Availability Equivalence Factors of a General Repairable Parallel-Series System

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1 ppled Mathematcs Publshed Ole Jue 04 cres. valablty Equvalece Factors of a Geeral Reparable Parallel-eres ystem bdelfattah Mustafa mmar M. arha epartmet of Mathematcs Faculty of cece Masoura Uversty Masoura Egypt epartmet of Mathematcs & tatstcs alhouse Uversty Nova cota aada Emal: abdelfatah_mustafa@yahoo.com asarha@yahoo.com Receved February 04; revsed prl 04; accepted 8 prl 04 opyrght 04 by authors ad cetfc Research Publshg Ic. Ths work s lcesed uder the reatve ommos ttrbuto Iteratoal Lcese ( Y). bstract The avalablty equvalece factors of a geeral reparable parallel-seres system are dscussed ths paper cosderg the avalablty fucto of the system. The system compoets are assumed to be reparable ad depedet but ot detcal. The lfe ad repar tmes of the system compoets are expoetally dstrbuted wth dfferet parameters. Two types of avalablty equvalet factors of the system are derved. The results derved ths paper geeralze those gve the lterature. umercal example s troduced to llustrate how the dea of ths work ca be appled. Keywords Relablty Egeerg uplcato Methods Reparable ystems. Itroducto I relablty aalyss there are two ma methods to mprove o-reparable system desg. These methods are the reducto ad redudacy methods []. I the reducto method t s assumed that the system desg ca be mproved by reducg the falure rate(s) of a set of system compoets by a factor ρ 0< ρ < []-[4]. The redudacy method assumes that the system ca be mproved by creasg ts compoets [5]. There are more tha oe redudacy methods such as hot warm cold ad cold wth mperfect swtch redudacy amed respectvely as hot warm cold ad cold wth mperfect swtch duplcato methods [6]. The redudacy methods ca be appled o reparable systems as well. I addto to the reducto method the reparable system ca be mproved by creasg the repar rate of some of the system compoet(s) by a factorσ σ > [7] [8]. Usg the redudacy method may ot be a practcal soluto for a system whch the mmum sze ad ow to cte ths paper: Mustafa. ad arha.m. (04) valablty Equvalece Factors of a Geeral Reparable Parallel-eres ystem. ppled Mathematcs

2 . Mustafa. M. arha weght are overrdg cosderatos: for example satelltes or other space applcatos well-loggg equpmet ad pacemakers ad smlar bomedcal applcatos [9]. I such applcatos space or weght lmtatos may dcate a crease compoet performace rather tha redudacy. The more emphass must be placed o better desg maufacturg qualty cotrol ad o cotrollg the operatg evromet. Therefore the cocept of relablty/avalablty equvalece takes place. I such cocept the desg of the system that s mproved accordg to reducto or creasg method should be equvalet to the desg of the system mproved accordg to oe of redudacy methods. That s ths cocept oe may say that the performace of a system ca be mproved through a alteratve desg [0]. I ths case dfferet system desgs should be comparable based o a performace characterstc such as ) the relablty fucto or mea tme to falure the case of o repars or ) the avalablty the case of reparable systems. The cocept of comparg dfferet desgs s appled the lterature order to: ) mprove the relablty of a o-reparable system []; ) determe a represetatve servce provder ad create equvalet elemets []; 3) derve the relablty equvalece factors of some o-reparable systems [] ad the refereces there; ad 4) derve the avalablty equvalece factors of a reparable system [7] [8]. The relablty equvalece cocept appled o varous o-reparable systems [] [] [4] [3]-[7]. I ths work the relablty fucto ad mea tme to falure are used as characterstc measures to compare dfferet system desgs to derve relablty/mea tme equvalece factors. Reparable system dcates a system that ca be repared to operate ormally the evet of ay falure such as automobles arplaes computer etwork maufacturg system sewage systems power plat or fre preveto system. valablty comprses relablty ad recovery part of urelablty after repar dcatg the probablty that reparable systems maches or compoets mata the fucto at a specfc momet [8]. It s geerally expressed as the operable tme over total tme. Parallel-seres system dcates sub-systems whch several compoets are coected seres ad the parallel or sub-systems that several compoets are coected parallel ad the seres [9]. The relablty/avalablty of a parallel-seres system has draw cotuous atteto both problem characterstcs ad soluto methodologes [] [9] ad [0]. Recetly [7] [8] dscussed the avalablty equvalece factors of a reparable seres-parallel system wth depedet ad detcal (o-detcal) compoets. Our goal ths paper s to derve the avalablty equvalece factors of a reparable parallel-seres system wth depedet ad o-detcal compoets. The avalablty fucto of the system wll be used as a performace measure to compare dfferet system desgs of the orgal system ad other mproved systems order to derve these factors. The structure of ths paper s orgazed as follows. ecto troduces the llustrato of the parallel-seres system ad the system avalablty. ecto 3 presets the avalabltes of the systems mproved accordg to fve dfferet methods that ca be appled to mprove the performace of the orgal system. I ecto 4 two types of avalablty equvalece factors of the system are dscussed. umercal example s troduced ecto 5 to llustrate how the dea of ths work ca be appled. Fally ecto 6 s devoted to the coclusos whch hadle the ma results derved throughout ths work.. Geeral Reparable Parallel-eres ystem The system cosdered here cossts of subsystems coected parallel ad wth subsystem cosstg of m depedet reparable ad odetcal compoets coected seres for. e refer to such system as a geeral reparable parallel-seres system. Fgure shows the dagram of that system. Let T ad Y be the lfetme ad repar tme respectvely of compoet j subsystem j m. It s assumed that the lfe ad repar tmes of compoet j subsystem j m follow expoetal dstrbutos wth falure rate λ ad repar rate μ. Let N be the total umber of the system compoets that s N m. pecal ases: Ths system geeralzes the followg cases: ) Reparable parallel-seres system wth detcal compoets whe λ λ µ µ j m ad. ) Reparable parallel system wth o-detcal compoets whe m ad. 3) Reparable seres system wth o-detcal compoets whe ad j m. Let be the avalablty of the compoet j subsystem ad be the avalablty of the subsystem 74

