Active redundancy allocation in systems. R. Romera; J. Valdés; R. Zequeira*

Transcription

1 Wrkig Paper -6 (3) Statistics ad Ecmetrics Series March Departamet de Estadística y Ecmetría Uiversidad Carls III de Madrid Calle Madrid, Getafe (Spai) Fax (34) Active redudacy allcati i systems R. Rmera J. Valdés R. Zequeira* Abstract A effective way f imprvig the reliability f a system is the allcati f active redudacy. Let X, X be idepedet lifetimes f the cmpets C ad C, respectively, which frm a series system. Let dete U = mi( max( X, X ), X ) ad U = mi( X,max( X, X )), where X is the lifetime f a redudacy (say S ) idepedet f X ad X. That is U ( U ) dete C as a active redudacy. the lifetime f a system btaied by allcatig S t ( ) Sigh ad Misra (994) csidered the criteri where C is preferred t C fr redudacy allcati if P ( U > U ) P( U > U ). I this paper we use the same criteri f Sigh ad Misra (994) ad we ivestigate the allcati f e active redudacy whe it differs depedig the cmpet with which it is t be allcated. We fid sufficiet cditis fr the ptimizati which deped the cmpets ad redudacies prbability distributis. We als cmpare the allcati f tw active redudacies (say S ad S ) i tw differet ways, that is, S with C ad S with C ad viceversa. Fr this case the hazard rate rder plays a imprtat rle. We btai results fr the allcati f mre tha tw active redudacies t a k- ut-f- systems. C Keywrds: active redudacy, stchastic rder, hazard rate rderig. *Rsari Rmera, Research supprted by DGES (Spai) Grat PB mrrmera@est-ec.uc3m.es. Pstal Address: Departamet de Estadística y Ecmetría, Uiversidad Carls III de Madrid, C/Madrid, Getafe, Madrid, Spai. Jsé Valdés. vcastr@matcm.uh.cu. Pstal Address: Facultad de Matemática y Cmputació, Uiversidad de La Habaa, Sa Lázar y L, CP 4, La Habaa, Cuba. Rómul Zequeira. rzequeir@ig.uc3m.es. Pstal Address: Departamet de Igeiería Mecáica, Uiversidad Carls III de Madrid, Avda. Uiversidad 3, Legaés 89, Madrid, Spai. Nte: This wrk has bee submitted t the IEEE fr pssible publicati. Cpyright may be trasferred withut tice, after which this versi may lger be accesible.

2 Active-Redudacy Allcati i Systems Rsari Rmera, Uiversidad Carls III de Madrid Jsé Valdés Uiversidad de La Habaa Rómul Zequeira 3 Uiversidad Carls III de Madrid Abstract A effective way f imprvig the reliability f a system is the allcati f active redudacy. Let X, X be idepedet lifetimes f the cmpets C ad C, respectively, which frm a series system. Let dete U = mi(max(x X)X ) ad U = mi(x max(x X)), where X is the lifetime f a redudacy (say S) idepedet fx ad X. That is U (U ) dete the lifetime f a system btaied by allcatig S t C (C ) as a active redudacy. Sigh ad Misra (994) csidered the criteri where C is preferred t C fr redudacy allcati if P (U >U ) P (U > U ). I this paper we use the same criteri f Sigh ad Misra (994) ad we ivestigate the allcati f e active redudacy whe it differs depedig the cmpet with which it is t be allcated. We fid sufficiet cditis fr the ptimizati which deped the cmpets ad redudacies prbability distributis. We als cmpare the allcati f tw active redudacies (say S ad S )itw differet ways, that is, S with C ad S with C ad viceversa. Fr this case the hazard rate rder plays a imprtat rle. We btai results fr the allcati f mre tha tw active redudacies t a k-ut-f- systems. Keywrds: active redudacy, stchastic rder, hazard rate rder Itrducti A effective way f imprvig the reliability f a system is the allcati f active redudacies. This prblem has bee studied by differet authrs usig differet criteria (see [], [] ad [3]). Sigh ad Misra [4] csidered the fllwig criteri. Let X, X be idepedet lifetimes f cmpets C ad C which frm a series system. Let U ad U dete the lifetime f tw systems such that U = ^ (_ (X X) X ) ad U = ^ (X _ (X X)), where X is the Research supprted by DGES (Spai) Grat PB96-. Pstal Address: Departamet de Estad stica y Ecmetr a, Uiversidad Carls III de Madrid, C/Madrid Getafe, Madrid, Spai. Pstal Address: Facultad de Matemática y Cmputació, Uiversidad de La Habaa, Sa Lázar y L, CP 4, La Habaa, Cuba. 3 Pstal Address: Departamet de Igeier a Mecáica, Uiversidad Carls III de Madrid, Avda. Uiversidad 3, Legaés 89, Madrid, Spai.

