Allocations of Cold Standbys to Series and Parallel Systems with Dependent Components
aa r X i v : . [ s t a t . O T ] N ov Allocations of Cold Standbys to Series and ParallelSystems with Dependent Components ∗ Xiaoyu ZhangSchool of Mathematics and StatisticsJiangsu Normal University, Xuzhou 221116, P. R. ChinaYiying ZhangSchool of Statistics and Data Science, LPMC and KLMDASRNankai University, Tianjin 300071, P. R. China
E-mail: [email protected]
Rui FangDepartment of MathematicsShantou University, Shantou 515063, P. R. ChinaNovember 25, 2019
Abstract
In the context of industrial engineering, cold-standby redundancies allocation strategyis usually adopted to improve the reliability of coherent systems. This paper investigatesoptimal allocation strategies of cold standbys for series and parallel systems comprised of de-pendent components with left/right tail weakly stochastic arrangement increasing lifetimes.For the case of heterogeneous and independent matched cold standbys, it is proved thatbetter redundancies should be put in the nodes having weaker [better] components for series[parallel] systems. For the case of homogeneous and independent cold standbys, it is shownthat more redundancies should be put in standby with weaker [better] components to en-hance the reliability of series [parallel] systems. The results developed here generalize andextend those corresponding ones in the literature to the case of series and parallel systemswith dependent components. Numerical examples are also presented to provide guidance forthe practical use of our theoretical findings.
Keywords:
Cold standby; Series system; Parallel system; LWSAI; RWSAI; Stochastic or-ders. ∗ Corresponding author: Yiying Zhang. E-mail: [email protected] SC 2010:
Primary 90B25, Secondary 60E15, 60K10
In reliability engineering and system security, one common way to optimize system performanceis to introduce redundancies (or spares) to the components. Three types of redundancies areusually used to improve the performance of reliability systems, i.e., the hot standby, the coldstandby, and the warm standby. For the hot standby, available spares are put in parallel with theoriginal components and function simultaneously with them. For the cold standby, concernedspares are attached to the components of the system in such a way that the redundanciesstart to work immediately after the failures of the original components. The warm standbyis a redundancy type between the hot standby and the cold standby. For the case of warmstandby, the redundant components may fail to work even before its operation; indeed, thewarm redundancies are more likely to be damaged than the cold standbys. Moreover, the failurerate of an inactive warm standby is less than its actual failure rate, and then it is switched tothe active state immediately after the failure of the original component and its failure rate willincrease. Interested readers may refer to Yun and Cha [43], Hazra and Nanda [23], Finkelsteinet al. [17], and Hadipour et al. [19] for more detailed discussions. For all of these three types ofstandbys, the system performance under different allocation policies can be effectively evaluatedvia stochastic comparisons in terms of various stochastic orders.A system is said to be coherent if each component is relevant and the structure function isincreasing in its components (c.f. Barlow and Proschan, [2]). The past several decades havewitnessed comprehensive developments on investigating optimal allocation policies of hot stand-bys (i.e., active redundancies) for coherent systems (especially k -out-of- n systems) consistingof independent components; see for example Boland et al. [5, 6], Singh and Misra [36], Vald´esand Zequeira [38], Vald´es et al. [37], Brito et al. [7], Misra et al. [30], Hazra and Nanda [22],Zhao et al. [48, 49, 50, 51], Da and Ding [12], Ding et al. [13], and Zhang [44]. On the otherhand, some research work has appeared on redundancies allocation for coherent systems withdependent components. For instance, Belzunce et al. [3] and Belzunce et al. [4] consideredhot standbys allocation for k -out-of- n systems comprised of statistically dependent componentswith