An Overview of Various Importance Measures of Reliability System

Transcription

1 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces A Overvew of Varous Importace Measures of Relablty System Kalpesh P Amrutkar, Krtee K Kamalja Departmet of Statstcs School of Mathematcal Sceces North Maharashtra Uversty, Jalgao, Ida * Correspodg author: amrutkarkp@gmalcom (Receved November 15, 2016; Accepted Jauary 3, 2017) Abstract Oe of the purposes of system relablty aalyss s to detfy the weakesses or the crtcal compoets a system ad to quatfy the mpact of compoet s falures Varous mportace measures are beg troduced by may researchers sce 1969 These compoet mportace measures provde a umercal rak to determe whch compoets are more mportat to system relablty mprovemet or more crtcal to system falure I ths paper, we overvew varous compoets mportace measures ad brefly dscuss them wth examples We also dscuss some other exteded mportace measures ad revew the developmets study of varous mportace measures wth respect to some of the popular relablty systems Keywords: Relablty systems Importace measures 1 Itroducto Relablty evaluato has a vtal mportace at all stages of processg ad cotrollg egeerg systems Apart from relablty evaluato of the systems, the study of compoet mportace s also a mportat part The compoet mportace (relablty mportace dex) s valuable establshg the drecto ad prortzato of actos related to a upgradg effort (relablty mprovemet) system desg or suggestg the most effectve way to operate ad mata system status The ma purpose of system relablty aalyss s to detfy the weakesses/crtcal compoets a system ad to quatfy the mpact of compoet s falures Sce 1969, varous compoet mportace measures are beg troduced by the researchers These measures provde a umercal rak to determe whch compoets are more mportat (more crtcal) to system relablty mprovemet (system falure) I ths paper, we overvew varous compoet mportace measures for coheret systems Bergma et al (1985) provdes a terestg survey o relablty theory Bolad ad El-Neweh (1995) overvewed the measures of compoet mportace bary coheret systems ad suggested some ew compoet mportace measures whch mght be useful to aalyze whe the system s about to udergo a relablty mprovemet Kuo ad Zhu (2012a, 2012b, 2012c) surveyed the recet advaces of mportace measures relablty ad Zhu ad Kuo (2014) gave the mplcatos to mathematcal programmg, sestvty ad ucertaty aalyss, probablstc rsk aalyss ad probablstc safety assessmet The revew of partcular research topc usually helps the researchers the sese of avalablty of all related lterature collectve form Wth ths realzato, we motvated to preset the varety of mportace measures altogether The dea s to brg the mportace measures wth dfferet bases/ backgrouds collectve form Brbaum (1969) was frst to troduce the cocept of mportace measures He classfed mportace measures to three classes based o the kowledge eeded for determg them 150

2 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces These three classes are as; structure mportace measures, relablty mportace measures ad lfetme mportace measures We brefly dscuss these three classes of mportace measures (a) Structure Importace Measures Structure mportace measures are defed uder the assumpto that the system structure s kow It measures the relatve mportace of varous compoets wth respect to ther postos a system It s relevat to buldg a system whe several compoets wth dstct relabltes ca be arbtrarly assged to several locatos the system Presumably, oe would lke to assg the most relable compoet to the more mportat locato (b) Relablty Importace Measures Relablty mportace measures deped o both the system structure ad relablty of compoets over a explct ad fxed msso tme It measures the chage the system relablty wth respect to the chage relablty of a specfc compoet (c) Lfetme Importace Measures The lfetme mportace measures, depeds o both, the posto of the compoet the system ad compoet lfetme dstrbuto Kuo ad Zhu (2012c) classfed the lfetme mportace measures to two classes as Tme-Depedet Lfetme (TDL) mportace ad Tme Idepedet Lfetme (TIL) mportace, depedg o whether they are a fucto of tme I ths paper, we overvew the varous mportace measures based o the above three classes ad also dscuss some other exteded mportace measures The paper s orgazed as follows I Secto 2, we dscuss varous structure mportace measures Secto 3 covers the probablty based mportace measures ad presets examples for some of the popular systems Secto 4 cludes a revew of lfetme mportace measures I secto 5, we overvew some other mportace measures studed the lterature Secto 6, cludes troducto to some exteded mportace measures Secto 7, revews the study of mportace measures of some weghted systems Fally, Secto 8 cocludes the overvew of mportace measures 2 Structure Importace Measures I ths secto, we dscuss some structure mportace measures, whch actually evaluate the relatve mportace of varous compoets wth respect to ther postos a system The compoet structure mportace geerally represets the mportace of the postos the system that the compoets occupy The structure mportace measure s used to assg a rak to the compoets whe the compoet relabltes are ot avalable We dscuss some of the structure mportace measures the followg Cosder a -compoet bary coheret system wth structure fucto φ( ) Let x dcates a state (faled/workg) of the th compoet of the system That s, x = { 1 f th compoet s workg 0 f th compoet s faled x = (x 1, x 2,, x ): the vector of states of compoets p the relablty of compoet e P(X = 1) = p = 1 P(X = 0) 151

3 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces p φ(x) R(p) = (p 1, p 2,, p ): the vector of relabltes of -compoets the system state whch s defed as, φ(x) = 1(0) f the system s workg (faled) system relablty wth compoet relablty vector p (a) Brbaum Structure (B-Structure) Importace A structure mportace measure s troduced by Brbaum (1969) The compoet structure mportace geerally represets the mportace of the postos the system that the compoets occupy Brbaum (1969) defes the B-structure mportace I B (φ)of compoet as, I B (φ) = x[ φ(1, x) φ(0, x)] where (1, x) : x wth x = 1 (0, x) : x wth x = 0 φ(1, x) : System state whe th compoet s workg φ(0, x) : System state whe th compoet s fal Ths formula further smplfes as follows I B (φ) = R ( 1 2, 1 2, 1 2, 1, 1 2,, 1 2 ) R (1 2, 1 2, 1 2, 0, 1 2,, 1 2 ) = R(1, p) R(0, p) where (1, p) : p wth p = 1 ad p j = 1, j = 1,2,, ; j 2 (0, p) : p wth p = 0 ad p j = 1, j = 1,2,, ; j 2 R(1, p) : System relablty whe th compoet s workg R(0, p) : System relablty whe th compoet faled The B-structural mportace establshes the probablty of the system falure due to a gve compoet whe all compoets are assumed to be equally relable e each compoet has relablty 1 2 (b) Fassell-Vesely (FV) Structure Importace The FV-mportace s troduced by Fussell ad Vesely (1972) whch s based o the cut set ad path set The FV-mportace s classfed to c-type ad p-type FV-mportace The c-type FVmportace, referred to as c-fv mportace, s based o cut set whle the p-type FV-mportace, referred to as p-fv mportace, s based o the path set of the system Let N = {1,2,, } be a set of compoets of system The c-fv mportace of compoet s deoted by I FV c(φ) ad defed as, I FV c (φ) = {x C C such that C N 0 (x); φ(x) = 0} {x φ(x) = 0} 152

