Mean residual life of coherent systems consisting of multiple types of dependent components

Transcription

1 Mea residual life f cheret systems csistig f multiple types f depedet cmpets Serka Eryilmaz, Frak P.A. Cle y ad Tahai Cle-Maturi z February 20, 208 Abstract Mea residual life is a useful dyamic characteristic t study reliability f a system. It has bee widely csidered i the literature t ly fr sigle uit systems but als fr cheret systems. This paper is ccered with the study f mea residual life fr a cheret system that csists f multiple types f depedet cmpets. I particular, the survival sigature based geeralized mixture represetati is btaied fr the survival fucti f a cheret system ad it is used t evaluate the mea residual life fucti. Furthermre, tw mea residual life fuctis uder di eret cditial evets cmpets lifetimes are als de ed ad studied. Key wrds. Depedece; Mea residual life; Miimal survival sigature; Reliability; Survival sigature Itrducti The study f mea residual life f a cheret system has attracted a great deal f atteti i reliability thery. Csider a system with cmpets which has tw pssible states; (x ; ; x ) if the system is fuctiig ad (x ; ; x ) 0 if the system has failed, where x i if the ith cmpet is fuctiig ad x i 0 if the ith cmpet has failed. The fucti (x ; ; x ) is called the structure fucti. A system with structure fucti (x ; ; x ) is cheret if it is decreasig i each argumet, ad each cmpet i is relevat t the perfrmace f the system, i.e. (x ; ; x i ; 0; x i+ ; ; x ) 0 ad (x ; ; x i ; ; x i+ ; ; x ) fr sme states x ; ; x i ; x i+ ; ; x f ther cmpets ; 2; ; i ; i + ; ;. Besides the classical de iti f the mea residual life, di eret mea residual life fuctis Departmet f Idustrial Egieerig, Atilim Uiversity, 06836, Icek, Akara, Turkey, serka.eryilmaz@atilim.edu.tr y Departmet f Mathematical Scieces, Durham Uiversity, Durham, Uited Kigdm z Durham Uiversity Busiess Schl, Durham Uiversity, Durham, Uited Kigdm

2 have bee de ed ad studied i the literature fr a cheret system. Fr a cheret system with lifetime T ad cmpets lifetimes T ; ; T ; the usual mea residual life is de ed by E(T t j T ): Navarr ad Heradez (2008) studied the mea residual life fucti f a system whse reliability fucti ca be writte as a geeralized mixture. The mea residual life f a cheret system has als bee studied uder di eret cditial evets, e.g. whe all cmpets are fuctiig at time t. The latter mea residual life ca be de ed as E(T t j T : ); where T r: detes the rth smallest lifetime amg T ; ; T (Asadi ad Bayramglu (2006)). See als Navarr (206) ad Navarr ad Durate (207) fr sme recet results the mea residual fuctis E(T t j T ) ad E(T t j T : ). Asadi ad Glifrushai (2008) studied the mea residual life f a system csistig f cmpets havig the prperty that if it is kw that at mst r cmpets (r < ) have failed, the system is still peratig with prbability, i.e. E(T t j T r: ). The ccept f sigature (see, e.g. Samaieg (2007)) has bee used t evaluate the latter mea residual life fuctis. Fr a cheret system that csists f exchageable cmpets, the survival fucti ca be writte as P ft g i P ft :i g ; () i where the vectr f ce ciets ( ; ; ) satisfyig P i i is called miimal sigature ad ly depeds the structure f the system (Navarr et al. (2007)). The equati () is a geeralized mixture represetati fr the survival fucti f a cheret system that csists f a sigle type f cmpets. With a sigle type, we mea that all cmpets withi the system have a cmm failure time distributi. The mixture represetati give by () is useful t study limitig behavir f the mea residual life fucti E(T t j T ) (Navarr ad Eryilmaz (2007), Navarr ad Heradez (2008)). Ather well-kw represetati fr the survival fucti f a cheret system that csists f sigle type f cmpets is give by P ft g (l)p fc(t) lg ; l0 where C(t) is the umber f wrkig cmpets at time t; ad (l) is the survival sigature de ed by (l) r (l) ; l where r (l) detes the umber f path sets f size l (Cle ad Cle-Maturi (202)). A path set is a set f cmpets whse simultaeus fuctiig esures the fuctiig f the system. 2

