad L, L. (25) Redudacy llocao a Compoe Level veru Syem Level. Europea Joural of Operaoal Reearch, 24, 42-4. hp://do.org/.6/j.ejor.24.8.4 [9] ve,. ad Jee, U. (999) Sochac Model Relably. Sprger Verlag, New York. hp://do.org/.7/b97596 [] remaud, P. (98) Po Procee ad Queue: Margale Dyamc. Sprger Verlag, New York. hp://do.org/.7/978--4684-9477-8 [] rja, E. ad Yah,. (988) Noe o Radom Iee ad Codoal Survval Fuco. Joural of ppled Probably, 25, 63-635. 5
V. da Coa ueo hp://do.org/.7/s2924346 [2] ueo, V.C. (26) Sgaure Po Procee. Lamber cademc Publhg, Saarbrucke, Germay. [3] Samaego, F.J. (985) O Cloure of he IFR Cla uder Formao of Cohere Syem. IEEE raaco o Relably, 34, 69-72. hp://do.org/.9/r.985.522935 [4] Shaked, M. ad Shahkumar, J.G. (994) Sochac Order ad her pplcao. cademc Pre, New York. [5] Kweck,. ad Szekl, R. (99) Compeaor Codo for Sochac Orderg of Po Procee. Joural of ppled Probably, 28, 75-756. hp://do.org/.7/s29242662 Subm or recommed ex maucrp o SCIRP ad we wll provde be ervce for you: ccepg pre-ubmo qure hrough Emal, Facebook, LkedI, wer, ec. wde eleco of joural (cluve of 9 ubjec, more ha 2 joural) Provdg 24-hour hgh-qualy ervce Uer-fredly ole ubmo yem Far ad wf peer-revew yem Effce ypeeg ad proofreadg procedure Dplay of he reul of dowload ad v, a well a he umber of ced arcle Maxmum demao of your reearch work Subm your maucrp a: hp://paperubmo.crp.org/ Or coac ajor@crp.org 5