4 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces = {(0,x): C C such that C N 0 ((0,x))} C where N 0 (x) = { N x = 0} : a cut set assocated wth a cut vector x P : set of mmal path vectors P : the mmal paths vectors cotag compoet P : mmal path from mmal path set P Smlarly, the p-fv structure mportace of compoet s deoted by I FV p(φ) ad s defed as, I FV p(φ) = {x P P such that P N 1 (x); φ(x) = 1} {x φ(x) = 1} = {(1,x): P P such that P N 1 ((1,x))} P where N 1 (x) = { N x = 1} : N 1 (x) s a path set assocated wth path vector x C : set of mmal cut vector C : the mmal cut vector cotag compoet C : mmal cut for a mmal cut set C (c) Barlow-Proscha (BP) Structure Importace The BP-structure mportace s troduced by Barlow ad Proscha (1975) The BP-structure mportace of compoet s deoted by I BP (φ) ad defed as, I BP 1 (φ) = [R(1, p) R(0, p)] dp 0 The BP-structure mportace ca be expressed terms of B-structural mportace as follows I BP 1 (φ) = I B (φ) dp 0 Comparso betwee I B (φ) ad I BP (φ) The Brbaum structure mportace measure assumes the relablty of each compoet to be 1 e 2 p = 1 2, = 1,2,, whle the BP-structure mportace s average of I B (φ) over all p [0,1] Thus, both measures favor ether hgh or low compoet relabltes (d) Cut Importace ad Path Importace Butler (1979) proposes the two structure mportaces, amely, cut mportace ad path mportace The cut mportace s a complete rakg of all compoets relatve to ther mportace to the system ad s defed o the bass of mmal cut sets whle the path mportace s defed o the bass of the mmal path set The addtoal coverage of cut ad path mportaces s dscussed by Kuo ad Zhu (2012c) 153

5 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces (e) Permutato Importace ad Permutato Equvalece The permutato mportace ad permutato equvalece measures are troduced by Bolad et al (1989) The defto of the permutato mportace s based o the mmal cut set ad mmal path set Compoet s more permutato mportat tha compoet j (deoted by > pe j) for structure fucto φ f φ(1, 0 j, x (,j) ) φ(0, 1 j, x (,j) ) holds for all x (,j) where, x (,j) = (x 1, x 2,, x 1, x +1,, x j 1, x j+1,, x ) That s, x (,j) s a vector of 2 compoet states excludg th ad j th compoet If equalty holds for some x (,j), the compoet ad compoet j are sad to be permutato equvalet ad s deoted by, = pe j Compoet ad compoet j are sad to be permutato equvalet f ad oly f they are structurally symmetrc ad φ(x) s sad to be a permutato symmetrc x ad x j Koutras et al (1994) proved the followg propertes for permutato mportace ad permutato equvalece () If > pe j ad j > pe k the > pe k (trastvty property) () If > pe j for system structure φ, the > pe j for ts dual system structure φ D (dual of φ) ad vce versa (dual relato) () If compoet s seres (parallel) wth the rest of the system, the (v) φ(0, 1 j, x (,j) ) = 0; φ(1, 0 j, x (,j) ) = 1, for all x (,j) Ths mples that compoet s more permutato mportat tha compoet j e pe j, for all j (specal case property) I ext secto, we dscuss the varous relablty based mportace measures 3 Relablty Importace Measures Relablty mportace measures deped o both the system structure ad relablty of compoets over a explct ad fxed msso tme It measures the chage the system relablty wth respect to the chage relablty of a specfc compoet I ths secto, we dscuss the relablty mportace measures whch are cosdered whe the msso tme of a system s mplct ad fxed, ad cosequetly, the compoets are evaluated by ther relablty at a fxed tme pot, e the probablty that compoet fuctos properly durg the msso tme (Kuo ad Zhu, 2012c) (a) Brbaum Relablty Importace Brbaum relablty mportace (B-mportace) measure s troduced by Brbaum (1969) It s used whe we are provded the survval probabltes of all the compoets a coheret system It s the rate of crease the system relablty wth respect to the crease compoet relablty 154

6 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces Brbaum (1969) defes the B-mportace of system wth compoet relabltes p = (p 1, p 2,, p ) through the followg The B-mportace of th compoet for the fuctog of system s deoted by I Bs (1, p) ad defed as, I Bs (1, p) = P{φ(x) = 1 X = 1} P{φ(x) = 1} The B-mportace of th compoet for the falure of system s deoted by I Bf (0, p) ad defed as, I Bf (0, p) = P{φ(x) = 0 X = 0} P{φ(x) = 0} Further Brbaum (1969) defes the B-mportace I B (p)of th compoet compoet system as addto of B-mportace for system fuctog ad system falure Thus I B (p) ca be gve as, I B (p) = I Bs (1, p) + I Bf (0, p) = P{φ(x) = 1 X = 1} P{φ(x) = 1} + P{φ(x) = 0 X = 0} P{φ(x) = 0} = P{φ(x) = 1 X = 1} + P{φ(x) = 0 X = 0} 1 = P{φ(x) = 1 X = 1} [1 P{φ(x) = 0 X = 0}] = P{φ(x) = 1 X = 1} P{φ(x) = 1 X = 0} A equvalet defto of B-mportace s gve by Barlow ad Proscha (1975) I B (p) = E(φ(1, x) φ(0, x)) R( p) = R(1, p) R(0, p) p (31) where (1, p) ((0, p)) s p wth p = 1(0), φ(1, x) (φ(0, x)) R(1, p) (R(0, p)) s a structure fucto wth th compoet workg (faled), s a relablty whe th compoet s workg (faled) Specfcally, (31) s gve uder the assumpto that compoet relabltes are depedet Brbaum measure s thus obtaed by partal dfferetato of the system relablty wth respect to p whe the compoet relabltes are depedet Chag et al (2002) refers ths measure as a Combatoral Brbaum Importace Accordg to the equvalet defto (31), I B (p) s a rate at whch system relablty mproves, whe, the relablty of compoet mproves 155