their lifetimes characterized by joint stochastic orders (c.f. Shanthikumar and Yao, [35]).Interested readers may refer to You and Li [41], You et al. [40], and Zhang et al. [45] for morestudy along this direction.On account of the complexity of distribution theory, there is not much work on studyingoptimal allocation strategies of cold-standby redundancies. Boland et al. [6] might be the firstto investigate how to optimally assign cold-standby redundancies to series and parallel systems.They showed that the optimal allocation strategy for a series system is opposite to that for aparallel system when the original components and the redundancies are i.i.d. After that, many2esearchers have paid attention to the allocation problem of cold-standby redundancies in seriesand parallel systems; see for instance Singh and Misra [36], Li et al. [29], Misra et al. [31],Zhuang and Li [52], Doostparast [14], and Chen et al. [10]. Another research stream focuseson the effects of cold-standby redundancies on the performance of coherent systems. For moredetails, readers are referred to da Costa Bueno [11], Ardakan and Hamadani [1], Eryilmaz [15],and Gholinezhad and Hamadani [18], Eryilmaz and Erkan [16], and the references therein.To the best of our knowledge, rare work exists on studying optimal allocation of cold-standbyredundancies for systems with interdependent components except Belzunce et al. [4] and Jeddiand Doostparast [25]. Belzunce et al. [4] established the optimal allocation policy for onecold-standby redundancy in series and parallel systems by means of the stochastic precedenceorder and the usual stochastic order when the lifetimes of the original components are orderedvia joint stochastic orders. Jeddi and Doostparast [25] studied the same allocation problem byemploying quadratic dependence orderings (see Shaked and Shanthikumar, [34]). However, noneof these results treats the case of more than two cold-standby redundancies. The objective of thepresent paper is to fill this gap through pinpointing optimal allocation strategies of cold-standbyredundancies for series and parallel systems with heterogeneous and dependent components.It should be remarked that our method is quite different with those of Belzunce et al. [4] andJeddi and Doostparast [25]. At the preliminary working stage of a system with cold-standbyredundancies, it is reasonable to assume that the original components are positively interde-pendent (due to the external stress or shock and common environment), while the cold-standbyredundancies are assumed to be independent since they are not activated before the failures oforiginal components. In this paper, we assume the components of series/parallel systems arepositively dependent and have left tail weakly stochastic arrangement increasing (LWSAI) orright tail weakly stochastic arrangement increasing (RWSAI) lifetimes. The concerned cold-standby spares are assumed to be statistically independent and they are also independent of theoriginal components. Several stochastic orders including the usual stochastic order, the increas-ing convex order, and the increasing concave order are employed to derive the optimal allocationpolicies. More explicitly, for the case of heterogeneous and independent matched cold-standbyredundancies, we prove that the better redundancy should be put in the node with weaker[better] component for a series [parallel] system. For the case of homogeneous and independentcold-standby redundancies, we show that more redundancies should be allocated to the weaker[better] component to enhance the reliability of a series [parallel] system. The results developedhere generalize and extend those related ones in Singh and Misra [36], Misra et al. [31], Belzunceet al. [4], and Jeddi and Doostparast [25].The remainder of the paper is rolled out as follows. Section 2 recalls some pertinent notionsand definitions used in the sequel. In Section 3, optimal allocation strategies of heterogeneousand independent matched cold-standby redundancies are presented for series and parallel systemscomprised of LWSAI or RWSAI components. In Section 4, optimal allocations are investigated3or the case of a batch of i.i.d. cold-standby redundancies in series and parallel systems. Section5 concludes the paper. Throughout, the terms increasing and decreasing are used in a non-restrict sense. Let R =( −∞ , + ∞ ), R + = [0 , + ∞ ), and N = { , , , , . . . } . All random variables are assumed to benon-negative, and all expectations are well defined whenever they appear. Definition 2.1