3 . Mustafa. M. arha Fgure. Parallel-seres system structure. j m. Oe ca easly derve ad respectvely as see [8] µ λ where η µ + λ + η µ ad m m. j j + Therefore the system avalablty deoted ca be derved as 3. fferet esgs of Improved ystem () m s. j + ( ) (3) The system ca be mproved accordg to oe of the followg three dfferet methods: ) Reducto method. I ths method t s assumed that the compoet ca be mproved by reducg ts falure rate by a factor ρ 0< ρ <. ) Icreasg method. It s assumed ths method that the compoet ca be mproved by creasg ts repar rate by a factor σ σ >. 3) tadby redudacy method: a) ot duplcato method: ths method we assume that the compoet s duplcated by a detcal hot stadby compoet. b) arm duplcato method: ths method we assume that the compoet s duplcated by a detcal warm stadby compoet. c) old duplcato method: ths method we assume that the compoet s duplcated by a detcal cold stadby compoet. I the followg sectos we derve the avalablty of the system mproved accordg to the methods metoed above. 3.. The Reducto Method It s assumed the reducto method that the system ca be mproved by reducg the falure rates of a set R compoets by a factor ρ 0 < ρ <. e assume that R where R s a set of the subsystem com poets. lso we assume that r 0 r m ad r r ( ) r N. Let ρ be the avalablty of the compoet j subsystem mproved by reducg ts falure rate λ by the factor ρ. Oe ca easly derve ρ λ where η. + ρη µ Therefore the avalablty of subsystem mproved by reducg the falure rates of a set R compoets by the factor ρ deoted R ρ ca be wrtte as () (4) 75