7 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces Observe that, the B-structure mportace s a specal case of B-mportace wth p = 1 2, = 1,2,, e I B (φ) = I B ( 1 2, 1 2,, 1 2 ) The B-mportace s referred as margal relablty mportace by Hog ad Le (1993), Armstrog (1995), Hsu ad Yuag (1999), Lu ad Jag (2007), Gao et al (2007) B-mportace s the most popular mportace measure We demostrate the evaluato of B- mportace for seres, parallel ad k-out-of- systems through followg examples Example 1: A seres system works (fals) f ad oly f all (at least oe) compoets works (fal) The structure fucto φ(x) ad relablty fucto R(p) of a -compoet seres system s gve by, =1 φ(x) = =1 x ad R(p) = p The B-mportace of th compoet s, j=1 I B (p) = j=1,j p j = p j /p Thus, the most (least) relable compoet has the smallest (largest) B-mportace Example 2: A parallel system, fuctos (fals) f ad oly f at least oe (all) compoet fuctos (fals) The structure ad relablty fucto for ths -compoet system s gve by, φ(x) = 1 =1 (1 x ) ad R(p) = 1 (1 p ) The B-mportace of th compoet of ths system s, I B (p) = j=1,j q j = j=1 q j /q ; (q = 1 p ) =1 That s, the th compoet wth the greatest relablty has the hghest B-mportace (e most relable compoet s most mportat) Example 3: A k-out-of- system fuctos (fals) f ad oly f at least k (1 k ) of the compoets fucto (fal) If k =, the k-out-of- system reduces to seres system ad f k = 1, t reduces to the parallel system The structure fucto of -compoet k-out-of- system s, 1f ( x) 0f 1 1 x k x k For a k-out-of- system, Kuo ad Zhu (2012c) observed that, 156

8 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces φ(1, x) φ(0, x) = 1 φ(1, x) = 1 ad φ(0, x) = 0 exactly k 1 of the 1 compoets (excludg compoet ) fucto Now, B-mportace of th compoet for k-out-of- system s I B (p) = p 1 p 2 p k 1 (1 p k )(1 p k+1 ) (1 p 1 ), where the sum s exteded over all permutatos ( 1, 2,, 1 ) of the subscrpts {1,2,, 1, + 1, } Further, for > 2 ad k 1, assumg p 1 p 2 p, Bolad ad Proscha (1983) ad Chadjcostatds ad Koustras (1999) proved the followg p k 1 1 for = 1,2,,, I B 1 (p) I B 2 (p) I B (p), p k 1 1 for = 1,2,,, I B 1 (p) I B 2 (p) I B (p) Revew of B-Importace of Cosecutve-Systems A lear cosecutve-k-out-of-: F (C(k, : F)) system s troduced by Kotoleo (1980), ad fals f ad oly f at least k cosecutve compoets the system fal Zuo (1993) studes the B- mportace of C(k, : F) ad C(k, : G) system ad clames that the comparso of B-mportace betwee two compoets for a C(k, : G) system s the same as that for the C(k, : F) system Hwag et al (2000) showed that the comparso gve by Zuo s ot correct ad provded a correct relato betwee the B-mportace of compoets of the two systems Chag et al (1999) studed the B-mportace of the C(k, : F) system Chadjcostatds ad Koutras (1999) studed the B- mportace of compoets for the wde class of Markov Cha Imbeddable Systems (MIS) ad provded formulae for the evaluato of the B-mportace of the compoets of a MIS through products of trasto probablty matrces ad multple recurrece relatos Chag et al (2002) troduced the cocept of uform, half-le ad combatoral B-mportace of compoets for cosecutve-k systems Meg (1996) compares the mportace of system compoets by the structural characterstcs of the system Kamalja (2012) studes the B-mportace of cosecutvetype systems as C(k, : F), m-cosecutve-k-out-of-: F ad r-wth-cosecutve-k-out-of-: F system She ad Cu (2015) studed B-mportace for sparsely coected crcular cosecutve-k systems L et al (2016) preseted the relablty modelg o cosecutve-k r -out-of- r : F lear zgzag structure ad crcular polygo structured systems Ca et al (2016) proposed a B- mportace-based geetc algorthm to search the ear global optmal soluto for lear cosecutve-k-out-of- system (b) Improvemet Potetal Importace The B-mportace of th compoet for the system fuctog s also kow as mprovemet potetal mportace of th compoet (Hoylad ad Rausad, 1994; Ave ad Jese, 1999; Frexas ad Pos, 2008) or rsk achevemet mportace (va der Borst ad Schooakker, 2001) 157

9 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces I Bs (1, p) s the crease systems relablty whe compoet s perfect e whe p = 1 ad represets the maxmum potetal mprovemet systems relablty that ca be obtaed by mprovg the relablty of compoet The mprovemet potetal mportace I IP (S) of the th compoet of the system S s the dfferece betwee system relablty wth perfect compoet (e compoet wth ts relablty p = 1) ad the system relablty wth actual compoet That s, I IP (S) = R(S compoet s perfect) R(S) = P(φ(x) = 1 X = 1) P(φ(x) = 1) (c) Crtcalty Relablty Importace Kuo ad Zuo (2003) proposed the crtcalty relablty mportace measure for system falure (fuctog) ad s the probablty that compoet fals (fuctos) ad s crtcal for system falure (fuctog) gve that the system fals (fuctos) The crtcalty relablty mportace I Cf (S) of the th compoet of system falure gve that the system fals s defed as follows I Cf (S) = P(X = 0) P(φ(x) = 0) (R(1, p) R(0, p)) = P(X =0) 1 R(S) I B (S) The crtcalty relablty mportace I Cs (S) of the th compoet of system (S) fuctog gve that the system fuctos s defed as follows I Cs (S) = P(X = 1) P(φ(x) = 1) (R(1, p) R(0, p)) = P(X =1) I R(S) B (S) (d) Bayesa Relablty Importace Brbaum (1969) proposes the Bayesa relablty mportace, whch s defed as the probablty that compoet fals gve that the system fals Thus the Bayesa mportace I Bay (S) of the th compoet of system S wth structure fucto φ(x) s defed as follows I Bay (S) = P(X = 0 φ(x) = 0) = P(φ(x) = 0 X = 0)P(X =0) P(φ(x)=0) Kuo ad Zhu (2012c) provded a smplfed formula for Bayesa mportace of -compoet system S wth depedet compoet relabltes p 1, p 2,, p as follows 158