3 I this paper, we study mea residual life fuctis E(T t j T ), E(T t j T ; T r: ) ad E(T t j T ; T r () : ; ; T r (K) K : K ) fr a cheret system detes the which is cmpsed f K 2 types f depedet cmpets, where T (i) r: i rth smallest amg the failure times f i cmpets f type i; i ; ; K. Uder this geeral setup, the radm failure times f cmpets f the same type are exchageable ad depedet ad the radm failure times f cmpets f di eret types are depedet. The ccept f survival sigature has bee fud t be very useful t study reliability prperties f such systems (see, e.g. Cle ad Cle- Maturi (202), Samaieg ad Navarr (206)). By utilizig the ccept f the survival sigature, we btai a geeralized mixture represetati fr the survival fucti f a cheret system that csists f K types f depedet cmpets. The btaied mixture represetati geeralizes the represetati give by () ad is used t study the limitig behavir f E(T t j T ). The survival sigature based represetatis fr E(T t j T ; T r: ) ad E(T t j T ; T r () : > t; ; T r (K) K : K ) are als btaied. Sadegh (20) exteded the results f Asadi ad Glifrushai (2008) whe the lifetimes f the system cmpets are idepedet radm variables but t ecessarily idetically distributed ad whe the jit distributi f the cmpet lifetimes is exchageable. Zhag ad Meeker (203) btaied mixture represetatis f the reliability fuctis f the residual life ad iactivity time f a cheret system with idepedet ad idetically distributed cmpets, give that befre time t, exactly r (r < ) cmpets have failed ad at time t 2, the system is either still wrkig r has failed. Sme recet discussis the mea residual life f systems ca be fud i Navarr ad Gmis (206), Bayramglu ad Ozkut (206), Bayramglu Kavlak (207). The paper is rgaized as fllws. I Secti 2, we btai a geeralized mixture represetati fr the survival fucti f a cheret system csistig f multiple types f depedet cmpet. Secti 3 is devted t study di eret mea residual life fuctis. 2 Miimal survival sigature Csider a cheret system with K 2 types f cmpets. Let i dete the umber f cmpets f type i, i ; 2; ; K; where P K i i. It is assumed that the radm failure times f cmpets f the same type are exchageable ad depedet, ad that the radm failure times f cmpets f di eret type are depedet. Withut lss f geerality, the assumpti cmpets lifetimes ca be writte as (T ; ; T ) d ( ; ; T () ); fr ay permutati such that (i) 2 f; ; g fr all i 2 f; ; g ; (i) 2 f + ; ; + 2 g fr all i 2 f + ; ; + 2 g ; ad s, where d 3 detes

4 equality i distributi. It shuld be pited ut that this is a quite strg assumpti. If C i (t) detes the umber f cmpets f type i wrkig at time t, the the survival fucti f the system ca be writte as P ft g l 0 K 0 (l ; ; )P fc (t) l ; ; C K (t) g ; (2) where (l ; ; ) represets the survival sigature ad is de ed by (l ; ; ) r ;; K (l ; ; ) ; (3) K (Cle ad Cle-Maturi (202, 205)). I (2), r ;; K (l ; ; ) detes the umber f path sets f the system icludig exactly l cmpets f type,..., exactly cmpets f type K. The cmputati f survival sigature is a challegig prblem. Reed (207) prpsed a e ciet algrithm t cmpute survival sigature f a system. Patelli et al. (207) preseted a simulati methd fr system reliability usig the survival sigature. Let T (i) j dete the failure time f the jth cmpet f type i; i ; 2; ; K. The frm Therem f Eryilmaz (207), the jit distributi f C (t); ; C K (t) ca be writte as P fc (t) l ; ; C K (t) g l l K S ;; K (t; l ; ; ); (4) where S ;; K (t; l ; ; ) l i 0 P K i K 0 ( ) i ++i K l K i ; ; l +i ; ; T (K) ; ; T (K) +i K i K :(5) We rst btai the fllwig geeralized mixture represetati fr the survival fucti f a cheret system which will be useful i the sequel. Therem The survival fucti f a cheret system csistig f i cmpets f type i, i ; 2; ; K ca be writte as P ft g m 0 K m K 0 (m ; ; m K )P mi( :m ; ; T (K) :m K ) ; (6) 4