3 lifetime f a redudacy V, idepedet fx ad X ad the symbls _ ad ^ dete the max ad mi, respectively. If we are gig t cmpare the ttal lifetimes f these systems, the it is better t allcate V as a active redudacy with C istead f with C if the fllwig iequality hlds, P (U >U ) P (U >U ) : () I sme cases it is mre realistic t csider the active redudacy differs depedig the cmpet with which it culd be allcated. Rade [5] btais results i this regard fr sme series-parallel systems whe the cmpets are expetially distributed. Let Y ad Y be idepedet lifetimes f spares V ad V. Let w U = ^ (_ (X Y ) X ) ad U = ^ (X _ (X Y )). Recall a radm variable X is said t be stchastically greater tha a radm variable Y, writte X st Y, if P (X >t) P (Y >t) fr all real value t. As is pited ut by Sigh ad Misra, the cditi U st U may t always imply () sice U ad U are depedet radm variables. S it wuld be f iterest t fid ut sufficiet cditis fr the lifetimes f cmpets ad redudacies such that () hlds. Fr the case Y = Y i [4] it is shw that if X st X the () hlds ad this result is exteded t k-ut-f- systems. The structure f this paper is as fllws. I secti we establish sme results that will be used i the prfs f the ext tw sectis. I secti 3 we fid sufficiet cditis the distributi fuctis f the lifetimes f cmpets ad redudacies fr () t hld whe it is allcated e active redudacy that differs depedig the cmpet with which it culd be allcated, extedig i this way the results give i [4]. I secti 4 we cmpare the allcati f tw redudacies i tw differet ways, i.e, V with C ad V with C ad viceversa. We als give results the allcati f mre tha tw active redudacies. I sectis 3 ad 4 we csider i the aalysis k-ut-f- systems. Recall the fllwig defiitis we will use. Let X ad Y be egative radm variables ad F (t) ad G (t) dete the respective survival fuctis f X ad Y. X is said t be greater tha Y i the hazard rate rderig, writte X hr Y, if F (t) =G (t) is -decreasig fr all t where this qutiet is defied. If the desity fuctis f X ad Y,say f(t) ad g(t), exist the the rderig X hr Y ca be equivaletly expressed as f (t) F (t)» g (t) G (t) : Fllwig [3] we will say that X is greater tha Y i the prbability rder, writte X pr Y,ifP(X >Y) P (Y >X) hlds. Fr a geeral referece i stchastic rderig see [6].