10 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces I Bay (S) = q {1 + p I 1 R(p) B (S)}, = 1,2,,, where, I B (S) s B-mportace of th compoet of system S (e) Fussell-Vesely (FV) Importace Fussell ad Vesely (1972) troduced a compoet mportace measure usg cut set ad path set ad s referred as FV-cut-mportace ad FV-path-mportace respectvely The FV-cut-mportace (path-mportace) of compoet, deoted by I FV c(p) s defed as the probablty that a compoet state vector has correspodg cut (path) that causes (makes) system falure (fucto), ad cotas a mmal cut C (path P ) e cotag compoet Mathematcally, I FV c(p) = P{ C C such that C N 0 (X) φ(x) = 0} = q P{(0,x): C C X j =0 j C}, 1 R(p) ad I FV p(p) = P{ P P such that P N 1 (X) φ(x) = 1} = p P{(0,x): P P X j =0 j P} R(p) We demostrate t for 2-compoet seres system through followg example Example 4: Cosder a seres system wth two compoets wth depedet relabltes, p 1 = 098 ad p 2 = 096 The relablty of the seres system s, R(p) = p 1 p 2 = There s oly oe cut set cotag compoet 1 ad 2, e C = {(1,2)} Now, FV-cut-mportace of compoet 1 ad 2, s as follows I c FV (1) = 1 p 1 1 R(p) = 1 p 1 = p 1 p 2 ad c (2) = 1 p 2 I FV 1 R(p) = 1 p 2 1 p 1 p 2 = Ths rakg agrees wth the rakg of B-mportace measure The weakest compoet a seres system s the most mportat Example 5: Cosder a -compoet parallel system For ths system costtutes a mmal cut set s C = {1,2,, } Hece, FV cut-mportace for all compoets s same A weakess of ths measure s that the same level of mportace s calculated for all compoets a parallel system regardless of ther relabltes 159

11 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces 4 Lfetme Importace Measures The lfetme mportace measures deped o both, the posto of the compoet the system ad compoet lfetme dstrbuto A lfetme mportace measures are dvded to two subclasses as Tme-Depedet Lfetme (TDL) mportaces ad Tme Idepedet Lfetme (TIL) mportaces, depedg o whether they are a fucto of tme (Kuo ad Zhu, 2012c) A TDL mportace evaluates the mportace of compoets at ay tme ad the rakgs of compoet mportace duced may vary wth tme Whle TIL mportace takes to accout the compoet lfetme over the log term, as such, perhaps gves a more global vew of compoet mportace Every type of relablty mportace measure ca be trasformed to a correspodg TDL mportace measure by substtutg F (t) (relablty dstrbuto of compoet ) for p, = 1,2,,, wthout chagg the probablstc terpretato of the mportace measure Here we dscuss some of the lfetme mportace measures Cosder the followg otatos eeded to defe lfetme mportace measures X (t) state of th compoet at tme t X(t) = (X 1 (t), X 2 (t),, X (t)): vector of states of compoets at tme t F (t) lfetme dstrbuto of th compoet F (t) = 1 F (t) = E[X (t)]: relablty of compoet at tme t F(t) = (F 1(t), F 2(t),, F (t)): relablty vector of -compoets at tme t R(F(t)) = P{φ(X(t)) = 1} = E[φ(X(t)) (a) Brbaum Lfetme (B-Lfetme) Importace Lambert (1975) ad Natvg (1979) exteded the B-mportace to the B-lfetme mportace at tme t by usg compoet lfetme dstrbuto The B-lfetme mportace of compoet at tme t s deoted by I B (F (t)) ad s defed as the probablty that the system s a state at tme t whch compoet s crtcal for the system That s, the probablty that at tme t the falure ad fuctog of compoet cocdes wth system falure ad fuctog respectvely Mathematcally, I B (F (t)) = P{φ(1, X(t)) φ(0, X(t)) = 1} = R(1, F (t)) R(0, F (t)) Xe (1987) provdes the followg propertes for the system wth structure fucto φ, lfetme dstrbuto F φ (t) ad relablty dstrbuto F φ(t) at tme t whch a compoet s seres or parallel wth the rest of a system Xe (1988) gves a upper boud for the B-lfetme mportace for coheret system wth structure fucto φ as I B (F (t)) m { F φ(t), F φ (t) F (t) F (t) } If compoet s seres wth the rest of a system, the I B (F (t)) = F φ(t) F (t) If compoet s seres wth the rest of a system, the I B (F (t)) = F φ (t) F (t) 160

12 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces (b) Crtcalty Lfetme Importace Lambert (1975) defes the crtcalty lfetme mportace for system falure at tme t, whch s relevat to falure dagossthe crtcalty lfetme mportace for the system falure of compoet at tme t s deoted by I Cf (F (t)) ad s defed as, the probablty that compoet has faled by tme t ad compoet s crtcal for the system at tme t, gve that the system has faled by tme t Mathematcally, I Cf (F (t)) = P{φ(1, X(t)) φ(0, X(t)) = 1 ad X (t) = 0 φ(x(t) = 0)} = F (t) I B (F (t)) 1 R(F (t)) Smlarly, the crtcalty lfetme mportace for system fuctog ca be defed (c) Fussell-Vesely (FV) Lfetme Importace Fussell ad Vesely (1972) troduced a compoet mportace measure usg mmal cut/path set Lambert (1975) defes the FV-lfetme cut-mportace through a mmal cut set ad Kuo ad Zhu (2012c) exteded the FV-lfetme cut-mportace to the FV-lfetme path-mportace usg a mmal path set The FV-lfetme cut-mportace of compoet at tme t s deoted by I FV c (F(t)) ad s defed as the probablty that at least oe mmal cut cotag compoet fals at tme t gve that the system fals at tme t That s, I FV c (F(t)) = P{ C C such that X j (t) = 0 j C φ(x(t) = 0)} = P{ C C such that X j (t)=0 j C} 1 R(F(t)) Smlarly, the FV lfetme path-mportace of compoet at tme t s deoted by I FV p s defed as, I FV p(f(t)) = P{ P P such that X j (t) = 0 j P φ(x(t)) = 0)} = P{ P P such that X j (t)=0 j P} R(F(t)) (F(t)) ad (d) Barlow-Proscha (BP) Lfetme Importace Barlow ad Proscha (1975) troduced a ew mportace measure assumg that the relabltes of all compoets of the system are s-depedet ad compoet have lfetme dstrbuto F (t), = 1,2,, at tme t The BP-lfetme mportace of compoet at tme t s deoted by I BP (F(t)) ad s defed as the probablty that compoet s crtcal for the system over a fte msso tme That s, 161

