Redundancy schemes for engineering coherent systems via a signature-based approach
aa r X i v : . [ s t a t . O T ] J u l Redundancy schemes for engineering coherentsystems via a signature-based approach
Mahdi Doostparast ∗ Department of Statistics, Faculty of Mathematical Sciences,Ferdowsi University of Mashhad, Mashhad, Iran
Abstract
This paper proposes a signature-based approach for solving redundancy allocation prob-lems when component lifetimes are not only heterogeneous but also dependent. The twocommon schemes for allocations, that is active and standby redundancies, are considered. Ifthe component lifetimes are independent, the proposed approach leads to simple manipula-tions. Various illustrative examples are also analysed. This method can be implemented forpractical complex engineering systems.
Key Words:
Coherent system; Redundancy; System signature; Stochastic orders.
Mathematics Subject Classification:
Redundancy policies are usually used to increase the reliabilities of engineering systems, if exist. Incommon, there are two schemes to allocate redundant components to the system, called “active”and “standby” redundancy allocations. In the former, the spars are put in parallel to the com-ponents of the system while in the later, the spars start functioning immediately after componentfailures. Determination of an optimal redundancy allocation in engineering systems is great ofinterest.There are many researches on the redundancy allocation problem (RAP) and deriving optimalallocations. For example, Boland et al. [5] considered some stochastic orders for a k -out-of- n sys-tem, which works whenever at least n ´ k ` n components work. They provedthat the optimal active redundancy policy always allocates the spare to the weakest component inseries systems if the component lifetimes are independent. For the standby redundancy and underthe likelihood ratio ordering, Boland et al. [5] provided also sufficient conditions in which forseries systems, the spare should be allocated to the weakest component while in parallel systems,it allocates the spare component to the strongest one. For recent results on RAP, see Singh andMisra [26], Singh and Singh [27], Mi [17], Valdes and Zequeira [28], Romera et al. [21], Hu andWang [9], Jeddi and Doostparast [10] and references therein.The concept of “signature” was introduced by Samaniego [22] and it is a useful tool for analysingstochastic behaviours of systems from a theoretical view of point. Precisely, let X “ p X , ¨ ¨ ¨ , X n q stand for absolutely continuous component lifetimes in a coherent system of order n and T “ φ p X q is the system lifetime and φ p . q stands for the “structure system function”. If the component life-times are independent and identically distributed (IID), then the system signature is the vector s “ p s , ¨ ¨ ¨ , s n q where s i “ P p T “ X i : n q and X i : n denotes i -th order statistics among X , ¨ ¨ ¨ , X n . ∗ Email address: [email protected] & [email protected] F T p t q “ P p T ą t q “ ř ni “ s i ¯ F i : n p t q where ¯ F i : n p t q “ P p X i : n ą t q for t ą
0. This paper suggests a signature-basedapproach for RAP. Therefore, the rest of this paper is organized as follow: In Section 2, the pro-posed signature-based is proposed for systems with independent but heterogeneous componentlifetimes. The RAP for systems with dependent component lifetimes is also studied in Section 3.Finally, Section 4 concludes. Illustrative examples are given throughout the paper.
In this section, we assume that the component lifetimes X , ¨ ¨ ¨ , X n are independent but hetero-geneous with respective reliability functions ¯ F , ¨ ¨ ¨ , ¯ F n , i.e.¯ F i p t q “ P p X i ą t q , @ t ą , i “ , ¨ ¨ ¨ , n. Let ¯ G p t q “ h ´ p H p ¯ F p t q , ¨ ¨ ¨ , ¯ F n p t qqq , @ t ą , (1)where H p p , ¨ ¨ ¨ , p n q is a multinomial expression, called “the structure reliability function” and h p p q “ H p p, ¨ ¨ ¨ , p q is the diagonal section of H p p , ¨ ¨ ¨ , p n q . Navarro et al. [19] proved that thereliability function of the system lifetime, ¯ F T p t q “ P p T ą t q , can be expressed as¯ F T p t q “ H p ¯ F p t q , ¨ ¨ ¨ , ¯ F n p t qq “ n ÿ i “ s i ¯ G i : n p t q , @ t ą , (2)where ¯ G i : n p t q stands for the reliability function of the i -th order statistics on the basis of a randomsample of size n from the distribution function (DF) G p t q “ ´ ¯ G p t q for t ą