4 Prelimiary results Let dete by z avalue f a real radm variable Z. Give a real umber t let defie z t as z t =ifz t ad z t =ifz<t: Csider w the values x ad y f tw radm variables X ad Y respectively. Observe that the iequality x>yis valid if ad ly if there exists a real umber t such that x t >y t : This equivalece allws us t reduce the treatmet f iequalities betwee real valued radm variables t the treatmet f Blea iequalities. I the fllwig i place f the variables z t we will simply write z. Fr a set f radm variables fz Z ::: Z g let (Z Z ::: Z ) [k] dete the kth largest rder statistics, s that (Z Z ::: Z ) [] ::: (Z Z ::: Z ) []. Let csider the radm variables X X ::: X, Y Y = 3::: ad dete U (k) =(_ (X Y ) X X 3 ::: X ) [k] U (k) =(X _ (X Y ) X 3 ::: X ) [k] () k =3 ::: Prpsiti. The fllwig equivalecies hld: a) U >U if ad ly if X < ^(X Y ). b) Fr >, U () >U () if ad ly if X < ^(Y X X 3 ::: X ). c) Fr > ad <k<, U (k) >U (k) if ad ly if e f the fllwig + excludig iequalities is satisfied: k k where _ X X j ::: X jk < ^ X Y X i ::: X ik _ X X Y X r ::: X rk < ^ Y X v ::: X vk fi ::: i k g f3 ::: g fj ::: j k g f3 ::: g fr ::: r k g f3 ::: g fv ::: v k g f3 ::: g ad fi ::: i k g fj ::: j k g =χ ad fv ::: v k g fr ::: r k g =χ: Prf. We will ly prve b) ad c), sice a) fllws i a similar fashi. Iequality U (k) >U (k) (3) hlds if ad ly if the fllwig system f Blea iequalities is satisfied _ (x y )+x + x 3 + ::: + x k (4) 3

5 x + _ (x y )+x 3 + ::: + x» k : (5) Suppse x =. The _ (x y ) = ad it is easy t see that i this case (4) ad (5) d t hld simultaeusly, csequetly, x =. Subtractig w (5) frm (4) we btai y + x +_ (x y ) ad this implies y =, sice x»_(x y ) ad, therefre _ (x y )=x : (6) Substitutig this last equality ad the values x = ad y = i (4), (5) we btai x + x 3 + ::: + x = k : (7) The the system (4), (5) is satisfied ly if x =,y = ad the system (6), (7) is satisfied. It is straightfrward t verify that, cversely, if these cditis hld, the system (4), (5) is satisfied. Fr k = the system (6), (7) has fr all value f y the uique sluti x = x 3 = ::: = x = ad the (3) is equivalet t X < ^(Y X X 3 ::: X ): Csider w the case k <. Observe that if x = the equati (7) has slutis ad, the ther had, frm (6) we btai y k =. I the case x = the equati (7) has slutis ad the value f y k may be arbitrary. The it ca be easily see that the equivalet iequalities fr (3) stated i the prpsiti are btaied. This cmpletes the prf. Let w dete V = ^ (_ (X Y ) _(X Y )) V = ^ (_(X Y ) _ (X Y )) k =3 :::. V (k) =(_ (X Y ) _(X Y )X 3 ::: X ) [k] V (k) =(_(X Y ) _ (X Y ) X 3 ::: X ) [k] (8) Prpsiti. The fllwig equivaleces hld: a) V > V if ad ly if e f the fllwig tw excludig iequalities is satisfied _(X Y ) < ^(X Y ) _(X Y ) < ^(X Y ): b) Fr >, V () > V () if ad ly if e f the fllwig tw excludig iequalities is satisfied _(X Y ) < ^(X Y X 3 ::: X ) _(X Y ) < ^(X Y X 3 ::: X ): 4