13 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces (F(t)) = [R (1, F(t)) R (0, F(t))] df (t) I BP 0 = 1,2,, I BP (F(t)) ca be expressed terms of B-lfetme mportace as follows (F(t)) = I B (F (t)) df (t) I BP 0 Thus, I BP (F(t)) s a weghted average of the I B (F(t)), over a fte msso tme Bolad ad El-Neweh (1995) provded a precse justfcato that I BP (F(t)) s exactly the probablty that the system lfetme cocde wth the lfe of compoet, that s I BP (F(t)) = P{T φ = T } where T φ s lfetme of system wth structure fucto φ ad T s lfetme of compoet Iyer (1992) exteds the Barlow-Proscha dex to the more geeral case whe the compoet lfetmes are jotly absolutely cotuous but ot ecessarly depedet I ths settg the dex I BP may deped ot oly o the structure fucto φ but also o the dstrbuto fucto F Propertes of BP-Lfetme Importace Measure a) 0 I BP (F(t)) 1 b) =1 I BP (F(t)) = 1 c) If 2 ad the tersecto of supports F j (t) = P(T j < t); (j = 1,2, ) has postve probablty wth respect to the product dstrbuto j=1 F j (t) the 0 < I BP (F(t)) < 1 BP-lfetme mportace shows that whe a compoet s seres (parallel) wth the rest of the j system ad t s stochastcally the weakest (strogest) compoet, the I BP (F(t)) I BP (F(t)) for all j (e) Natvg Importace Natvg (1979) troduces a ew compoet mportace measure kow as Natvg s measure A Natvg s measure has the same characterstcs as the FV-lfetme mportace If Z s the reducto remag lfetme due to the falure of the th compoet the a Natvg measure of mportace I N of compoet s defed as, j=1 I N = E(Z ) E(Z j ), Xe (1988) exteds the relatoshp of BP-lfetme measure to the Natvg measure as, 162

14 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces I BP (F(t)) I j BP (F(t)) for all j, the I N I j N for all j Ave (1986) compares I BP (F(t)) ad I N by restrctg atteto to a fte terval of tme assumg compoet lfetmes Webull dstrbuted Calculato of I N s qute feasble for seres ad parallel systems where proportoal hazard modelg for compoets s approprate 5 Other Importace Measures I ths secto, we overvew some other mportace measures whch are apart from the 3-classes defed by Brbaum (1969) ad brefly dscuss t (a) Hwag Idex Hwag (2001) proposes a ew dex of compoet mportace based o the cut set, path set, cut ad path absoluteess ad mmal cut ad mmal path set Hwag dex for compoet s deoted by I h ad s defed as, I h = { C (d) }, d = 1,2, where C (d) s the set of cut sets of cardalty d that cotas compoet (b) Rsk Achevemet Worth Rsk Achevemet Worth (RAW) has bee troduced by Cheok et al (1998a, 1998b) for rsk aalyss termology Rsk Achevemet Worth maly used as a rsk mportace measure probablstc safety assessmets of uclear power statos RAW of compoet at tme t s deoted by I RAW (t) ad defed as the rato of the system urelablty wth compoet faled at tme t to the actual system urelablty at tme t Mathematcally, I RAW (t) = 1 R(0,p(t)) 1 R(p(t)) (c) Rsk Reducto Worth Cheok et al (1998a, 1998b) troduced Rsk Reducto Worth (RRW) by usg rsk aalyss termology RRW of compoet at tme t s deoted by I RRW (t) ad defed as the rato of the actual system urelablty to the system urelablty wth compoet replaced by a perfect compoet e p (t) 1 Mathematcally, I RRW (t) = 1 R(p(t)) 1 R(1,p(t)) (d) Falure Crtcalty Idex A Falure Crtcalty Idex (FCI) s troduced by Wag et al (2004) FCI s a relatve dex showg the percetage of tmes that a falure of the compoet caused a system falure The FCI of th compoet at tme t s deoted by I FCI (t) ad s defed as, 163

15 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces I FCI (t) = Number of system falures caused by compoet (0,t) Number of system falures (0,t) (e) Restore Crtcalty Idex Restore Crtcalty Idex (RCI) s troduced by Wag et al (2004) for the reparable system RCI of compoet s the percetage of tmes that system restorato results from the restorato of compoet tme terval (0, t) The RCI of th compoet at tme t s deoted by I RCI (t) ad defed as, I RCI (t) = Number of actos o compoet that restored the system (0,t) Number of tmes the system was restored (0,t) Ths dex gves the percetage of tmes that a restorato of the compoet wll result restorg the system from a dow state tme terval (0, t) (f) Operatoal Crtcalty Idex Operatoal Crtcalty Idex (OCI) has bee troduced by Wag et al (2004) OCI s the percetage of compoet s dow tme over the system dowtme or the percetage of a compoet s up tme over the system up tme The OCI of th compoet at tme t s deoted by I OCI (t) ad defed as, I OCI (t) = Or equvaletly, I OCI (t) = Total dow tme of compoet whe the system s dow (0,t) Total system dow tme (0,t) Total up tme of compoet whe the system s up (0,t) Total system up tme (0,t) 6 Geeralzed (Pars ad Group of Compoets) Importace Measures I ths secto, we dscuss the geeralzed or pars/ group of mportace measures ad also dscuss codtoal relablty mportace troduced the lterature The mportace measures that are preseted Secto 2 to Secto 5, evaluate the stregth of a dvdual compoet Here, we dscuss mportace measures for a par or group of compoets, cosderg the effects ad hgher order teractos of compoets o system performace It cludes Jot Relablty Importace (JRI) ad Jot Falure Importace (JFI) troduced by Hog ad Le (1993) Note that the order of compoets does ot matter for ay mportace measure of the par or the group (a) JRI ad JFI of Two Compoets JRI (JFI) of two compoets s a measure of teracto of two compoets a system cotrbuto to the system relablty (falure), 164