0; that is¯ G i : n p t q “ i ´ ÿ j “ ˆ nj ˙ G p t q j ¯ G p t q n ´ j “ ´ E p n, i, G p t qq , t ą , i “ , ¨ ¨ ¨ , n. (3)where E p n, i, a q “ ř nj “ i ` nj ˘ a j p ´ a q n ´ j . In other words, for every given coherent system withindependent and heterogeneous component lifetimes, one can construct an equivalent coherentsystem with IID component lifetimes with the common reliability function (1).As a suggested procedure for comparing various redundancy allocation policies, one may derive anequivalent system with IID component lifetimes with the common reliability function (2) and thenuse the signature-based results for comparing systems. More precisely, assume one has k spars withlifetimes Y , ¨ ¨ ¨ , Y k and plan to allocate the spars to the n (original) components. She has also twopossible policies, say Policy I and Policy II. Under Policies I and II, the improved systems wouldhave lifetimes T r I s and T r II s with signatures s r I s “ p s r I s , ¨ ¨ ¨ , s r I s n ` k q and s r II s “ p s r II s , ¨ ¨ ¨ , s r II s n ` k q ,respectively. Analogously to Equation (1), let¯ G r I s p t q “ h ´ , r I s ´ H r I s p ¯ F p t q , ¨ ¨ ¨ , ¯ F n ` k p t qq ¯ , t ą , and¯ G r II s p t q “ h ´ , r II s ´ H r II s p ¯ F p t q , ¨ ¨ ¨ , ¯ F n ` k p t qq ¯ , t ą , where ¯ F n ` p t q , ¨ ¨ ¨ , ¯ F n ` k p t q call for the reliability functions of the spare lifetimes Y , ¨ ¨ ¨ , Y k . Here, H r I s ( H r II s ) and h r I s ( h r II s ) denote the structure system function and the reliability function of theimproved system, respectively, under Policy I (Policy II). Equation (2) yields for all t ą F T r I s p t q “ H r I s p ¯ F p t q , ¨ ¨ ¨ , ¯ F n ` k p t qq “ n ` k ÿ i “ s r I s i ¯ G r I s i : n ` k p t q , (4)2nd¯ F T r II s p t q “ H r II s p ¯ F p t q , ¨ ¨ ¨ , ¯ F n ` k p t qq “ n ` k ÿ i “ s r II s i ¯ G r II s i : n ` k p t q . (5)Now one can use the signature-based results for comparing the system lifetimes T r I s and T r II s withthe reliability functions (4) and (5), respectively. Here, some results which are useful in sequel arementioned. For more information, see Chapter 4 of Samaniego [23]. Definition 2.1.
Let X and Y be two random variables with reliability functions ¯ F and ¯ G , respec-tively. Then X is said to be smaller than Y in stochastic order (hazard rate order, likelihood rationorder), denoted by X ď st pď hr , ď lr q Y if ¯ F p t q ě ¯ G p t q for all t ( ¯ G p t q{ ¯ F p t q , ¯ g p t q{ ¯ f p t q is increasing in t ). Here, f and g are density functions of X and Y , respectively.In the sequel and for all orderings above-mentioned, the statements “ X ď Y ” and “ F ď G ”are used interchangeably. Theorem 2.2 (Samaniego [23], Chapter 4) . Let s i , i “ , , denote the i -th system signature withIID component lifetimes and the common reliability function ¯ G i .1. If ¯ G p t q “ ¯ G p t q for t ą and s ď st,hr,lr s then T ď st,hr,lr T ;2. If G ď st G and s “ s then T ď st T ; Example 2.3.
Consider a 2-component series system and k “ T r I s “ min t max t X , Y u , X u and T r II s “ min t X , max t X , Y uu . Boland etal. [5] proved that if the component lifetimes are independent and X ď st X then T r I s ě st T r II s .Notice that in this case, s r I s “ s r II s “ p { , { , q , H r I s p p , p , p q “ p ´ p ´ p qp ´ p qq p , H r II s p p , p , p q “ p p ´ p ´ p qp ´ p qq . (6)and hence h r I s p p q “ h r II s p p q “ p p ´ p ´ p q q for 0 ă p ă
1. Moreover,¯ F T r I s p t q “ H r I s p ¯ F p t q , ¯ F p t q , ¯ F p t qq “ p ´ F p t q F p t qq ¯ F p t q , (7)and¯ F T r II s p t q “ H r II s p ¯ F p t q , ¯ F p t q , ¯ F p t qq “ ¯ F p t qp ´ F p t q F p t qq . (8)The mathematical package MAPLE version 18 with procedure “SOLVE” gives h ´ , r I s p p q “ h ´ , r II s p p q“ b ´ p ` a p ´ p `
83 1 b ´ p ` a p ´ p ` , @ ă p ă . (9)From Equations (1) and (6), one can see that for all t ą G r I s p t q “ h ´ , r I s ´ H r I s p ¯ F p t q , ¨ ¨ ¨ , ¯ F n ` k p t qq ¯ “ h ´ , r I s ` p ´ F p t q F p t qq ¯ F p t q ˘ , ¯ G r II s p t q “ h ´ , r II s ´ H r II s p ¯ F p t q , ¨ ¨ ¨ , ¯ F n ` k p t qq ¯ “ h ´ , r II s ` ¯ F p t qp ´ F p t q F p t qq ˘ . Assume now that X ď st X or ¯ F p t q ď ¯ F p t q for all t . Since h ´ , r I s p p q “ h ´ , r II s p p q are increasingin p and after some algebraic calculations p ´ F p t q F p t qq ¯ F p t q ě ¯ F p t qp ´ F p t q F p t qq for all t ,then ¯ G r I s ě st ¯ G r II s and the above-mentioned result of Boland et al (1999) follows also by Part 2of Theorem 2.2. l Example 2.4.