5 where T (i) :m i mi(t (i) ; ; T (i) m i ); i ; ; K; ad (m ; ; m K ) m l 0 m K ( ) m l ++m K l K 0 l m l ad fr cveiece P T (i) :0 : K m K l K (l ; ; ); (7) Prf Let l +i ; +i K P ; ; l +i ; ; T (K) ; ; T (K) +i K ; the frm (2), (4) ad (5) we have l i 0 P ft g m 0 K i K 0 l 0 K 0 (l ; ; ) ( ) i ++i K l K i l 0 m K 0 l 0 K 0 (l ; ; ) l l i K K K K ( ) m l++mk lk l K m l m K m l m K K m m K ( ) m l ++m K l K m 0 K m K 0 m 0 l m l m 0 (m ; ; m K )P K m K 0 0 K m K K m K 0 l +i ;i K +i K l (l ; ; ) m ;m K (m ; ; m K ) m ;m K m ;m K K ; ; T m () ; ; T (K) ; ; T m (K) K (m ; ; m K )P :m ; ; T (K) :m K ; 5

6 where (m ; ; m K ) m l 0 m K ( ) m l ++m K l K 0 l m l K m K l K (l ; ; ): Clearly, the ce ciets (m ; ; m K ) i (6) satisfy m 0 K m K 0 (m ; ; m K ) ; but they may take egative values. Therefre equati (5) is a geeralized mixture f series systems. Similar t systems with a sigle type f cmpets, we will call the (m ; ; m K ) miimal survival sigature f the system that csists f multiple types f cmpets. Crllary If the system csists f idepedet cmpets such that the cmm failure time distributi f type i cmpets is F i (t); i ; 2; ; K, the P ft g m 0 K m K 0 (m ; ; m K ) F m (t) F m K(t): (8) The geeralized distrted distributi crrespdig t distributi fuctis G ; G 2 ; ; G is represeted as F Q (t) Q(G (t); ; G (t)); where the icreasig ctiuus fucti Q : [0; ]! [0; ] is called multivariate distrti fucti ad satis es Q(0; ; 0) 0 ad Q(; ; ). Fr the survival fucti we have F Q (t) Q( G (t); ; G (t)); where Q(u ; ; u ) Q( u ; ; u ) is called multivariate dual distrti fucti. The fucti Q is als a multivariate distrti fucti ad it satis es the same prperties as Q (Navarr et al. (206)). Prpsiti Let ^C be a survival cpula crrespdig t ; ; T () ; ; T (K) ; ; T (K) K ; i.e. P () ; ; T () () ; ; T (K) (K) ; ; T (K) K (K) K ^C( F (t () ); ; F (t () ); ; F K (t (K) ); ; F K (t (K) K )): 6 K

7 The the lifetime T S f a cheret system that csists f K types f depedet cmpets has a geeralized distrted distributi whse survival fucti is P ft g Q( F (t); ; F K (t)); (9) where the multivariate distrti fucti is give by Q(u ; ; u K ) m 0 K m K 0 (m ; ; m K ) ^C(u ; ; u {z } ; ; ; ; ; u {z } K ; ; u K ; ; ; ): (0) {z } {z } m m m K K m K Prf The prf is immediate frm (6) sice P mi( :m ; ; T (K) :m K ) P ; ; T m () ; ; T (K) ; ; T m (K) K ^C( F (t); ; F (t); ; ; ; ; F {z } {z } K (t); ; F K (t); ; ; ): {z } {z } m m m K m K Althugh Navarr et al. (206) have represeted the system s lifetime distributi as a geeralized distrted distributi whe cmpets lifetimes are depedet, their represetati was implicit. I particular, they ted that P ft g H( F (t); ; F (t)); where H Q is a fucti which depeds the miimal path sets f the cheret system structure ad the survival cpula ^C (see als Navarr et al. (207), Miziula ad Navarr (207)). Our represetati give by (9)-(0) is explicit as a fucti f the survival sigature which fully characterizes the system structure ad ca be cmputed thrugh equati (3). As a direct csequece f Prpsiti, fr a cheret system that csists f idepedet cmpets such that the cmm failure time distributi f type i cmpets is F i (t); i ; 2; ; K; we have Q(u ; ; u K ) m 0 K m K 0 K (m ; ; m K )u m u m K K : I the special case, if the system csists f sigle type f idepedet cmpets, the Q(u) (m)u m m0 which has bee called dmiati fucti by Navarr ad Spizzichi (205). It shuld be ted that (0) ca be used jitly with the results i Navarr et al. (206) t cmpare di eret systems. 7