16 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces Mathematcally, JRI of compoet c 1, c 2 s deoted by JRI(c 1, c 2 ) ad for -compoet system wth relablty fucto R(p) of statstcally depedet compoet relabltes p 1, p 2,, p s defed as, JRI(c 1, c 2 ) = 2 R(p) p 1 p 2 Whle JFI of c 1, c 2 s deoted by JFI(c 1, c 2 ) ad for a system wth statstcally depedet compoet relabltes s defed as, ad JFI(c 1, c 2 ) = 2 F(q) q 1 q 2 Armstrog (1995) exteds the cocept of JRI to clude a system wth s-depedet compoets ad proved that JRI s always o-zero for some classes of systems Example 6: Cosder the -compoet seres system (S 1 ) ad parallel system (S 2 ) wth s - depedet compoet relabltes p 1, p 2,, p =1, R(S 1 ) = =1 p, R(S 2 ) = 1 (1 p ) JRI u1,u 2 (S 1 ) = =3 p > 0, JRI u1,u 2 (S 2 ) = =1 (1 p ) < 0 Bref Revew of JRI of k-out-of-: G System Hog et al (2002) detfed the sg of the JRI of two compoets wthout fdg ts value usg Schur-covexty of relablty fucto ad studed JRI for k-out-of-: G system Hog et al (2002) compared the JRI of 2-out-of- ad 2-out-of-( + 1) system ad stated the equalty about JRI of par of compoets as JRI 2, (p) < JRI 2,+1 (p) whe p > 1 2 where, JRI 2, (p) s the JRI of c 1, c 2 of 2-out-of- system wth d compoet relabltes p The formula for JRI of 2-compoets of the k-out-of- system wth d compoet relabltes for 3 ad 2 k gve by Hog et al (2002) s as follows JRI k, (c, c j ) = p k 2 (1 p) k 1 [( 2 k 2 ) ( 1 k 1 )p] Hog et al (2002) stated the results about the sgs of JRI of a k-out-of- system wth d compoet relabltes for 3 ad 2 k ad are as follows ) JRI k, (p) > 0 f 0 < p < k 1 ) JRI k, (p) = 0 ) JRI k, (p) < 0 1 f p = k 1 1 f k 1 < p <

17 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces From above results, t ca be see that a system wth d compoet relabltes p, the JRI of two compoets s postve for smaller values of p ad egatve for larger values of p Moreover the threshold s p = k 1 1 Chag ad Ja (2006) re-state the propertes of JRI of the 2-out-of- system ad correct the errors of Hog et al (2002) The propertes are as follows ) JRI 2, (p) = (1 p) 3 [1 ( 1)p] ) lm JRI 2, (p) = 1 ad lm JRI 2, (p) = 0 p 0 p 1 ) M{JRI 2, (p)} = ( 3 1 ) 3 whe p = 2 1 v) For 5, the graph of JRI 2, (p) has a pot of flecto at ( 3, 2 ( ) 3 ) JRI of Three Compoets The JRI of three compoets c 1, c 2 ad c 3 for -compoet system wth relablty fucto R(p) ad compoet relabltes p 1, p 2,, p s gve by, JRI(c 1, c 2, c 3 ) = 3 R(p) p 1 p 2 p 3 For depedet compoet relabltes, JRI(c 1, c 2, c 3 ) smplfes as follows JRI(c 1, c 2, c 3 ) = R(1 1, 1 2, 1 3, p) R(1 1, 1 2, 0 3, p) R(1 1, 0 2, 1 3, p) R(0 1, 1 2, 1 3, p) + R(1 1, 0 2, 0 3, p) + R(0 1, 1 2, 0 3, p) +R(0 1, 0 2, 1 3, p) R(0 1, 0 2, 0 3, p), where, R(l 1, l 2, l 3, p) : Relabltes of -compoet system wth compoet relabltes (p 1, p 2,, l 1,, l 2,, l 3,, p ) Example 7: The JRI of 3 compoets for k-out-of-: G system wth d compoet relabltes p ad k 3 (gve by Gao et al (2007) s, JRI(c 1, c 2, c 3 ) = p k 3 q k 2 (( 3 k 3 ) q2 2 ( 3 3 ) pq + ( k 2 k 1 ) p2 ) JRI of More Tha Three Compoets The defto of JRI of 3 compoets to a group of compoets s exteded as follows The JRI for r (r < ) compoets for c 1, c 2,, c r s defed as, JRI(c 1, c 2,, c r ) = r R(p) r =1 p For depedet compoet relabltes, JRI(c 1, c 2,, c r ) smplfes to JRI(c 1, c 2,, c r ) = R(1 1, 1 2,, 1 r, p) R(1 1, 1 2,, 0 r, p) ± 166

18 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces ± R(0 1, 0 2,, 1 r, p) ± R(0 1, 0 2,, 0 r, p) Zhu et al (2015) studed the jot relablty mportace of cosecutve-k-out-of-: F ad mcosecutve-k-out-of-: F system for Markov depedet compoets (b) Codtoal Relablty Importace The codtoal relablty mportace s troduced by Gao et al (2007) Uder the codto that some compoets are workg or some compoets are faled, the relablty mportace of other compoets s evaluated Based o the defto of Margal Relablty Importace (MRI) ad JRI, we cosder the relablty mportace whe the states of certa compoets are kow It s assumed that the system also works uder these codtos Here we preset the deftos of Codtoal MRI ad Codtoal JRI gve by Gao et al (2007) The codtoal MRI of compoet c j, whe compoet c s workg or faled (e p = z, z = 0,1) s deoted by MRI j (p = z ) ad s defed as, MRI j (p = z ) = R(p 1,p 2,,p 1,z,p +1,,p ) p j,, j = 1,2,,, j For depedet compoet relabltes, MRI j (p = z ) smplfes to MRI j (p = z ) = R(p 1,, z,, p j 1, 1 j, p j+1,, p ) R(p 1,, z,, p j 1, 0 j, p j+1,, p ), where z = 1 (0) whe the compoet c s workg (faled) JRI ca be expressed terms of MRI as follows JRI(c, c j ) = MRI j (p = 1) MRI j (p = 0) The codtoal JRI of compoets c ad c j, whe the state of compoet c k s deoted by JRI(c, c j )(p k = z k ) ad s defed as, JRI(c, c j )(p k = z k ) = 2 R(p 1,,z k,,p ) p p j, k, j For depedet compoet relabltes, JRI(c, c j )(p k = z k ) smplfes to JRI(c, c j )(p k = z k ) = R(z k, 1, 1 j, p) R(z k, 1, 0 j, p) R(z k, 0, 1 j, p) +R(z k, 0, 0 j, p) Ra et al (2011) ad Ja et al (2014) obtaed expressos for codtoal MRI ad JRI whe the compoet relabltes are depedet but eed ot be detcally dstrbuted for seres-parallel ad seres-parallel systems 167