Consider the five-component bridge system in Figure 1. We also have k “ s r i s , i “ , ¨ ¨ ¨ ,
5, denote the system signature when the spare has beenredundant with the i -th component. One can verify that s r s “ s r s “ s r s “ s r s “ ˆ , , , , , ˙ , s r s “ ˆ , , , , , ˙ . The reliability function of the bridge system (without the spare) is given by H p p , ¨ ¨ ¨ , p q “ p p ` p p ` p p p ` p p p ´ p p p p ´ p p p p ´ p p p p ´ p p p p ´ p p p p ` p p p p p . (10)Then the reliability function of the system with an active spare which has been allocated to the i -th component, for i “ , ¨ ¨ ¨ ,
5, is derived from (10) by replacing p i Y p : “ ´ p ´ p i qp ´ p q instead of p i . Let H r i s p p , ¨ ¨ ¨ , p q denote the reliability function of the system when the activespare is allocated to the i -th component. Then from (1), the common reliability function of theequivalent system with IID components is derived as¯ G r i s p t q “ h r i s , ´ p H r i s p ¯ F p t q , ¨ ¨ ¨ , ¯ F p t qqq , @ t ą , (11)where h r i s p p q “ H r i s p p, ¨ ¨ ¨ , p q . Using the mathematical package MAPLE version 18, we derived h r i s p p q “ ´ p ` p ´ p ` p ` p , i “ , ¨ ¨ ¨ , . (12)Now let X i , i “ , ¨ ¨ ¨ , F i p t q “ ´ exp t´ λ i t u and there is one active spare component ( k “
1) with the lifetime X . Similar toBoland et al. [5], the question is where to place the standby redundancy in order to make “best”improvement in the bridge system. The answer depends to the relative values of λ i , i “ , ¨ ¨ ¨ , F p t q “ ´ exp t´ λ t u for t ą λ i , i “ , ¨ ¨ ¨ ,
6. In some cases,one can not determine the optimal policy and another reliability index may used such as the meantime to failure (MTTF) of the improved system. For example, let W p α, λ q stand for the Weibulldistribution with DF F p t q “ ´ exp t´p λt q α u , t ą , λ ą , α ą . Figure 2 pictures the reliability function of the bridge system with one spare when the componentlifetime X i follows the Weibull distribution W p α i , q for i “ , ¨ ¨ ¨ ,
6. Here, α “ α “ α “ α “ α “ . α “
1. The calculations have been done by the mathematical package MAPLEversion 18. The corresponding program is given in the appendix. l The proposed signature-based approach may be applied also to standby redundancy policies.The next example illustrates this approach. In the example and hereafter, F ˚ G p t q means convo-lution of two DFs F and G , defined by F ˚ G p t q “ ż t ´8 G p t ´ x q dF p x q , λ λ λ λ λ λ optimal component for allocation1 1 1 2 2 1.5 41 1 1.5 2 2 1.5 41 1 2 2 2 1.5 41 1 1 2 2 1 41 1 1 2 2 1.5 41 1 1 2 2 2 41 2 1 1 2 1.5 21 2 1.5 1 2 1.5 21 2 2 1 2 1.5 22 1 1 1 2 1.5 22 1 1 2 1 1.5 1 or 43 1 1 2 1 1.5 1and F ˚ G p t q “ ´ F ˚ G p t q for all t . Example 2.5.