For any two non-negative random variables X and Y , let f X and f Y , F X and F Y , F X and F Y be their density, distribution, and survival functions, respectively. Then, X issaid to be smaller than Y in the(i) likelihood ratio order (denoted by X ≤ lr Y ) if f Y ( x ) /f X ( x ) is increasing in x ∈ R + ;(ii) hazard rate order (denoted by X ≤ hr Y ) if F Y ( x ) /F X ( x ) is increasing in x ∈ R + ;(iii) reversed hazard rate order (denoted by X ≤ rh Y ) if F Y ( x ) /F X ( x ) is increasing in x ∈ R + ;(iv) usual stochastic order (denoted by X ≤ st Y ) if F X ( x ) ≤ F Y ( x ) for all x ∈ R + ;(v) increasing convex order (denoted by X ≤ icx Y ) if E [ φ ( X )] ≤ E [ φ ( Y )] for any increasingand convex function φ ;(vi) increasing concave order (denoted by X ≤ icv Y ) if E [ φ ( X )] ≤ E [ φ ( Y )] for any increasingand concave function φ . It is well known that X ≤ lr Y = ⇒ X ≤ hr [or rh] Y = ⇒ X ≤ st Y = ⇒ X ≤ icx [or icv] Y. One may refer to Shaked and Shanthikumar [34] for comprehensive discussions on the propertiesand applications of above mentioned stochastic orders.Next, we recall several useful dependence notions of arrangement increasing (AI). For anyfunction g : R n R and any pair ( i, j ) such that 1 ≤ i < j ≤ n , let G i,js ( n ) = { g ( x ) : g ( x ) ≥ g ( τ i,j ( x )) for any x i ≤ x j } , G i,jl ( n ) = { g ( x ) : g ( x ) − g ( τ i,j ( x )) is decreasing in x i ≤ x j } , and G i,jr ( n ) = { g ( x ) : g ( x ) − g ( τ i,j ( x )) is increasing in x j ≥ x i } , where τ i,j ( x ) denotes the permutation of x with its i -th and j -th components exchanged. Definition 2.2
A random vector X = ( X , . . . , X n ) is said to be i) stochastic arrangement increasing (SAI) if E [ g ( X )] ≥ E [ g ( τ i,j ( X ))] for any g ∈ G i,js ( n ) and all pair ( i, j ) such that ≤ i < j ≤ n ;(ii) left tail weakly stochastic arrangement increasing (LWSAI) if E [ g ( X )] ≥ E [ g ( τ i,j ( X ))] forany g ∈ G i,jl ( n ) and all pair ( i, j ) such that ≤ i < j ≤ n ;(iii) right tail weakly stochastic arrangement increasing (RWSAI) if E [ g ( X )] ≥ E [ g ( τ i,j ( X ))] for any g ∈ G i,jr ( n ) and all pair ( i, j ) such that ≤ i < j ≤ n . Since G i,js ( n ) ⊃ G i,jl ( n ) [ G i,jr ( n )], SAI implies both LWSAI and RWSAI. Multivariate versionsof Dirichlet distribution, inverted Dirichlet distribution, F distribution, and Pareto distributionof type I (see Hollander et al., [24]) are all SAI and hence are RWSAI and LWSAI whenever thecorresponding parameters are arrayed in the ascending order. SAI, LWSAI, and RWSAI wereproposed by Cai and Wei [8, 9] and have been applied in the fields of financial engineering andactuarial science; see for example Cai and Wei [9], Zhang and Zhao [47], You and Li [42], andZhang et al. [46]. According to Cai and Wei [8, 9], the following chain of implications alwaysholds: SAI = ⇒ LWSAI [RWSAI] = ⇒ X ⊥ ≤ st · · · ≤ st X ⊥ n , (1)where X ⊥ , . . . , X ⊥ n are the independent version of X . In this paper, we shall employ these usefulnotions to characterize the dependence structure of components lifetimes in series and parallelsystems.The joint multivariate likelihood ratio order and the joint multivariate reversed hazard rateorder , which were introduced by Shanthikumar and Yao [35], compare random variables bytaking into account the statistical dependence. These two types of joint stochastic orders areequivalent to SAI and LWSAI, respectively. Therefore, the results established in Theorems 3.2and 3.7 cover Theorem 3.3(b) and Theorem 3.5 of Belzunce et al. [4], respectively. Also, forabsolutely continuous random vectors, LWSAI coincides with LTPD according to Proposition3.7 of Cai and Wei [9]. For comprehensive treatments on other interesting higher joint stochasticorders, please refer to Wei [39].For a random vector X = ( X , . . . , X