8 Example Csider the system i Figure which has bee csidered i Feg et al. (206). The system has six cmpets with K 2 types with 3 ad 2 3: Type ad type 2 cmpets are represeted respectively by blak ad black bxes. Table displays the miimal survival sigature f the system. Nte that the miimal survival sigature is cmputed usig the relati (7) ad the survival sigature f the system preseted i Table f Feg et al. (206). Figure. System with tw types f cmpets Usig the etries i Table ad the equati (5), the survival fucti f the system ca be represeted as P ft g P : ; T (2) :2 + 2P +P :3 3P :2 ; T (2) :2 + 2P :3 ; T (2) :2 2P :3 ; T (2) :3 Usig survival cpula, the survival fucti ca be represeted as where the distrti fucti is give by P ft g Q( F (t); F 2 (t)); :2 ; T (2) :3 : () Q(u ; u 2 ) ^C(u ; u 2 ; u 2 ) + 2 ^C(u ; u ; u 2 ; u 2 ) 2 ^C(u ; u ; u 2 ; u 2 ; u 2 ) + ^C(u ; u ; u ) 3 ^C(u ; u ; u ; u 2 ; u 2 ) + 2 ^C(u ; u ; u ; u 2 ; u 2 ; u 2 ): If the cmpets are idepedet, the the distrti fucti becmes Q(u ; u 2 ) u u u 2 u 2 2 2u 2 u u 3 3u 3 u u 3 u 3 2: 8

9 m m 2 (m ; m 2 ) m m 2 (m ; m 2 ) Table. Miimal survival sigature f the system i Figure 3 Mea residual life fuctis Usig Therem, the MRL f the system that csists f multiple types f cmpets ca be cmputed frm m(t) E(T t j T ) K R (m ; ; m K ) P m 0 m K 0 m 0 K m K 0 0 (m ; ; m K )P :m + x; ; T (K) :m K + x dx :m ; ; T (K) :m K (2) : The fllwig result f Navarr ad Heradez (2008) is useful t examie the limitig behavir f the MRL fucti. Therem 2 (Navarr ad Heradez (2008)) Let S be a survival fucti such that S(t)! i S i (t); i fr all t 0, where S (t); ; S (t) are survival fuctis, ad! ; ;! are real umbers such that P i! i : Let m i (t) be MRL fucti crrespdig t S i (t); i ; ; ; i.e. m i (t) (S i (t)) R S t i (u)du: If lim if m (t) t! m i (t) > ; lim sup m (t) t! m i (t) < ; fr i 2; 3; ;, the the MRL fucti m f S satis es lim t! m(t) m (t) : 9