Consider again the 2-component series system and k “ T r I s “ min t X ` Y , X u and T r II s “ min t X , X ` Y u . Theorem3.2 of Boland et al. [5] states that if the component lifetimes are independent and X ď st X then T r I s ě st T r II s provided that the component lifetimes possess “the reverse rule of order 2property”; For more information, see the appendix and also Karlin [11]. Notice that in this caseone has two two-component series systems with the component reliability functions p F ˚ F , ¯ F q and p ¯ F , F ˚ F q under Policies I and II, respectively. So s r I s “ s r II s “ p , q , H r I s p p , p q “ H r II s p p , p q “ p p , h r I s p p q “ h r II s p p q “ p and then h ´ , r I s p p q “ h ´ , r II s p p q “ ? p . Equation (2)gives¯ F T r I s p t q “ H r I s p F ˚ F p t q , ¯ F p t qq “ F ˚ F p t q ¯ F p t q , t ą , (13)¯ F T r II s p t q “ H r II s p ¯ F p t q , F ˚ F p t qq “ ¯ F p t q F ˚ F p t q , t ą . (14)5quation (1) yields¯ G r I s p t q “ b F ˚ F p t q ¯ F p t q , (15)¯ G r II s p t q “ b ¯ F p t q F ˚ F p t q . (16)Therefore, for comparison purposes, one just needs to consider the reliability functions given byEquations (15) and (16) since both systems have identical system signatures. Then, if ¯ G r I s p t q ě ¯ G r II s p t q , for all t ą
0, then T r I s ě st T r II s by Part 2 of Theorem 2.2.For example, suppose X follows the exponential distribution with the DF F p t q “ ´ exp t´ t u , t ą X has the Pareto distribution with the DF F p t q “ ´ p ` t q ´ , t ą
0. Moreover, thespare lifetime Y „ F p t q “ ´ exp t´ t u , for t ą
0. Using the mathematical software MAPLEversion 18, the graphs of ¯ G r I s p t q and ¯ G r II s p t q , given by Equations (15) and (16), are pictured inFigure 3.Figure 3: Graphs of ¯ G r I s p t q (Dashed line) and ¯ G r II s p t q (Solid line) in Example 2.5.As one can see from Figure 3, G r I s ě st G r II s then T r I s ě st T r II s by Part 2 of Theorem 2.2.Note that in this example the family t F , F u does not posses the RR2 property. To see this, therespective densities are f p t q “ p ` t q ´ and f p t q “ exp t´ t u , for t ą
0. Let x “ x “ f p x q f p x q “ . ğ f p x q f p x q “ . l In practice, the system components may share the same environmental factors such as tempera-ture, pressure, loading and etc. Then, the component lifetimes are not independent, but ratherare “associated” and exhibit some dependency. Examples include structures in which componentsshare the load, so that failure of one component results in increased load on each of the remainingcomponents. For more information, see Barlow and Proschan [1] and Nelsen [20]. The RAP forsystems with dependent component lifetimes has not been extensively studied in literature. Amongfew works, Kotz et al. [12] investigated the increase in the mean lifetime for parallel redundancywhen the two component lifetimes are positive (negative) dependent. da Costa Bueno [6] definedthe concept of “minimal standby redundancy” and used the reverse rule of order 2 property be-tween component lifetimes to study the problem of RAP for k -out-of- n systems with dependentcomponents using a martingale approach. See also da Costa Bueno and Martins do Carmo [7].Belzunce et al. [2, 3] considered the RAP with dependent component lifetimes. For modelling6he dependency among component lifetimes and comparison purposes, they used the concept of“joint stochastic orders”. You and Li [29] extended the result of Boland et al. [5] from indepen-dent components to allocating m independent and identically distributed (IID) active redundancylifetimes to k -out-of- n system with components lifetimes having an arrangement increasing (AI)joint density. They proved that more redundancies should be allocated to the weaker componentto increase