6 c) Fr > ad <k<, V (k) >V (k) if ad ly if e f the fllwig excludig iequalities is satisfied k where _ X Y X j ::: X jk < ^ X Y X i ::: X ik _ X Y X j ::: X jk < ^ X Y X i ::: X ik fi ::: i k g f3 ::: g fj ::: j k g f3 ::: g ad fi ::: i k g fj ::: j k g =χ: Prf. We csider the cases b) ad c), sice a) fllws i a similar maer. Iequality V (k) >V (k) (9) hlds if ad ly if the fllwig system f iequalities hlds _(x y )+_(x y )+x 3 + ::: + x k: () _(x y )+_(x y )+x 3 + ::: + x» k () If _(x y ) = _(x y )=frm () we have x 3 + ::: + x» k 3, but the () is t satisfied. Let _(x y ) = _(x y ) =. I this case frm () ad () we btai the ctradictry iequalities x 3 + ::: + x k ad x 3 + ::: + x» k. Suppse w that _(x y ) = ad _(x y ) =, r _(x y ) = ad _(x y )=. I these cases, frm () wehave x 3 +:::+x» k. Subtractig this iequality frm () we btai _(x y )+_(x y ) = ad als frm () we have x 3 + ::: + x k. The the system (), () is satisfied ly if x 3 + x 4 + ::: + x = k ad, _(x y )=ad^(x y )=,r_(x y )= ad ^(x y )=. Cversely, if these cditis hld, the system (), () is satisfied. This prves the prpsiti. 3 Allcati f a active redudacy Whe assumed t exist we will dete the prbability desities ad hazard rates f X ad X by f (x), f (x), (x) ad (x), respectively. We will dete the distributi fuctis f Y Y, X X ::: X by G (x)g (x), F (x)f (x) ::: F (x), respectively. Fr ay distributi fucti G we will dete G μ (x) = G (x). 5

7 Lemma 3. Let X, X, Y, Y ad Z be egative idepedet radm variables. Suppse i) X ad X have prbability desities ad (x) μ G (x) (x) μ G (x) x () r The ad ii) X» st X ad μ F (x) μ G (x) μ F (x) μ G (x) x : (3) a) P (X < ^ (Y X )) P (X < ^ (X Y )) b) P (X < ^ (Y X Z)) P (X < ^ (X Y Z)) : Prf. We ly prve part b), sice a) fllws i a similar fashi. Let H (x) dete the distributi fucti f Z. = P (X < ^ (Y X Z)) P (X < ^ (X Y Z)) = R R μf (x) G μ (x) H μ (x) df (x) μf (x) G μ (x) H μ (x) df (x) : But frm ii) fllws Z μf (x) μ G (x) μ H (x) df (x) Z μf (x) μ G (x) μ H (x) df (x) sice μ F (x) μ G (x) μ H (x) is a -icreasig fucti ad F (x) F (x) [6]. This prve b). Observe w that if X ad X have prbability desities = R The b) fllws frm i). R μf (x) F μ (x) G μ (x) (x) H μ (x) dx μf (x) F μ (x) G μ (x) (x) H μ (x) dx: Prpsiti 3. Let X, X,...,X, Y ad Y be idepedet lifetimes. Suppse i) X ad X have prbabilities desities ad (x) μ G (x) (x) μ G (x) x (4) r The ii) X» st X ad μ F (x) μ G (x) μ F (x) μ G (x) x : (5) U pr U ad U () pr U () : (6) 6