n ) with joint distribution function H and univariatemarginal distribution functions F , . . . , F n , its copula is a distribution function C : [0 , n [0 , H ( x ) = C ( F ( x ) , . . . , F n ( x n )) . Similarly, a survival copula is a distribution function C : [0 , n [0 , H ( x ) = P ( X > x , . . . , X n > x n ) = C ( F ( x ) , . . . , F n ( x n )) , where F i = 1 − F i , for i = 1 , . . . , n , are marginal survival functions and H ( x ) is the joint survivalfunction. The copula does not include any information of marginal distributions, and thus itprovides us a particularly convenient way to impose a dependence structure on predetermined5arginal distributions in practice. Archimedean copulas are rather popular due to its mathe-matical tractability and the capability of capturing wide ranges of dependence. By definition, fora decreasing and continuous function φ : [0 , + ∞ ) [0 ,
1] such that φ (0) = 1 and φ (+ ∞ ) = 0, C φ ( u , · · · , u n ) = φ n X i =1 φ − ( u i ) ! , for u i ∈ [0 , i = 1 , , . . . , n, is called an Archimedean copula with the generator φ if ( − k φ ( k ) ( x ) ≥ k = 0 , . . . , n − − n − φ ( n − ( x ) is decreasing and convex. The Archimedean family contains many well-known copulas, including the independence (or product) copula, the Clayton copula, and theAli-Mikhail-Haq (AMH) copula. For more detailed study on the properties of copulas, one mayrefer to Nelsen [32].In accordance with Theorem 5.7 of Cai and Wei [8], X is RWSAI if X ≤ hr · · · ≤ hr X n andare connected with an Archimedean survival copula with log-convex generator. Proposition 4.1of Cai and Wei [9] shows that X is LWSAI if X ≤ rh · · · ≤ rh X n and share an Archimedeancopula with log-convex generator. According to Corollary 8.23(b) of Joe [26], X is positive lowerorthant dependent (PLOD) if the generator is log-convex, which indicates some kind of positivedependence structure. For two non-negative n -dimensional real vectors x and y , we denote min( x + y ) = min( x + y , . . . , x n + y n ), max( x + y ) = max( x + y , . . . , x n + y n ), and the sub-vector x { i,j } = ( x , . . . , x i − , x i +1 , . . . , x j − , x j +1 , . . . , x n ) , for 1 ≤ i < j ≤ n .Consider a system consisting of heterogeneous components C , . . . , C n with C i having lifetime X i ,for i = 1 , . . . , n . Let Y i be the lifetime of cold-standby redundancy R i allocated to component C i ,for i = 1 , . . . , n . Then, the lifetimes of the resulting series and parallel systems can be denotedby min( X + Y ) and max( X + Y ), respectively. Hereafter, it is assumed that X i ’s are dependentthrough SAI, LWSAI or RWSAI, while the lifetimes Y i ’s are assumed to be independent andthey are also independent of X i ’s. This subsection deals with optimal allocations of n heterogeneous and independent redundanciesto a series system consisting of n dependent components. To begin with, let us introduce a usefullemma presenting functional characterizations of LWSAI and RWSAI bivariate random vectors. Lemma 3.1 (You and Li [42])
A bivariate random vector ( X , X ) is LWSAI [RWSAI] if andonly if E [ g ( X , X )] ≥ E [ g ( X , X )] for all g and g such that i) g ( x , x ) − g ( x , x ) is decreasing [increasing] in x ≤ x [ x ≥ x ] for any x [ x ];(ii) g ( x , x ) + g ( x , x ) ≥ g ( x , x ) + g ( x , x ) for any x ≥ x . Now, we present the first main result.
Theorem 3.2
Suppose that Y ≥ lr Y ≥ lr · · · ≥ lr Y n . For any ≤ i < j ≤ n ,(i) if X is LWSAI, then min( X + Y ) ≥ icv min( X + τ i,j ( Y )) ;(ii) if X is RWSAI, then min( X + Y ) ≥ st min( X + τ i,j ( Y )) . Proof.