10 Because Therem presets a geeralized mixture represetati fr a cheret system that csists f multiple types f depedet cmpets, Therem 2 eables us t ivestigate the limitig behavir f the MRL fucti fr such systems. Applicati f Therem 2 eeds a multivariate distributi r survival fucti fr mdelig lifetimes f cmpets. Suppse that the jit survival fucti f ; ; T () ; ; T (K) ; ; T (K) K is give by () ; ; T () () ; ; T (K) (K) ; ; T (K) K (K) K P " + i t () i + + K K i t (K) i # ; (3) fr t (j) i 0, i ; ; j ; j ; ; K; i > 0; > 0. It shuld be ted that the survival cpula crrespdig t (3) is h ^C(u ; u 2 ; ; u ) u + u u ( )i ; ad P () ; ; T () () ; ; T (K) (K) ; ; T (K) K (K) K ^C( F (t); ; F (t); ; F {z } K (t); ; F K (t)); {z } K with F i (t) ( + i t) ; i ; 2; ; K. I the fllwig, we preset the limitig behavir f (2) fr the mdel (3). Prpsiti 2 Fr the multivariate Paret mdel give by (3), let C f(i ; ; i K ) : i + + i K < j + + j K ad (i ; ; i K ) > 0 fr all j 0; ; ; ; ; j K 0; ; ; K g : If v + + v K K i + + i K K ; (4) fr all (i ; ; i K ) 2 C; the lim t! Prf The MRL crrespdig t mi( :i ; ; T (K) :i K ) is P :i ; ; T (K) :i K [ + i t + + K i K t] t + m(t) h i : (5) t + v ++v K K Z 0 Z 0 i + + K i K P :i + x; ; T (K) :i K + x dx [ + i (t + x) + + K i K (t + x)] ; 0 dx

11 fr > : Fr (v ; ; v K ) 2 C satisfyig (4), the cditis i Therem 2 hld true fr the MRL f mi( :v ; ; T (K) :v K ). Thus the prf is cmplete. Example 2 Fr the system i Figure, let P () ; 2 () 2 ; 3 () 3 ; T (2) (2) ; T (2) 2 (2) 2 ; T (2) 3 (2) 3 " # t () i + 2 t (2) i : i i Frm Table, it is easy t see that C f(; 2); (3; 0)g. Thus frm Prpsiti 2, if 2 ; the m(t) lim h i ; t! t ad if 2 ; the lim t! m(t) h i : t + 3 It shuld be ted here that the limitig result i (5) depeds determiati f the ce ciets v ; ; v K de ed by (4). As it is clear frm Example 2, these ce ciets heavily deped the relati betwee the parameters ad 2 : Csider a cheret system that has the prperty that if at mst r cmpets (r < ) have failed, the system is still peratig with prbability. The, the cditial expected value E(T t j T r: ) represets the mea residual lifetime fucti f a cheret system give that at least r + cmpets f the system are wrkig at time t (Asadi ad Bayramglu (2006), Sadegh (20)). Fr a cheret system that csists f multiple types f cmpets, de e the fllwig mea residual life. E(T t j T ; T r: ) Z 0 P ft + x j T ; T r: g dx: (6) Fr a cheret system csistig f K 2 types f cmpets, it is easy t see that P ft ; T r: g (l ; ; )P fc (t) l ; ; C K (t) g l ++ r+ l ++ r+ (l ; ; ) l K S ;; K (t; l ; ; ); (7) where S ;; K (t; l ; ; ) is give by (5). I the fllwig Therem, we preset the cditial survival fucti f T give ft ; T r: g.

12 Therem 3 Fr a cheret system csistig f i cmpets f type i, i ; 2; ; K, P ft > s j T ; T r: g P ft S ; T r: g l 0 K (l ; ; )N(j ; l ; ; ; j K ; ; K ) 0 (j ;;j K )2U P j f s t; j l f s 2 (t; s] ; l f s > s; ; K j K f T (K) s t; j K f T (K) s 2 (t; s] ; f T (K) s > s ; (8) where U f(j ; ; j K ) : j + + j K r + ; l j ; ; j K K g, ad K N(j ; l ; ; ; j K ; ; K ) : j ; j l ; l K j K ; j K ; Prf By cditiig the umber f wrkig cmpets f each type at time t ad s; P ft > s; T r: g K (l ; ; ) l 0 0 (j ;;j K )2U P fc (s) l ; ; C K (s) ; C (t) j ; ; C K (t) j K g : (9) Thus the prf fllws tig that P fc (s) l ; ; C K (s) ; C (t) j ; ; C K (t) j K g K j ; j l ; l K j K ; j K ; P j f s t; j l f s 2 (t; s] ; l f s > s; ; K j K f T (K) s t; j K f T (K) s 2 (t; s] ; f T (K) s > s ; fr s ad j l ; ; j K. I equati (9), it is quite iterestig t bserve that the survival sigature depeds ly the umber f wrkig cmpets f each type at time s (later time pit) ad idepedet f j ; ; j K which dete the umber f wrkig cmpets f each type at a previus time pit t. Crllary 2 If the system csists f idepedet cmpets such that the cm- 2