8 Prf. Accrdigly t Prpsiti., part b), U () pr U () hlds if ad ly if P (X < ^ (Y X X 3 ::: X )) P (X < ^ (X Y X 3 ::: X )) : The the result straightfrwardly fllws frm part b) f Lemma 3. takig Z = ^ (X 3 ::: X ). It is bvius that the case U pr U fllws i a similar way. Cditis i) ad ii) f Prpsiti 3: give us criteria fr the ptimal allcati i the sese f prbability rderig f a redudacy which differs depedig the cmpet with which it is allcated. Suppse, fr example, that Y st Y ad X» hr X r X» st X, the it is ptimal i prbability rder t allcate the strger redudacy t the weaker cmpet. If G = G, cditi i) reduces t hazard rate rder betwee lifetimes X ad X ad cditi ii) reduces t stchastic rder betwee lifetimes X ad X. This case is cvered i [4], where is als give a cuterexample that i ur case allws t shw that cditi ii) it is t ecessary fr prbability rderig (6) t hld. Nte that Fi μ (x) Gj μ (x), i j = is the survival fucti f a series system frmed by cmpets with lifetimes X i ad Y j. The cditi ii) ca be stated i the fllwig way. If the series system frmed by cmpet C with the redudacy V is stchastically greater tha the series system frmed by cmpet C with redudacy V, ad X» st X, the it is better t allcate redudacy V with cmpet C tha t allcate redudacy V with cmpet C. Remark 3. If X X Y ad Y are idepedet expetial radm variables with meas = = =μ ad =μ,respectively, the it is see that P (^f_(x Y ) X g > ^fx _ (X Y )g) = + + μ P (^f_ (X Y ) X g = ^fx _ (X Y )g) = μ + μ + μ μ ( + + μ )( + + μ ) : (7) Lemma 3. will be useful i extedig the result f Prpsiti 3. t k-utf- systems. Result b) i Lemma 3. is stated i Lemma. f [4]. Lemma 3. Let X X Y Y Z ad Z be egative idepedet radm variables ad Z 3 Z 4 egative radm variables idepedet f Y ad Y. Suppse that X» st X ad Y st Y. The a) P (_ (X Z ) < ^ (X Y Z )) P (_ (X Z ) < ^ (X Y Z )) : b) P (_ (Y Z 3 ) < ^ (Y Z 4 )) P (_ (Y Z 3 ) < ^ (Y Z 4 )) : (8) 7

9 Prf. Let H (x) ad H (x) dete the distributi fuctis f Z ad Z, respectively. =P (_ (X Z ) < ^ (X Y Z )) P (_ (X Z ) < ^ (X Y Z )) = R R μf (_ (x y)) G μ (_ (x y)) H μ (_ (x y)) df (x) dh (y) μf (_ (x y)) G μ (_ (x y)) H μ (_ (x y)) df (x) dh (y) : R R Sice μ G (x) μ G (x) ad μ F (x) μ F (x) the R R R R μf (_ (x y)) G μ (_ (x y)) H μ (_ (x y)) df (x) dh (y) μf (_ (x y)) G μ (_ (x y)) H μ (_ (x y)) df (x) dh (y) : Usig the same argumet as i the prf f Lemma 3. it ca be btaied that ad the a) fllws. Prpsiti 3. Let X ::: X Y ad Y be idepedet lifetimes. Suppse that X» st X ad Y st Y. The fr <k<, > U (k) pr U (k) : (9) Prf. It is sufficiet t use part c) f Prpsiti. with the same tati ad cditis stated there ad t take i Lemma 3.. Z = _ Xj ::: X jk Z = ^ Xi ::: X ik Z 3 = _ X X X r ::: X rk Z4 = ^ Xv ::: X vk 4 Allcati f mre tha e redudacy We csider w the allcati f tw active redudacies. I what fllws we will dete the prbability desities f Y ad Y by g (x) ad g (x), respectively. Lemma 4. Let X Y X Y Z ad Z be idepedet radm variables. Suppse X Y X ad Y have prbability desities. Let X» hr X ad Y» hr Y. The ad a) P (^ (X Y ) > _ (X Y ) OR ^ (X Y ) > _ (X Y )) P (^ (X Y ) > _ (X Y ) OR ^ (X Y ) > _ (X Y )) b) P (^ (X Y Z ) > _ (X Y Z ) OR ^ (X Y Z ) > _ (X Y Z )) P (^ (X Y Z ) > _ (X Y Z ) OR ^ (X Y Z ) > _ (X Y Z )) : Prf. We will ly prve b) sice a) fllws i a similar way. It is sufficiet t prve that =P (^ (X Y Z ) > _ (X Y Z )) + P (^ (X Y Z ) > _ (X Y Z )) P (^ (X Y Z ) > _ (X Y Z )) P (^ (X Y Z ) > _ (X Y Z )) 8