4 . Mustafa. M. arha R ρ ρ j R j R j R ρ j R + + (5) M s the set of all subsystem compoets { } where M \ R M m. Fally the avalablty of the system mproved by reducg the falure rates of a set R compoets by the same factor ρ deoted R ρ ca be derved as 3.. The Icreasg Method. R ρ j R + ρη j R + η (6) It s assumed the creasg method that the system ca be mproved by creasg the repar rates of a set compoets by a factor σ σ >. e assume that where s a set of the subsystem compoets. lso we assume that s 0 s m ad s s s N. Let σ be the avalablty of compoet j subsystem after creasg ts repar rate µ by the factor σ σ > ad σ be the avalablty of subsystem whch s mproved by creasg the repar rates of a set compoets by the same factor σ ; ad σ be the avalablty of the system mproved by creasg the repar rates of a set compoets by the same factor σ. Oe ca derve these avalabltes the followg forms where M \ for. σµ + λ σ + η σ σµ σ σ σ σ j j j σ j + + σ 3.3. The ot uplcato Method (8) σ j σ j + + (9) It s assumed the hot duplcato method that the system ca be mproved by coectg every elemet a set compoets wth a detcal compoet parallel. e assume that where s a set of the subsystem compoets. lso we assume that h 0 h m ad h h h N. Let be the avalablty of the subsystem whch s mproved by mprovg a set M compoets ; ad be the avalablty of the system mproved by mprovg a set compoets accordg to the hot duplcato method. Oe ca derve where M \ for The arm uplcato Method η j j j j + + ( ) (0) η j j + + () e say that a compoet j subsystem s warm duplcated f t s coected parallel wth a o-detcal compoet havg a falure rate ν parallel va a perfect swtch. I the warm duplcato method t s as- (7) 76

5 . Mustafa. M. arha sumed that the system ca be mproved whe every compoet a set compoets s warm duplcated. e assume that where s a set of the subsystem compoets. lso we assume that w 0 w m ad w w. w N Let be the avalablty of the compoet j the subsystem whe t s mproved accordg to the warm duplcato method. Usg Markov process ca be obtaed as follows see [] + η + ξ () + η + ξ + η + ηξ where ξ ν µ for j m ad. Let be the avalablty of the subsystem mproved by mprovg subsystem compoets accordg to the warm duplcato method. Therefore oe ca derve + η + ξ j η j η ξ η ηξ (3) Fally let be the avalablty of the system mproved by mprovg a set compoets accordg to the warm duplcato methods. Usg Equato (3) we get 3.5. The old uplcato Method + η + ξ. j η j η ξ η ηξ (4) It s assumed the cold duplcato method that each compoet of set compoets s coected parallel wth a detcal compoet va a perfect swtch. e assume that where s a set of the subsys- tem compoets. lso we assume that c 0 c m ad c c. c N Let s the avalablty of the compoet j subsystem whe t s mproved accordg to the cold duplcato method; be the avalablty of subsystem whch s mproved accordg to cold duplcato method; ad be the avalablty of the system mproved by mprovg set compoets accordg to the cold duplcato method. Usg Markov process theory s see [] µ + λµ + η. µ + λµ + λ + η + η Usg Equato (5) ad the ature of the seres subsystem oe ca derve + η. j η j η η (5) (6) Fally usg Equato (6) ad the ature of the parallel coecto of the subsystems we get 4. valablty Equvalece Factors + η. j η j η η (7) I ths secto we derve the avalablty equvalece factors of a reparable parallel-seres system wth depedet o-detcal ad reparable compoets. Two types of avalablty equvalece factors wll be dscussed. These two types are referred as avalablty equvalet reducg factor ad avalablty equvalet creasg factor. Followg the defto of relablty equvalece factors troduced []. 77

6 . Mustafa. M. arha 4.. valablty Equvalece Reducg Factor valablty equvalece reducg factor short ERF referred as ρ ρ R for hot warm ad cold respectvely s defed as the factor ρ by whch the falure rate of a set R compoets should be reduced order to get equalty of the avalablty of aother better desg whch ca be obtaed from the orgal system by assumg hot warm ad cold duplcatos of a set compoets. That s ρ ρ R for s the soluto of the followg equatos ρ R ρ. (8) I what follows we gve the o-lear equatos eeded to be solved to get the three possble ERF s. ) ot avalablty equvalece reducg factor (ERF): ubsttutg Equatos (6) ad () to Equato (8) ρ ρ R s the soluto of the followg o-lear equato ρ η. + ρ (9) j R j R j j ) arm avalablty equvalece reducg factor (ERF): ubsttutg Equatos (6) ad (4) to Equato (8) ρ ρ R s the soluto of the followg o-lear equato ρ + η + ξ. + ρ + + η + ξ + η + ηξ + (0) j R j R j j 3) old avalablty equvalece reducg factor (ERF): ubsttutg Equatos (6) ad (7) to Equato (8) ρ ρ R satsfes the followg o-lear equato + η. j R + ρη j R + η j + η + j + η () Equatos (9)-() have o closed solutos therefore a umercal techque method s eeded to get ther solutos. 4.. valablty Equvalece Icreasg Factor valablty equvalece creasg factor short EIF referred as σ σ for hot warm ad cold respectvely s defed as the factor σ by whch the falure rate of a set compoets should be reduced order to get equalty of the avalablty of aother better desg whch ca be obtaed from the orgal system by assumg hot warm ad cold duplcatos of a set compoets. That s σ σ for s the soluto of the followg equatos σ. σ. () I what follows we gve the o-lear equatos eeded to be solved to get the three possble EIF s. ) ot avalablty equvalece creasg factor (EIF): ubsttutg Equatos (9) ad () to Equato () σ σ s the soluto of the followg o-lear equato σ η. σ (3) j j j j ) arm avalablty equvalece creasg factor (EIF): ubsttutg Equatos (9) ad (4) to Equato () σ σ s the soluto of the followg equato σ σ + η + ξ. j σ η j η + + j + η j η + ξ + η + ηξ + (4) 78