13 m failure time distributi f type i cmpets is F i (t); i ; 2; ; K, the P ft > s j T ; T r: g P ft ; T r: g KY i i l 0 i j i ; j i l i ; l i K (l ; ; ) 0 (j ;;j K )2U F i j i i (t)(f i (s) F i (t)) j i l i ( F i (s)) l i : (20) Crllary 3 Let r i Therem 3. The the cditial survival fucti f the system uder the cditi that all cmpets are wrkig at time t ca be represeted as P ft > s j T : g P ; ; T () ; ; T (K) ; ; T (K) K l 0 K 0 (l ; ; ) P fc (s) l ; ; C K (s) ; C (t) ; ; C K (t) K g ; (2) fr s : I the fllwig, we btai a expressi fr the jit prbability ivlved i (8) whe K 2, i.e. the system csists f tw types f cmpets. The fllwig result is useful sice it ly ivlves jit survival prbabilities. Prpsiti 3 Fr a system that csists f tw types f cmpets, P fc (s) l ; C 2 (s) l 2 ; C (t) ; C 2 (t) 2 g 2 [p (s; t; l ; l 2 ) p 2 (s; t; l ; l 2 ) l l 2 p 3 (s; t; l ; l 2 ) + p 4 (s; t; l ; l 2 )] ; (22) where fr s ; p (s; t; l ; l 2 ) P > s; ; l > s; () l + ; ; T T (2) > s; ; T (2) l 2 > s; T (2) (2) l 2 + ; ; T 2 ; (23) p 2 (s; t; l ; l 2 ) l i ( ) i l i P > s; ; () () l +i > s; T l +i+ ; ; T T (2) > s; ; T (2) l 2 > s; T (2) (2) l 2 + ; ; T 2 ; (24) 3

14 p 3 (s; t; l ; l 2 ) 2 l 2 i ( ) i 2 l 2 i P T (2) > s; ; T (2) (2) (2) l 2 +i > s; T l 2 +i+ ; ; T 2 > s; ; l > s; () l + ; ; T ; (25) p 4 (s; t; l ; l 2 ) l i 2 l 2 j ( ) i+j 2 l i 2 l 2 j P () l +i+ ; ; T ; T (2) > s; ; T (2) l 2 +j > s; T (2) (2) l 2 +j+ ; ; T 2 > s; ; l +i > s; (26) I equatis (24)-(26), P b a 0 if a > b. Prf Clearly, P fc (s) l ; C 2 (s) l 2 ; C (t) ; C 2 (t) 2 g P 2 l l 2 > s; ; l > s; () l + ; ; T () l + s; ; T s; T (2) > s; ; T (2) l 2 > s; T (2) T (2) (2) l 2 + s; ; T 2 s : l 2 + ; ; T (2) 2 De e the evets The A A 2 B > s; ; l > s; () l + ; ; T T (2) > s; ; T (2) l 2 > s; T (2) (2) l 2 + ; ; T 2 [ il + i > s, B 2 [ 2 il 2 + T (2) i > s : P fc (s) l ; C 2 (s) l 2 ; C (t) ; C 2 (t) 2 g 2 [P (A \ A 2 ) P (A \ A 2 \ B ) l l 2 P (A \ A 2 \ B 2 ) + P (A \ A 2 \ B \ B 2 )] : The prf is w cmpleted usig the priciple f iclusi-exclusi. 4