10 But R R R R R R R R R R = R + R μf (_ (x y z)) G μ (_ (x y z)) H μ (_(x y z)) dg (x) df (y) dh (z) μf (_ (x y z)) G μ (_ (x y z)) H μ (_(x y z)) dg (x) df (y) dh (z) μf (_ (x y z)) G μ (_ (x y z)) H μ (_(x y z)) dg (x) df (y) dh (z) μf (_ (x y z)) G μ (_ (x y z)) H μ (_(x y z)) dg (x) df (y) dh (z) where H (x) ad H (x) dete the distributi fucti f Z ad Z, respectively. A sufficiet cditi fr is which ca be rewritte as μf (_ (x y z)) μ G (_ (x y z)) g (x) f (y) + μ F (_ (x y z)) μ G (_ (x y z)) g (x) f (y) μ F (_ (x y z)) μ G (_ (x y z)) g (x) f (y) + μ F (_ (x y z)) μ G (_ (x y z))g (x) f (y) g (x) μ G (_ (x y z)) f (y) μ F (_ (x y z)) f (y) μ F (_ (x y z)) Λ g (x) μ G (_ (x y z)) f (y) μ F (_ (x y z)) f (y) μ F (_ (x y z)) Λ : () () Observe w that if a b, the sice frm X» hr X fllws Likewise frm Y» hr Y fllws f (b) μ F (a) f (b) μ F (a) f (b) f (b) μ F (b) μf (b) f (b) μ F (a) μf (a) : g (b) μ G (a) g (b) μ G (a) : The () hlds ad the prf is cmplete. Prpsiti 4. Let X ::: X, Y ad Y be idepedet lifetimes. Suppse X Y X ad Y have prbability desities. Let X» hr X ad Y» hr Y. The V pr V ad V (k) pr V (k) () fr <k». Prf. We ly csider the case <k<>, sice the remaiig cases ca be prved i a similar way. The it is sufficiet t use part c) f Prpsiti. with the same tati ad cditis stated there ad t take Z = _ Xj ::: X jk Z = ^ Xi ::: X ik i Lemma 4.. 9

11 Crllary 4. Let X X ::: X Y Y ::: Y be idepedet lifetimes fr which the prbability desities exist. Suppse X» hr X» hr :::» hr X ad Y» hr Y» hr :::» hr Y. The fr» k», (_ (X Y ) _ (X Y ) ::: _ (X Y )) [k] X Y ß() _ X Y ß() ::: _ X Y ß() pr _ fr ay permutati ß =(ß()ß() ::: ß()) f f ::: g: Prf. Fr tw arbitrary permutatis ß ad ffi f f ::: g let '(ffi ß) =P f _(X Y ffi() ) _(X Y ffi() ) ::: (X Y ffi() ) > _(X Y ß() ) _(X Y ß() ) ::: (X Y ß() ) [k] g: Detig by ß the permutati ( ::: ) we ca rewrite the result that is required t prve as'(ß ß) '(ß ß ). Give a permutati ß =(ß() ß() ::: ß()) let csider the permutati ß i =(ß()ß() ::: ß(i )ß(i +)ß(i)ß(i +) ::: ß()), i = :::. If Y ß(i+) hr Y ß(i), frm Prpsiti 4. we btai '(ß i ß) '(ß ß i ) fr all i = :::. But uder the suppsitis that are made this result implies '(ß ß) '(ß ß ) fr ay permutati ß. Csequetly, the crllary is prved. Crllary 4. meas that if we have ak-ut-f- system frmed by cmpets c c ::: c with respective failure rates (t) (t) ::: (t), ad r (r» ) redudacies c c ::: c r with respective failure rates μ (t) μ (t) ::: μ r (t), the if we are gig t allcate each redudacy t a cmpet as a active redudacy, the ptimal allcati regardig the prbability rderig is t allcate c r with c, c r with c, ad s. I [7] is csidered the decisi betwee t expad a k-ut-f- system ad imprvig the already existig system by meas f redudacy. I the fllwig prpsiti we csider this situati. Prpsiti 4. Let X ::: X ad Y Y ::: Y r (r» ) be lifetimes. The the fllwig iequality always hlds [k] [k] (X X ::: X Y Y ::: Y r ) [k] (_(X Y ) _(X Y ) ::: _(X r Y r )X r+ ::: X ) [k] : (3) Prf. Let suppse, the ctrary, that (3) des t hld. I this case usig the tati f secti it is hard t see that the system _(x y )+_(x y )+::: + _(x r y r )+x r+ + ::: + x k x + x + ::: + x + y + y + ::: + y r» k must be satisfied. Nevertheless this system has t sluti ad, csequetly, (3) always hlds. It is bvius that the result btaied i Prpsiti 4. implies that substitutig i (3) the symbl by the symbls st ad pr the iequality als hlds.