the reliability of the system. Recently, Jeddi and Doostparast [10] considered the RAPwithout any restriction to a special structure form for dependency among component lifetimes.Their conditions are expressed in terms of the joint distribution of the component lifetimes. Thissection deals with the problem of allocating spare components via the proposed signature-basedapproach for improving the system reliability in which the lifetimes of components are dependent.Navarro et al. [19] extended th representation (2) based on signatures to coherent systems withcomponent lifetimes that may be dependent. To describe the results, let T “ φ p X , ¨ ¨ ¨ , X n q be thelifetime of a coherent system with structure function φ and component lifetimes X , ¨ ¨ ¨ , X n withthe joint reliability function ¯ F X , ¨¨¨ ,X n p x , ¨ ¨ ¨ , x n q “ P p X ą x , ¨ ¨ ¨ , X n ą x n q . Sklar’s theorem(Nelsen, [20], p. 46) ensures¯ F X , ¨¨¨ ,X n p x , ¨ ¨ ¨ , x n q “ K ` ¯ F X p x q , ¨ ¨ ¨ , ¯ F X n p x n q ˘ , (17)where ¯ F X i p x i q “ P p X i ą x i q , for i “ , ¨ ¨ ¨ , n , is the marginal reliability function of componentlifetime X i and K is the survival copula. The coherent system lifetime T may be represented as T “ max ď j ď l X P j where X P j “ min i P P j X i and P , ¨ ¨ ¨ , P l stand for the all “minimal paths” ofthe system. For more information, see Barlow and Proschan [1]. Hence, the system reliabilityfunction can be written as (Navarro et al. [19])¯ F T p t q “ W ` ¯ F X p x q , ¨ ¨ ¨ , ¯ F X n p x n q ˘ , (18)where W p x , ¨ ¨ ¨ , x n q “ l ÿ j “ K p x P j q ´ ÿ i ă j K p x P i Ť P j q ` ¨ ¨ ¨ ` p´ q l ` K p x P Ť ¨¨¨ Ť P l q , and x P “ p z , ¨ ¨ ¨ , z n q with z i “ x i for i P P and z i “ i R P . Here W “ W p φ, K q is known as“structure-dependence function ”. In particular, if the component lifetimes are independent thenthe function W is equal to the structure reliability function H in Equation (2). Example 3.1 (Navarro et al. [19]) . Consider a system with lifetime T “ min p X , max p X , X qq .Then P “ t , u and P “ t , u . By Equation (18)¯ F T p t q “ W p ¯ F p t q , ¯ F p t q , ¯ F p t qq , (19)where W p x , x , x q “ K p x , x , q ` K p x , , x q ´ K p x , x , x q . Notice that if the componentlifetimes are independent, then K p x , x , x q “ x x x and W p x , x , x q “ x x ` x x ´ x x x . l Navarro et al. [19] proved that the system lifetime T is equal in law with T ‹ “ φ p Y , ¨ ¨ ¨ , Y n q where Y , ¨ ¨ ¨ , Y n are identically distributed with the joint reliability function P p Y ą x , ¨ ¨ ¨ , Y n ą x n q “ K p ¯ G W p x q , ¨ ¨ ¨ , ¯ G W p x n qq , (20)where ¯ G W p t q “ m W p ¯ F p t q , ¨ ¨ ¨ , ¯ F n p t qq and m W p x , ¨ ¨ ¨ , x n q is “the mean function” of W , definedby m W p x , ¨ ¨ ¨ , x n q “ δ ´ p W p x , ¨ ¨ ¨ , x n qq on the space r , s n and δ p x q “ W p x, ¨ ¨ ¨ , x q for x Pr , s . Moreover, if the survival copula K is exchangeable, then¯ F T p t q “ n ÿ i “ s i ¯ G i : n p t q , (21)7here ¯ G i : n p t q “ P p Y i : n ą t q and Y n ă ¨ ¨ ¨ ă Y n : n are the order statistics obtained from therandom variables Y , ¨ ¨ ¨ , Y n with the joint reliability function (20). The next theorem is valuablein RAP for coherent systems. Theorem 3.2 (Navarro et al. [19]) . If T is the lifetime of a coherent system with signature s “ p s , ¨ ¨ ¨ , s n q and with component lifetimes X , ¨ ¨ ¨ , X n having the structure-dependence function W , then ¯ F T p t q “ n ÿ i “ s i ¯ G i : n p t q , (22) where ¯ G i : n p t q “ P p Y i : n ą t q and Y n ă ¨ ¨ ¨ ă Y n : n stand for the order statistics obtained from theIID random variables Y , ¨ ¨ ¨ , Y n with the common reliability function ¯ G p t q “ h ´ p W p ¯ F X p t q , ¨ ¨ ¨ , ¯ F X n p t qqq and h is the reliability polynomial of the coherent system. Example 3.3.