19 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces 7 Revew of Importace Measures of Weghted-Systems I may egeerg systems, ts compoets cotrbute to the system ot oly through ts (workg or fal) status, but also through ther capacty/load (weght) The compoets cotrbute dfferetly to the system performace Thus, compoets of a system may be assocated wth a addtoal weght characterzato apart from ts status I real lfe, the weght of a compoet may be ts heatg/coolg capacty, voltage/wattage, test-score etc relatve to the whole system Ths leads to the cocept of weghted systems, whch cossts of compoets havg some postve teger weght Wu ad Che (1994a) troduced a weghted k -out-of- : G(F) system whch cossts of compoets, each oe havg postve teger weght w such that the total system weght s w = =1 w ad the system works (fal) f ad oly f the total weght of workg (faled) compoets s at least k, a prespecfed value Sce k s a weght, t may be larger tha Samaego ad Shaked (2008) referred these types of systems as Systems wth Weghted Compoets (SWCs) Further Wu ad Che (1994b) troduced ad studed a weghted verso of cosecutve-k-out-of- : F (C w (k, : F)) system lear as well as crcular case A weghted-cosecutve-k-out-of-: F system fals f ad oly f the total weght of faled cosecutve compoets s at least k Chadjcostatds ad Koutras (1999) provded formulae for the evaluato of the B-mportace ad the mprovemet potetal mportace of compoets of weghted k -out-of- : F ad C w (k, : F) system usg Markov cha mbeddg techque Amrutkar ad Kamalja (2014) developed formulae for evaluato of relablty ad the relablty mportace of weghted k-outof-: Fsystem through Weghted Markov Bomal Dstrbuto (WMBD) Erylmaz ad Bozbulut (2014) studed jot ad margal Brbaum ad Barlow-Proscha mportace measures of compoets of weghted k-out-of-: G system by usg the uversal geeratg fucto Rahma et al (2016) studed the mportace of compoets k-out-of- system wth compoets havg radom weghts Kamalja ad Amrutkar (2014) studed relablty mportace of C w (k, : F) system 8 Dscusso ad Coclusos I ths paper we preseted a overvew of varous mportace measures of coheret system Apart from the mportace measures dscussed the preset mauscrpt, there are may other mportace measures Ths paper s a effort to overvew the some of the popular mportace measures ad study of these for some commo systems such as, seres, parallel, k-out-of-, cosecutve ad ther weghted versos I ths paper, we overvewed the recet lterature o compoet mportace measures Further, we preseted these mportace measures for dfferet well-kow systems Ths survey would spread awareess amog readers regardg the mportace measures avalable as well as ecourage further research decdg, whch mportace measures ad whe t should be used Ackowledgemet () Ths work s supported by UGC, New Delh, Ida, through the Major Research Project (F No /2011 (SR)) () The frst author s thakful to the CSIR, New Delh, for awardg Seor Research Fellowshp (F No 09/728 (0033)/2014-EMR-I) () The authors would lke to thak Guest Edtor (s) for ther costructve commets that mproved the mauscrpt 168

20 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces Refereces Amrutkar, K P, & Kamalja, K K (2014) Relablty ad mportace measures of weghted-k-out-of-: F system Iteratoal Joural of Relablty, Qualty ad Safety Egeerg, 21(03), Armstrog, M J (1995) Jot relablty-mportace of compoets IEEE Trasacto o Relablty, 44(3), Ave, T (1986) O the computato of certa measures of mportace of system compoets Mcroelectrocs Relablty, 26(2), Ave, T, & Jese, U (1999) Stochastc models relablty Sprger, New York Barlow, R E, & Proscha, F (1975) Importace of system compoets ad fault tree evet Stochastc Processes ad ther Applcatos, 3(2), Barlow, R E, & Proscha, F (1975) Statstcal theory of relablty ad lfe testg Holt, Rheart ad Wsto, New York Bergma, B, Arjas, E, Rausad, M, Natvg, B, Doksum, K A, & Schweder, T (1985) O relablty theory ad ts applcatos [wth Dscusso ad Reply] Scadava Joural of Statstcs, 1-41 Brbaum, Z W (1969) O the mportace of dfferet compoets a mult-compoet system Multvarate Aalyss II, PR Krshaah (Edtor), Academc, New York, Bolad, P J, & El-Neweh, E (1995) Measures of compoet mportace relablty theory Computers & Operatos Research, 22(4), Bolad, P J, & Proscha, F (1983) The relablty of k-out-of- systems The Aals of Probablty, 11(3), Bolad, P J, Proscha, F, & Tog, Y L (1989) Optmal arragemet of compoets va parwse rearragemets Naval Research Logstcs, 36, Butler, D A (1979) A complete mportace rakg for compoets of bary coheret systems, wth extesos to mult state systems Naval Research Logstcs Quarterly, 26(4), Ca, Z, S, S, Su, S, & L, C (2016) Optmzato of lear cosecutve-k-out-of- system wth a Brbaum mportace-based geetc algorthm Relablty Egeerg & System Safety, 152, Chadjcostatds, S, & Koutras, M V (1999) Measures of compoet mportace for Markov cha mbeddable relablty structures Naval Research Logstcs (NRL), 46(6), Chag, G J, Cu, L, & Hwag, F K (1999) New comparsos Brbaum mportace for the cosecutvek-out-of- system Probablty the Egeerg ad Iformatoal Sceces, 13(02), Chag, H W, & Ja, S (2006) Jot relablty mportace of k-out-of- systems ad seres-parallel systems Proceedg Techques ad Applcatos & Coferece o Real-Tme Computg Systems ad Applcatos, PDPTA, Las Vegas, Nevada, USA Chag, H W, Che, R J, & Hwag, F K (2002) The structural Brbaum mportace of cosecutve-k systems Joural of Combatoral Optmzato, 6(2), Cheok, M C, Parry, G W, & Sherry, R R (1998a) Respose to Supplemetal vewpots o the use of mportace measures rsk-formed regulatory applcatos Relablty Egeerg & System Safety, 60(3), 261 Cheok, M C, Parry, G W, & Sherry, R R (1998b) Use of mportace measures rsk-formed regulatory applcatos Relablty Egeerg & System Safety, 60(3), Erylmaz, S, & Bozbulut, A R (2014) Computg margal ad jot Brbaum, ad Barlow Proscha mportaces weghted-k-out-of-: G systems Computers & Idustral Egeerg, 72,