12 We w examie the fllwig prblem. Suppse we have ak-ut-f- system ad there are R spares t be allcated i parallel with its cmpets. Suppse further that lifetimes f all the cmpets ad the spares are idepedet, the lifetimes f the cmpets are idetically distributed ad the lifetimes f the redudacies are idetically distributed. We are iterested i determiig the ptimal allcati f the spares. Fr a series system this prblem has bee csidered i [8] ad [9] frm the pit f view f stchastic ad failure rate rderig, respectively. I thse wrks it has bee fud that i rder t ptimize the lifetime f the system, the allcati f spares must be balaced amg the cmpets as much as pssible. We will shw that it is als the ptimal allcati i the sese f the prbability rderig fr a k-ut-f- system. Let (x x ::: x ) be a egative -dimesial vectr ad let x [], x [],..., x [] dete the crdiates f the vectr arraged i decreasig rder. Fr a egative vectr (y y ::: y ), similarly defie y [i] i= :::. Recall that x majrizes y (x > m y)if P j P i= x j [i] i= y [i] hlds fr all j = ::: ad mrever P i= x [i] = P i= y [i] []. We will dete by r = (r r ::: r ) ad l = (l l ::: l ) tw pssible arragemets f spares t be placed i parallel with the cmpets f the system such that r i (respectively l i ) spares are allcated with the i th cmpet, where r i l i f ::: Rg. Of curse P i= r i = P i= l i = R. Let X,X,...,X dete the lifetimes f the cmpets ad Y,Y,...,Y R dete the lifetimes f the spares. Let csider the sets f lifetimes fy (i) Y(i) (i) ::: Y i = :::, which cstitute a partiti f the set fy Y ::: Y R g crrespdig t the arragemet r. That is, Y (i) j detes the lifetime f the j th active redudacy allcated t the i th cmpet, j = ::: r i, i = :::. Similarly, we csider the partiti fz (i) Z(i) ::: Z(i) g, i = :::, f the set li fy Y ::: Y R g crrespdig t the arragemet l. Let w(rl k) =P X _ > X _ k = ::: _ l i= Z() i _ r i= Y () i X r i= Y () i ::: X _ X l i= Z() i ::: X l i= Z() i _ r i= Y () i Prpsiti 4.3 Suppse X X ::: X Y Y ::: Y R are idepedet lifetimes such that X X ::: X are idetically distributed ad Y Y ::: Y R are idetically distributed. Let r> m l. The w(rl k)» w(l r k): Prf. Let csider a arragemet f spares r =(r r ::: r ) ad dete by r(i) =(r r ::: r i r i +r i+ r i+ ::: r ), i = :::, where r i+ >, the arragemet btaied frm r chagig the spare with lifetime [k] ri g, [k]