15 As it is clear frm Prpsiti 3, t cmpute E(T evaluate the itegrati i the frm t j T : ); it is eugh t Z 0 P + x; ; a + x; a+ ; ; b ; T (2) + x; ; T c (2) + x; T (2) c+ ; ; T (2) d dx: Fr the multivariate Paret mdel give by (3), it ca be easily see that the later itegral equals t Z [ + ((t + x)a + (b a)t) + 2 ((t + x)c + (d c)t)] dx 0 [ + t( b + 2 d)] ; a + 2 c fr > : Thus, usig Prpsiti 3 the MRL f a cheret system whe all cmpets are fuctiig at time t ca be cmputed frm 2 E(T t j T : ) (l ; l 2 ) [ + t t] ( ) l 0 l 2 0 " 2 [ + t( ( l ) + 2 ( 2 l 2 ))] l l 2 l + 2 l 2 l ( ) i l [ + t( ( l i) + 2 ( 2 l 2 ))] i i (l + i) + 2 l 2 2 l 2 ( ) i 2 l 2 [ + t( ( l ) + 2 ( 2 l 2 i))] i i l + 2 (l 2 + i) l 2 l 2 ( ) i+j 2 l 2 l 2 i j i j # [ + t( ( l i) + 2 ( 2 l 2 j))] ; (27) (l + i) + 2 (l 2 + j) fr > : Ather MRL fucti that may be f practical iterest ca be de ed as m r ;;r K (t) E(T t j T ; r : ; ; T (K) r K : K ); (28) fr r i i ; i ; ; K. The fucti de ed by (28) represets the mea residual life f the system give that at least i r i + cmpets f type i are 5

16 wrkig at time t, i ; ; K: Clearly, fr s, P T > s; T r () : ; ; T r (K) K : K K (l ; ; ) l 0 0 (j ;;j K )2U P fc (s) l ; ; C K (s) ; C (t) j ; ; C K (t) j K g ; (29) where U f(j ; ; j K ) : max(l m ; m r m + ) j m m ; m ; ; Kg : O the ther had, P T ; T r () : ; ; T (K) l r + K K r K + r K : K (l ; ; )P fc (t) l ; ; C K (t) g : (30) The MRL fucti de ed by (28) ca be cmputed usig (29) ad (30) i m r ;;r K (t) E(T t j T ; T r () : ; ; T r (K) K : K ) P T ; r : ; ; T (K) r K : K Z 0 P T + x; r : ; ; T (K) r K : K dx: (3) Equati (3) crrespds t E(T t j T : ) whe r r K. Example (ctiued) I Figure 2, we plt m(t) E(T t j T ) (MRL), m ; (t) E(T t j T : ) (MRL) ad m 2;2 (t) E(T t j T ; 2: > t; T (2) 2: 2 ) (MRL2) fr the system i Figure uder the mdel (3) whe ; 2 2; 2: We have m(t) m 2;2 (t) m ; (t) with m(0) m 2;2 (0) 6

17 m ; (0) E(T ) 0:403: Figure 2. MRL fuctis f the system i Figure. 4 Discussi This paper has preseted geeral results survival fucti ad mea residual life fr cheret systems, with multiple types f cmpets, with the ly assumpti that the failure times f cmpets f the same type are exchageable. Hece, such cmpets ca be depedet, ad als depedece f cmpets f di eret types is allwed. The use f the survival sigature eabled derivati f a geeral expressi fr the mea residual life fr such scearis, i particular thrugh the itrducti f the miimal survival sigature fr such system, geeralizig this ccept that was itrduced by Navarr et al. (2007) fr systems with a sigle type f cmpets. Mai future research challeges related t this wrk iclude cmputatial issues, i particular fr large real wrld systems, ad the use f the mea residual life fr decisi supprt, where e ca thik abut aspects like maiteace but als issues f system desig. I additi t the miimal sigature, Navarr et al. (2007) als represeted the survival fucti f a cheret system as a geeralized mixture f survival fuctis f parallel systems ad called the crrespdig set f ce ciets as a maximal sigature. This ccept ca be geeralized t the maximal survival sigature alg 7