Suppose that the component lifetimes in Example 2.3 are dependent and followthe Mardia tri-variate Pareto distribution with the joint reliability function (Mardia [14])¯ F X ,X ,X p x , x , x q “ ˆ x σ ` x σ ` x σ ´ ˙ ´ α , x i ą σ i ą , i “ , , , α ą . (23)Jeddi and Doostparast [10] proved that T r I s ě st T r II s provided X ď st X . Here a signature-based approach is utilized. To do this note that similar to Example 3.1, the reliability systemfunctions under Policies I and II, respectively, are given by ¯ F T r I s p t q “ W r I s p ¯ F p t q , ¯ F p t q , ¯ F p t qq and¯ F T r II s p t q “ W r II s p ¯ F p t q , ¯ F p t q , ¯ F p t qq , where W r I s p x , x , x q “ K p x , x , q ` K p , x , x q ´ K p x , x , x q , (24) W r II s p x , x , x q “ K p x , x , q ` K p x , , x q ´ K p x , x , x q , (25)with the survival copula K p x , x , x q “ ´ x ´ { α ` x ´ { α ` x ´ { α ´ ¯ ´ α , which known as “Clytoncopula”; See Nelsen [20] for more information. Notice that the popular form for the Clyton copulais as follow: K p x , x , x q “ ` x ´ θ ` x ´ θ ` x ´ θ ´ ˘ ´ { θ , θ ě . (26)By Theorem 3.2, the equivalent systems under Policies I and II with IID component lifetimes havethe common component reliability functions¯ G r I s p t q “ h ´ , r I s ´ W r I s p ¯ F p t q , ¯ F p t q , ¯ F p t qq ¯ “ h ´ , r I s ` K p ¯ F p t q , ¯ F p t q , q ` K p , ¯ F p t q , ¯ F p t qq ´ K p ¯ F p t q , ¯ F p t q , ¯ F p t qq ˘ , (27)¯ G r II s p t q “ h ´ , r II s ´ W r II s p ¯ F p t q , ¯ F p t q , ¯ F p t qq ¯ “ h ´ , r II s ` K p ¯ F p t q , ¯ F p t q , q ` K p ¯ F p t q , , ¯ F p t qq ´ K p ¯ F p t q , ¯ F p t q , ¯ F p t qq ˘ , (28)respectively, where the functions h ´ , r I s p p q and h ´ , r II s p p q are given by Equation (9). Since s r I s “ s r II s , one solely needs to compare the component reliability functions (27) and (28) and then usesPart 2 of Theorem 2.2. Therefore, T r I s ě st T r II s if and only if ¯ G r I s p t q ě ¯ G r II s p t q for all t ą h ´ , r I s p p q “ h ´ , r II s p p q are increasing in p . From (24)-(28) and aftersimple algebraic calculations, one can see that ¯ G r I s p t q ě ¯ G r II s p t q for all t ą F p t q ď ¯ F p t q for all t ą
0. Hence, T r I s ě st T r II s if and only if X ď st X as proved by Jeddi andDoostparast [10]. l K p x , x , x q “ exp ´ tp´ log x q γ ` p´ log x q γ ` p´ log x q γ u { γ ¯ , x i ą , i “ , , . (29)where “log” stands for the natural logarithm. Moreover, let ¯ F i p t q “ p ` t q i , t ą i “ , , G r I s p t q and ¯ G r II s p t q are derived. A graph ofthese functions may be useful to derive the optimal policy. This paper dealt with a signature-based approach for redundancy allocation problems when com-ponent lifetimes are either heterogeneous or dependent. An important part to implement thederived results is derivation of the system signature. There are some researches to obtain systemsignatures; See, e.g., Gertsbakh et al. [8], Marichal and Mathonet [15] and Navarro and Rubino[18] and references therein.
Acknowledgement
This research was supported by a grant from Ferdowsi University of Mashhad (No MS94329DSP).
References [1] Barlow, R. E., Proschan, F. (1975).
Statistical theory of reliability and life testing . Holt,Rinehart and Winston, Inc., New York.[2] Belzunce, F., Martinez-Puertas, H. Ruiz, J.M. (2011). On optimal allocation of redundantcomponents for series and parallel systems of two dependent components,
Journal of StatisticalPlanning and Inference, , 3094-3104.[3] Belzunce, F., Martinez-Puertas, H. Ruiz, J.M. (2013). On allocation of redundant componentsfor systems with dependent components,
European Journal of Operational Research, , 573-580.[4] Boland, P.J., EI-Neweihi, E., Porschan, F. (1992). Stochastic order for redundancy alloca-tions in series and parallel systems,
Advances in Applied Probability, , 161-171.[5] Boland, P.J., EI-Neweihi, E., Porschan, F. (1992). Stochastic order for redundancy alloca-tions in series and parallel systems, Advances in Applied Probability, , 161-171.[6] da Costa Bueno, V., (2005). Minimal standby redundancy allocation in a k -out-of- n : F systemof dependent components. European Journal of Operational Research, , 786-793.[7] da Costa Bueno, V., Martins do Carmo, I. (2007). Active redundancy allocation for a k -out-of- n : F system of dependent components. European Journal of Operational Research, ,1041-1051.[8] Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2011). Signatures of coherent systems builtwith separate modules.
Journal of Applied Probability, , 843-855.99] Hu, T., Wang, Y. (2009). Optimal allocation of active redundancies in r-out-of-n systems. Journal of Statistical Planning and Inference, , 3733-3737.[10] Jeddi, M., Doostparast, M. (2016). Optimal redundancy allocation problems in engineeringsystems with dependent component lifetimes.
Applied Stochastic Models in Business andIndustry, , 199-208.[11] Karlin, S. (1968). Total Positivity . Stanford University Press.[12] Kotz, S., Lai, C. D., Xie, M., (2003). On the effect of redundancy for systems with dependentcomponents.