7 . Mustafa. M. arha 3) old avalablty equvalece creasg factor (EIF): ubsttutg Equatos (9) ad (7) to Equato () σ σ s the soluto of the followg equato σ σ + η. j σ η j η + + j + η j η + η + (5) The above Equatos (3)-(5) have o closed-form solutos σ so a umercal techque method to get the value of σ. 5. Numercal Results To expla how oe ca utlze the prevously obtaed theoretcal results we troduce a umercal example. I such example we calculate the two dfferet avalablty equvalece factors of a geeral reparable parallelseres wth subsystems. Each subsystem cossts of m o-detcal compoets uder the followg assumptos: ) The parallel-seres system has two subsystems ; ) The subsystems have the compoets m m the N m + m 3; 3) The values of the system parameters λ µ ad v ( j m ) are preseted Table. The objectve s to mprove the reparable parallel-seres system by mprovg the performace of some compoets stead of creasg the umber of these compoets. e gve the values of avalablty of the orgal system ad of the desg obtaed usg the duplcato methods for the example cosdered ths secto. Table shows the avalablty of the orgal ad mproved system obtaed from the orgal system by applyg hot warm ad cold duplcatos usg all possble set compoets where ad ϕ s the empty set. From the results show Table oe ca easly see that: ) < < < for all possble set compoets whe λ < ν ; ) < < < for all possble set compoets whe λ > ν ; Table. et values of the system parameters. j λ < ν λ > ν λ ν µ λ ν µ Table. The avalablty of the mproved system. λ < ν λ > ν { } φ φ { } φ { } { } { } { } { } φ { } { } { }

8 . Mustafa. M. arha 3) Improvg the oly oe compoet subsystem accordg to the duplcato method provdes a better desg tha that ca be acheved by mprovg oe compoet from the subsystem accordg to the same method; 4) uplcatg two compoets oe from each subsystem produces a better desg tha that ca be obtaed by duplcatg the two compoets subsystem accordg to the same method; ad 5) old duplcatg all compoets the system provdes the best desg the sese of havg the hghest avalablty. e used Mathematca Program ystem to calculate all possble avalablty equvalece factors of the studed system. Table 3 ad Table 4 gve the hot warm ad cold ( ) avalablty equvalece reducg factors ρ ρ R ad the hot warm ad cold avalablty equvalece creasg factors σ σ respectvely for all possble sets R ad. From the results preseted Table 3 Table 4 we ca mmedately coclude that: Table 3. The ERF ( ρ ) for dfferet R whe λ < ν. R R R R { } R R R R 3 { } φ φ φ { } { } ρ R { } { } { } φ { } { } { } { } φ φ { } N N N N N φ { } N N N N N { } { } { } { } R φ { } { } { } { } R R R ρ R φ φ { } N N N N N φ { } N N N N N { } { } { } { } R φ { } { } { } ρ R { } φ φ { } N N N N N φ { } N N N N N { } { } { } { } φ { } { } { }