21 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces Frexas, J, & Pos, M (2008) The fluece of the ode crtcalty relato o some measures of compoet mportace Operato Research Letters, 36(5), Fussell, J B, & Vesely, W E (1972) A ew methodology for obtag cut sets for fault trees Trasactos of the Amerca Nuclear Socety, 15, Gao, X, Cu, L, & L, J (2007) Aalyss for jot mportace of compoets a coheret system Europea Joural of Operatoal Research, 182(1), Hog, J S, & Le, C H (1993) Jot relablty-mportace of two edges a udrected etwork IEEE Trasactos o Relablty, 42(1), Hog, J S, Koo, H Y, & Le, C H (2002) Jot relablty mportace of k-out-of- systems Europea Joural of Operatoal Research, 142(3), Hoylad, A, & Rousad, M (1994) System Relablty Theory New York: Joh Wley & Sos Hsu, S J, & Yuag, M C (1999) Effcet computato of margal relablty mportace for reducble etworks etwork maagemet Proceedgs of the 1999 IEEE Iteratoal Coferece o Commucatos, Hwag, F K (2001) A ew dex of compoet mportace Operatos Research Letters, 28(2), Hwag, F K, Cu, L, Chag, J C, & L, W D (2000) Commets o relablty ad compoet mportace of a cosecutve-k-out-of- system by Zuo Mcroelectrocs Relablty, 40(6), Iyer, S (1992) The Barlow Proscha mportace ad ts geeralzatos wth depedet compoets Stochastc Processes ad ther Applcatos, 42(2), Ja, K, Dewa, I, & Ra, M (2014) Multcompoet jot relablty mportace of seres--parallel ad parallel--seres systems Iteratoal Joural of Qualty & Relablty Maagemet, 31(7), Kamalja, K K & Amrutkar, K P (2014) Computatoal methods for relablty ad mportace measures of weghted-cosecutve-systems, IEEE Trasacto o Relablty, 63(1), Kamalja, K K (2012) Brbaum mportace for cosecutve-k systems Iteratoal Joural of Relablty, Qualty ad Safety Egeerg, 19(4), (25 pages) Kotoleo, J M (1980) Relablty determato of a r-successve-out-of-: F system IEEE Trasactos o Relablty, 29(5), Koutras, M V, Papadopoylos, G, & Papastavrds, S G (1994) Note: Parwse rearragemets relablty structures Naval Research Logstcs, 41, Kuo, W, & Zhu, X (2012a) Relatos ad geeralzatos of mportace measures relablty IEEE Trasactos o Relablty, 61(3), Kuo, W, & Zhu, X (2012b) Some recet advaces o mportace measures relablty IEEE Trasactos o Relablty, 61(2), Kuo, W, & Zhu, X (2012c) Importace measures relablty, rsk ad optmzato: prcples ad applcatos Joh Wley & Sos, Chchester, UK 472 Kuo, W, & Zuo, MJ (2003) Optmal relablty modellg: Prcples ad Applcatos Joh Wley & Sos, New Jersey Lambert, H E (1975) Measure of mportace of evets ad cut sets fault trees Relablty ad Fault Tree Aalyss (eds Barlow, RE, Fussell, JB ad Sgpurwalla, ND) Socety for Idustral ad Appled Mathematcs, Phladelpha,

22 Iteratoal Joural of Mathematcal, Egeerg ad Maagemet Sceces L, C, Cu, L, Cot, D W, & Lv, M (2016) Relablty modelg o cosecutve-$ k_r $-out-of-$ _r $: F lear zgzag structure ad crcular polygo structure IEEE Trasactos o Relablty, 65(3), Lu, L, & Jag, J (2007) Jot falure mportace for ocoheret fault trees IEEE Trasactos o Relablty, 56(3), Meg, F C (1996) Comparg the mportace of system compoets by some structural characterstcs, IEEE Trasactos o Relablty, 45(1), Natvg, B (1979) A suggesto of a ew measure of mportace of system compoets Stochastc Processes ad ther Applcatos, 9(3), Rahma, R A, Izad, M, & Khaled, B E (2016) Importace of compoets k-out-of- system wth compoets havg radom weghts Joural of Computatoal ad Appled Mathematcs, 296, 1-9 Ra, M, Ja, K, & Dewa, I (2011) O codtoal margal ad codtoal jot relablty mportace Iteratoal Joural of Relablty, Qualty ad Safety Egeerg, 18(2), Samaego, F J, & Shaked, M (2008) Systems wth weghted compoets Statstcs & Probablty Letters, 78(6), She, J, & Cu, L (2015) Relablty ad Brbaum mportace for sparsely coected crcular cosecutve- $ k $ systems IEEE Trasactos o Relablty, 64(4), Va der Borst, M, & Schooakker, H (2001) A overvew of PSA mportace measures Relablty Egeerg & System Safety, 72(3), Wag, W, Loma, J, & Vasslou, P (2004) Relablty mportace of compoets a complex system, Reprted from 2004 Proceedgs Aual Relablty ad Mataablty Symposum, Los Ageles, Calfora, USA, Jauary Wu, J S, & Che, R J (1994a) A algorthm for computg the relablty of weghted-k-out-of- systems IEEE Trasactos o Relablty, 43(2), Wu, J S, & Che, R J (1994b) Effcet algorthms for k-out-of- ad cosecutve-weghted-k-out-of-: F system IEEE Trasactos o Relablty, 43(4), Xe, M (1987) O some mportace measures of system compoets Stochastc Process ad ther Applcatos, 25, Xe, M (1988) A ote o the atvg measure Scadava Joural of Statstcs, 15, Zhu, X, & Kuo, W (2014) Importace measures relablty ad mathematcal programmg Aals of Operatos Research, 212(1), Zhu, X, Boushaba, M, & Reghoua, M (2015) Jot relablty mportace a cosecutve-k-out-of-: F system ad a m-cosecutve-k-out-of-: F system for Markov-depedet compoets IEEE Trasactos o Relablty, 64(2), Zuo, M (1993) Relablty ad compoet mportace of a cosecutve-k-out-of- system Mcroelectrocs Relablty, 33(2),