IIE Transactions, , 1103-1110.[13] Lehmann, E. L. (1966). Some concepts of dependence. The Annals of Mathematical Statistics, , 1137-1153.[14] Mardia, K. V. (1962). Multivariate Pareto distributions. The Annals of Mathematical Statis-tics , 1008-1015.[15] Marichal, J.-L. and Mathonet, P. (2013). Computing system signatures through reliabilityfunctions. Statistics and Probability Letters , 710-717.[16] Meeker, W. Q., Escobar, L. A. (1998). Statistical Methods for Reliability Data . John Wiley& Sons, New York.[17] Mi, J. (1999). Optimal active redundancy,
Journal of Applied Probability, , 1004-1014.[18] Navarro, J., and Rubino, R. (2010). Computations of Signatures of Coherent Systems withFive Components. Communications in Statistics-Simulation and Computation, , 68-84.[19] Navarro, J., Samaniego, F. J., Balakrishnan, N., (2011). Signature-based representations forthe reliability of systems with heterogeneous components. Journal of Applied Probability, ,856-867.[20] Nelsen, R. B. (2006). An introduction to copulas . Springer, New York.[21] Romera, R., Valdes, J.E., Zequeira, R.I., (2003). Active-redundancy allocation in systems.
IEEE Transactions on Reliability, , 313-318.[22] Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Transactions on Information Theory, R- , 6972.[23] Samaniego, F. J. (2007). System signatures and their applications in engineering reliability .Springer Sciences+Business Media, LLC, New York.[24] Shanthikumar, J. G., Yao, D.D., 1991. Bivariate Characterization of some stochastic orderrelations.
Advances in Applied Probability, , 642-659.[25] Shaked, M., Shanthikumar, J. G. (2007). Stochastic Orders . Springer-Verlag, New York.[26] Singh, H., Misra, N. (1994). On redundancy allocations in systems,
Journal of AppliedProbability , , 1004-1014.[27] Singh, H., Singh, R. S. (1997). Note: optimal allocation of resources to nodes of series systemswith respect to failure rate ordering, Naval Research Logistics, , 147-152.[28] Valdes, J. E., Zequeira, R. I. (2003). On the optimal allocation of an active redundancy in atwo-component series system, Statistics and Probability Letters, , 325-332.[29] You, Y., Li, X. (2014). On allocating redundancies to k -out-of- n reliability systems, AppliedStochastic Models in Business and Industry, , 361-371.10 ppendix The reverse rule of order 2 property
Definition 4.1.
The function g p θ, x q has the reverse rule of order 2 (denoted by RR2) propertyin θ and x if for θ ą θ and x ă x , g p θ , x q g p θ , x q ě g p θ , x q g p θ , x q . Many one-parameter families of density functions t g p θ, x q : “ f θ p x q , θ P Θ u of life distributionspossess the RR2 property. Examples include: • Gamma distribution Γ p m, λ q with density f m,λ p x q “ λ m Γ p m q x m ´ exp t´ λx u , x ą . The families t Γ p m, θ q , θ ą u (fixed shape parameter) and t Γ p θ ´ , λ q , θ ą u (fixed scaleparameter); • Weibull distribution with density f θ p x q “ αθx α ´ exp t´ θx α u , for λ ą
0, is RR2 with fixedshape parameter α ; • Pareto family of densities t f θ p x q “ θ p ` x q ´p θ ` q , θ ą u . MAPLE codes for the bridge example • H(p1,p2,p3,p4, p5):=p1* p3+p2* p4+p1* p4* p5+p2* p3* p5 -p1* p2* p3* p4-p1* p3* p4*p5-p1* p2* p3* p5-p2* p3* p4* p5-p1* p2* p4* p5 +2 p1* p2* p3* p4* p5); h(p):=H(p,p,p,p,p); • H1(p1,p2,p3,p4, p5,p6):=H(1-(1- p1)*(1-p6),p2,p3,p4, p5); h1(p):=H1(p,p,p,p, p,p); • H2(p1,p2,p3,p4, p5,p6):=H(p1,1-(1- p2)*(1-p6),p3,p4, p5); h2(p):=H2(p,p,p,p, p,p) ; • H3(p1,p2,p3,p4, p5,p6):=H(p1,p2,1-(1- p3)*(1-p6),p4, p5); h3(p):=H3(p,p,p,p, p,p) ; • H4(p1,p2,p3,p4, p5,p6):=H(p1,p2,p3,1-(1- p4)*(1-p6), p5); h4(p):=H4(p,p,p,p, p,p); • H5(p1,p2,p3,p4, p5,p6):=H(p1,p2,p3,p4,1-(1- p5)*(1-p6)); h5(p):=H5(p,p,p,p, p,p); • simplify(h1(p)); simplify(h2(p)); simplify(h3(p)); simplify(h4(p)); simplify(h5(p)); • f := x Ñ ´ ˚ x ` ˚ x ´ ˚ x ` ˚ x ` ˚ x ;solve(x = f(y), y) assuming 0 ď y ď ď x ď • F1bar := t Ñ exp t´p λ t q α u ;F2bar := t Ñ exp t´p λ t q α u ;F3bar := t Ñ exp t´p λ t q α u ;F4bar := t Ñ exp t´p λ t q α u ;F5bar := t Ñ exp t´p λ t q α u ;F6bar := t Ñ exp t´p λ t q α u ; • n := 5; s1245 := (0, 1/15, 7/30, 1/2, 1/5, 0); s3 :=( 0, 2/15, 4/15, 7/15, 2/15, 0); λ : “ λ : “ λ : “ λ : “ λ : “ λ : “ α : “ α : “ . α : “ α : “ α : “ . α : “ • E(n,i,a):= ř nj “ i a j p ´ a q n ´ j Gbar1nPolicy1 := 1-E(n, 1, 1-g(H1(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar2nPolicy1 := 1-E(n, 2, 1-g(H1(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar3nPolicy1 := 1-E(n, 3, 1-g(H1(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar4nPolicy1 := 1-E(n, 4, 1-g(H1(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar5nPolicy1 := 1-E(n, 5, 1-g(H1(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar1nPolicy2 := 1-E(n, 1, 1-g(H2(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar2nPolicy2 := 1-E(n, 2, 1-g(H2(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar3nPolicy2 := 1-E(n, 3, 1-g(H2(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar4nPolicy2 := 1-E(n, 4, 1-g(H2(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar5nPolicy2 := 1-E(n, 5, 1-g(H2(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar1nPolicy3 := 1-E(n, 1, 1-g(H3(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar2nPolicy3 := 1-E(n, 2, 1-g(H3(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar3nPolicy3 := 1-E(n, 3, 1-g(H3(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar4nPolicy3 := 1-E(n, 4, 1-g(H3(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar5nPolicy3 := 1-E(n, 5, 1-g(H3(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar1nPolicy4 := 1-E(n, 1, 1-g(H4(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar2nPolicy4 := 1-E(n, 2, 1-g(H4(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar3nPolicy4 := 1-E(n, 3, 1-g(H4(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar4nPolicy4 := 1-E(n, 4, 1-g(H4(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar5nPolicy4 := 1-E(n, 5, 1-g(H4(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar1nPolicy5 := 1-E(n, 1, 1-g(H5(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar2nPolicy5 := 1-E(n, 2, 1-g(H5(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar3nPolicy5 := 1-E(n, 3, 1-g(H5(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar4nPolicy5 := 1-E(n, 4, 1-g(H5(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t))));Gbar5nPolicy5 := 1-E(n, 5, 1-g(H5(F1bar(t), F2bar(t), F3bar(t), F4bar(t), F5bar(t), F6bar(t)))) • FbarSystemPolicy1 := s1245[1]*Gbar1nPolicy1(t)+s1245[2]*Gbar2nPolicy1(t)+s1245[3]*Gbar3nPolicy1(t)+s1245[4]*Gbar4nPolicy1(t)+s1245[5]*Gbar5nPolicy1(t);FbarSystemPolicy2 := s1245[1]*Gbar1nPolicy2(t)+s1245[2]*Gbar2nPolicy2(t)+s1245[3]*Gbar3nPolicy2(t)+s1245[4]*Gbar4nPolicy2(t)+s1245[5]*Gbar5nPolicy2(t);FbarSystemPolicy3 := s3[1]*Gbar1nPolicy3(t)+s3[2]*Gbar2nPolicy3(t)+s3[3]*Gbar3nPolicy3(t)+s3[4]*Gbar4nPolicy3(t)+s3[5]*Gbar5nPolicy3(t);FbarSystemPolicy4 := s1245[1]*Gbar1nPolicy4(t)+s1245[2]*Gbar2nPolicy4(t)+s1245[3]*Gbar3nPolicy4(t)+s1245[4]*Gbar4nPolicy4(t)+s1245[5]*Gbar5nPolicy4(t);FbarSystemPolicy5 := s1245[1]*Gbar1nPolicy5(t)+s1245[2]*Gbar2nPolicy5(t)+s1245[3]*Gbar3nPolicy5(t)+s1245[4]*Gbar4nPolicy5(t)+s1245[5]*Gbar5nPolicy5(t) ••