9 . Mustafa. M. arha Table 4. The EIF ( σ ) for dfferet whe λ < ν. 3 { } φ φ φ { } { } { } φ φ φ { } { } { } φ { } σ φ φ { } N N N N N φ { } N N N N N { } { } { } { } φ { } N N N N N N N 3 { } { } { } σ φ φ { } N N N N N φ { } N N N N N { } { } { } { } φ { } N N N N N N N 3 { } { } { } σ φ φ { } N N N N N φ { } N N N N N { } { } { } { } φ { } N N N N N N N 3 { } { } ad φ creases the system avalablty from to see Table. The mproved system wth ca be acheved by performg oe of the followg: a) Reducg the falure rate(s) of (see Table 3): ) the oly compoet subsystem R R R where R { } ad R φ by the ERF ρ R ) the oly compoet subsystem ad the frst compoet subsystem R { } R { } by the ERF ρ R ) the oly compoet subsystem ad the secod compoet subsystem R { } R { } by the ERF ρ R v) the two compoets subsystem R φ ad R { } by the ERF ρ R v) all the three compoets R { } R { } by the ERF ρ R b) Icreasg the repar rate(s) of (see Table 4): ) the oly compoet subsystem where ad φ by the EIF σ ) the oly compoet subsystem ad frst compo- ) ot duplcato of the oly oe compoet subsystem { } { } 7

10 . Mustafa. M. arha et subsystem { } { } by the EIF ad secod compoet subsystem { } { } poets { } { } by the EIF σ ) arm duplcato of the oly compoet subsystem { } σ ) the oly compoet subsystem by the EIF σ 7.60 v) all the three com- ad φ creases the system avalablty from to see Table. The mproved system wth ca be acheved by performg oe of the followg: a) Reducg the falure rate(s) of (see Table 3): ) the oly compoet subsystem R R R where R { } ad R φ by the ERF ρ R ) the oly compoet subsystem ad the frst compoet of subsystem R { } R { } by the ERF ρ R ) the oly compoet subsystem ad the secod compoet of subsystem R { } R { } by the ERF ρ R 0.48 v) the two compoets subsystem R φ R { } by the ERF ρ R v) all three com- poets R { } R { } by the ERF ρ R b) Icreasg the repar rate(s) of (see Table 4): ) the oly compoet subsystem where { } ad φ by the EIF σ.0000 ) the oly compoet subsystem ad frst compoet of subsystem { } { } by the EIF σ 7.3 ) the oly compoet subsystem ad secod compoet of subsystem { } { } by the EIF σ v) all three compoets { } { } by the EIF σ ) old duplcato of the oly compoet subsystem { } ad φ creases the system aval- ablty from to see Table. The mproved system wth ca be acheved by performg oe of the followg: a) Reducg the falure rate(s) of (see Table 3): ) the oly compoet subsystem R R R where R { } ad R φ by the ERF ρ R ) the oly compoet subsystem ad frst compoet of subsystem R { } R { } by the ERF ρ R ) the oly compoet subsystem ad secod compoet of subsystem R { } R { } by the ERF ρ R v) the two compoets subsystem R φ R { } by the ERF ρ R v) all three compoets R { } R { } by the ERF ρ R b) Icreasg the repar rate(s) of (see Table 4): ) the oly compoet subsystem where { } ad φ by the EIF σ ) the oly compoet subsystem ad frst compoet of subsystem { } { } by the EIF σ ) the oly compoet subsystem ad secod compoet of subsystem { } { } by the EIF σ v) all three compoets { } { } by the EIF σ ) I the same maer we ca llustrate the rest of results show Table 3 ad Table 4. 5) The otato N meas that there s o possble equvalece betwee the two mproved systems that ca be acheved by reducg (creasg) the falure (repar) rates of the set R( ) of system compoets ad that ca be acheved by duplcatg elemets of set of system compoets. 6. oclusos Ths paper dscusses the avalablty equvalece factors of a geeral reparable parallel-seres system wth depedet but o-detcal compoets. The system studed here geeralzes several well-kow systems such as a reparable parallel-seres system wth depedet ad detcal compoets; reparable seres ad reparable parallel systems wth depedet ad o-detcal or detcal compoets. e derved two types of the avalablty equvalece factors of the system. e preseted a umercal example to llustrate how the theoretcal results derved the paper ca be appled. Ideed there are several possble extesos of ths work. s a example the case of a geeral reparable parallel-seres system wth o-costat falure rates ca be studed. Refereces [] arha.m. (00) Relablty Equvalece wth a asc eres/parallel ystem. ppled Mathematcs ad omputato [] arha.m. (009) Relablty Equvalece Factors of a Geeral eres-parallel ystem. Relablty Egeerg ad ystem afety [3] arha.m. l-ruzaza.. lwasel I.. ad El-Gohary.I. (004) Relablty Equvalece of a eres-parallel 7

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