13 Y (i+) ri+ frm the (i +) th t the i th cmpet. By the ature f majrizati it is sufficiet t shw that if r i +» r i+ the w(rr(i) k)» w(r(i)r k). The the prblem is reduced t aalyze the allcati f the redudacy with lifetime Y (i+) ri+ whe the arragemet fspares is r, betwee the i th ad the (i +) th cmpets. That is, we must cmpare the lifetime X _ X i r j= Y () j _ ri j= Y (i) j X _ X i+ r j= Y () j ::: _ ri+ j= Y (i+) j ::: X r j= Y () j [k] which is btaied whe the arragemet f the spares is r versus the lifetime X r j= Y () j X r j= Y () i ::: _(X i _ ri j= Y (i) j Y (i+) ri+ )X i+ ri+ j= Y (i+) j which is btaied whe the arragemet f the spares is r(i). Sice r i <r i+ _ X i Y (i) Y(i) (i) ::: Y ri ::: X r j= Y () i [k]» st _ X i+ Y (i+) Y (i+) ::: Y (i+) ri+ the the result fllws frm Prpsitis 3. ad 3.. Prpsiti 4.3 gives us a criteria fr the allcati f active redudacies t k-ut-f- systems, regardig prbability rderig. If we have m = r + r + ::: + r = p redudacies t be allcated as active redudacies t a k-ut-f- system, that is t allcate r i redudacies t the i th cmpet, i = :::, the the better allcati regardig prbability rderig is t take r i = p, i = :::. Fr a arbitrary umber f redudacies the best chice is t allcate the redudacies the mst uifrmly as pssible amg the cmpets. 5 Cclusis I this paper we have discussed the allcati f e active redudacy which differs depedig the cmpet with it is t be allcated ad we have aalyzed the allcati f mre tha e redudacy. I the e redudacy case, stchastic rderig tgether with restrictis the distributi fuctis f the cmpets ad the redudacy, are fud as sufficiet cditis fr the prbability rderig t hld. I the case f mre tha e redudacy allcati, the sufficiet cditis are expressed thrugh the hazard rate rder. Fially we have btaied results the allcati f mre tha tw redudacies. Ackwledgemet. The authrs are very grateful t a referee fr his useful cmmets which greatly imprved a earlier versi f this paper. Rmul Zequeira is very grateful t Fudació Uiversidad Carls III (Spai) which partially supprted his research byaschlarship.

14 Refereces [] P. Blad, E. El-Neweihi ad F. Prscha, "Stchastic Order fr Redudacy allcatis i Series ad Parallel Systems," Adv. Appl. Prb, 4, pp. 6-7, 99. [] Jie Mi, "Blsterig Cmpets fr Maximizig System Lifetime," Naval Research Lgistics, 45, pp , 998. [3] Jie Mi, "Optimal Active Redudacy Allcati i k-ut-f- system," J. Appl. Prb, 36, pp97-933, 999. [4] H. Sigh ad N. Misra, "O redudacy allcati i systems," J. Appl. Prb, 3, pp 4-4, 994. [5] L. Rade, "Expected Time t Failure f Reliability Systems," Math. Scietist, 4, pp 4-37, 989. [6] M. Shaked ad J. Gerge Shathikumar Stchastic Orders ad their Applicatis. Academic Press, 994. [7] P. Blad, E. El-Neweihi ad F. Prscha, "Redudacy Imprtace ad Allcati f Spares i Cheret Systems," Jural f Statistical Plaig ad Iferece, 9, pp 55-66, 99. [8] M. Shaked ad J.G. Shathikumar, "Optimal Allcati f Resurces t Ndes f Series ad Parallel Systems," Advaces i Applied Prbability, 4, pp , 99. [9] H. Sigh ad R.S. Sigh, "Nte: Optimal Allcati f Resurces t Ndes f Series Systems with Respect t Failure-Rate Orderig," Naval Research Lgistic, 44, pp 47-5, 997. [] A. Marshall ad I. Olki (979). Iequalities: Thery f Majrizati ad its applicatis. New Yrk: Academic Press. 3