18 similar lies as the miimal survival sigature preseted i this paper, ad may be useful fr varius reliability prblems, e.g. stchastic cmparis f tw di eret systems. Ackwledgmets The authrs thak the Assciate Editr ad tw reviewers fr supprtive ad cstructive cmmets ad suggestis. This wrk was de while the rst authr was visitig the Durham Uiversity, Departmet f Mathematical Scieces with the supprt prvided by the Scieti c ad Techlgical Research Cucil f Turkey (TUBITAK). Refereces [] M. Asadi, S. Glifrushai (2008) O the mea residual life fucti f cheret systems. IEEE Trasactis Reliability 57 (4), [2] I. Bayramglu, M. Ozkut (206) Mea residual life ad iactivity time f a cheret system subjected t Marshall-Olki type shcks.jural f Cmputatial ad Applied Mathematics, 298, [3] K. Bayramglu Kavlak (207) Reliability ad mea residual life fuctis f cheret systems i a active redudacy. Naval Research Lgistics, 64, [4] F.P. Cle, T. Cle-Maturi (202) Geeralizig the sigature t systems with multiple types f cmpets. I: Cmplex systems ad depedability. Spriger; Berli, p [5] F.P. Cle, T. Cle-Maturi (205) Mdellig ucertai aspects f system depedability with survival sigatures. I: Depedability prblems f cmplex ifrmati systems. Spriger; Berli, p [6] S. Eryilmaz (207) The ccept f weak exchageability ad its applicatis, Metrika, 80, [7] G. Feg, E. Patelli, M. Beer ad F.P.A. Cle (206) Imprecise system reliability ad cmpet imprtace based survival sigature. Reliability Egieerig & System Safety, 50, [8] P. Miziula, J. Navarr (207) Sharp buds fr the reliability f systems ad mixtures with rdered cmpets. Naval Research Lgistics, 64, [9] J. Navarr (206) Distributi-free cmpariss f residual lifetimes f cheret systems based cpula prperties. Statistical Papers, DOI: 0.007/s

19 [0] J. Navarr, S. Eryilmaz (2007) Mea residual lifetimes f csecutive-k-ut-f- systems. Jural f Applied Prbability, 44, [] J. Navarr, J.M. Ruiz, C.J. Sadval (2007) Prperties f cheret systems with depedet cmpets. Cmmuicatis i Statistics-Thery ad Methds, 36, [2] J. Navarr, P.J. Heradez (2008) Mea residual life fuctis f ite mixtures, rder statistics ad cheret systems. Metrika, 67, [3] J. Navarr, F. Spizzichi (200) Cmpariss f series ad parallel systems with cmpets sharig the same cpula. Applied Stchastic Mdels i Busiess ad Idustry, 26, [4] J. Navarr, M.C. Gmis (206) Cmpariss i the mea residual life rder f cheret systems with idetically distributed cmpets. Applied Stchastic Mdels i Busiess ad Idustry, 32, [5] J. Navarr, Y. del Águila, M.A. Srd, A. Suárez-Llres (206) Preservati f stchastic rders uder the frmati f geeralized distrted distributis. Applicatis t cheret systems, Methdlgy ad Cmputig i Applied Prbability, 8, [6] J. Navarr, F. Durate (207) Cpula-based represetatis fr the reliability f the residual lifetimes f cheret systems with depedet cmpets. Jural f Multivariate Aalysis, 58, [7] J. Navarr, F. Pellerey, M. Lgbardi (207). Cmparis results fr iactivity times f k-ut-f- ad geeral cheret systems with depedet cmpets. Test, 26, [8] E. Patelli, G. Feg, F.P.A. Cle, T. Cle-Maturi (207) Simulati methds fr system reliability usig the survival sigature. Reliability Egieerig & System Safety, 67, [9] S. Reed (207) A e ciet algrithm fr exact cmputati f system ad survival sigatures usig biary decisi diagrams. Reliability Egieerig & System Safety, 65, [20] M.K. Sadegh (20) A te the mea residual life fucti f a cheret system with exchageable r idetical cmpets, Jural f Statistical Plaig ad Iferece, 4, [2] F.J. Samaieg, System Sigatures ad Their Applicatis i Egieerig Reliability. New Yrk: Spriger,

20 [22] F.J. Samaieg, J. Navarr (206) O cmparig cheret systems with hetergeus cmpets. Advaces i Applied Prbability, 48, 88-. [23] Z. Zhag, W.Q. Meeker (203) Mixture represetatis f reliability i cheret systems ad preservati results uder duble mitrig Cmmuicatis i Statistics-Thery ad Methds, 42,