Let p i ( · ) be the density function of Y i , for i = 1 , . . . , n . For any 1 ≤ i < j ≤ n and anyintegrable function u , it is easy to verify that E [ u (min( X + Y ))] − E [ u (min( X + τ i,j ( Y )))]= Z · · · Z R n − n Y k = i,j p k ( y k )d y k Z Z y i ≤ y j E [ u (min( X + y ))] p i ( y i ) p j ( y j )d y i d y j + Z · · · Z R n − n Y k = i,j p k ( y k )d y k Z Z y i ≤ y j E [ u (min( X + τ i,j ( y )))] p i ( y j ) p j ( y i )d y i d y j − Z · · · Z R n − n Y k = i,j p k ( y k )d y k Z Z y i ≤ y j E [ u (min( X + τ i,j ( y )))] p i ( y i ) p j ( y j )d y i d y j − Z · · · Z R n − n Y k = i,j p k ( y k )d y k Z Z y i ≤ y j E [ u (min( X + y ))] p i ( y j ) p j ( y i )d y i d y j = Z · · · Z R n − n Y k = i,j p k ( y k )d y k Z Z y i ≤ y j [ E [ u (min( X + y ))] − E [ u (min( X + τ i,j ( y )))]] × [ p i ( y i ) p j ( y j ) − p i ( y j ) p j ( y i )]d y i d y j . (2)In the next, we prove that the integrand of (2) is non-negative. First, upon using the condition Y i ≥ lr Y j for i < j , we know that p i ( y ) /p j ( y ) is increasing in y ∈ R + . Thus, it must hold that p i ( y i ) p j ( y j ) ≤ p i ( y j ) p j ( y i ) , y i ≤ y j . (3)Now, it suffices to show that E [ u (min( X + y ))] ≤ E [ u (min( X + τ i,j ( y )))] , y i ≤ y j , (4)for increasing concave u under (i) and increasing u under (ii).Proof of (i): X is LWSAI. In light of Proposition 3.2 of Cai and Wei [9], the LWSAI propertyof X implies that [( X i , X j ) | X { i,j } ] is LWSAI. For any given X { i,j } = x { i,j } , let g ( x i , x j ) = u (min( x + y )) = u (min { x i + y i , x j + y j , min { ( x + y ) { i,j } }} )7nd g ( x i , x j ) = u (min( x + τ i,j ( y ))) = u (min { x i + y j , x j + y i , min { ( x + y ) { i,j } }} ) . Thus, for any y i ≤ y j and x i ≤ x j we have∆ ( x i , x j ) = g ( x i , x j ) − g ( x i , x j )= u (min { x i + y j , x j + y i , min { ( x + y ) { i,j } }} ) − u (min { x i + y i , min { ( x + y ) { i,j } }} ) . (5)On the one hand, it can be checked that, for any x j ≥ x i , g ( x i , x j ) + g ( x j , x i ) = g ( x i , x j ) + g ( x j , x i ) . (6)On the other hand, in order to prove that g ( x i , x j ) − g ( x i , x j ) is decreasing in x i ≤ x j for anyincreasing concave u , the following several cases are considered.Case 1: x i + y j ≥ x j + y i . For this case, it is clear that∆ ( x i , x j ) = u (min { x j + y i , min { ( x + y ) { i,j } }} ) − u (min { x i + y i , min { ( x + y ) { i,j } }} )is decreasing in x i ≤ x j for any x j and any increasing concave u .Case 2: x i + y j < x j + y i . For this case, we have∆ ( x i , x j ) = u (min { x i + y j , min { ( x + y ) { i,j } }} ) − u (min { x i + y i , min { ( x + y ) { i,j } }} ) . Note that x i + y j ≥ x i + y i for any y j ≥ y i . The proof can be obtained from the following threespecial cases.Subcase 1: min { ( x + y ) { i,j } }} ≤ x i + y i . Clearly, it holds that ∆ ( x i , x j ) = 0, and the proofis trivial.Subcase 2: x i + y i < min { ( x + y ) { i,j } }} ≤ x i + y j . For this case, it is clear to see that∆ ( x i , x j ) = u (min { ( x + y ) { i,j } } ) − u ( x i + y i )is decreasing in x i ≤ x j .Subcase 3: min { ( x + y ) { i,j } }} > x i + y j . Note that∆ ( x i , x j ) = u ( x i + y j ) − u ( x i + y i ) . By the increasing concavity of u , it holds that, for any x ′ i ≤ x i ≤ x j ,∆ ( x ′ i , x j ) = u ( x ′ i + y j ) − u ( x ′ i + y i ) ≥ u ( x i + y j ) − u ( x i + y i ) = ∆ ( x i , x j ) , which means that ∆ ( x i , x j ) is decreasing in x i ≤ x j for any x j .To sum up, we have shown that g ( x i , x j ) − g ( x i , x j ) is decreasing in x i ≤ x j for anyincreasing concave u . Now, upon applying Lemma 3.1, it follows that E [ u (min( X + y )) | X { i,j } = x { i,j } ] ≤ E [ u (min( X + τ i,j ( y ))) | X { i,j } = x { i,j } ] . (7)8y applying iterated expectation formula on inequality (7), (4) is obtained. Upon combining(4) with (3), (2) is non-negative for y i ≤ y j . Thus, the proof is finished.Proof of (ii): X is RWSAI. According to Proposition 3.9(ii) of Cai and Wei [8], the RWSAIproperty of X implies that [( X i , X j ) | X { i,j } ] is RWSAI. For any y i ≤ y j and x i ≤ x j , ∆ ( x i , x j )given in (5) is increasing in x j ≥ x i by using the increasing property of u . By adopting a similarproof method as in (i), the desired result can be reached.For a series system with components C , . . . , C n having LWSAI [RWSAI] lifetimes, Theorem3.2 suggests that the redundancy R i should be put in standby with C n − i +1 , for i = 1 , . . . , n , inthe sense of the increasing concave [usual stochastic] ordering if the redundancies lifetimes Y i ’sare such that Y ≥ lr · · · ≥ lr Y n .The following example adopts Monte Carlo method to validate the result of Theorem 3.2. n b E n,S ( X , Y ) b E n,S ( X , τ , ( Y )) (a) u ( x ) = x . n b E n,S ( X , Y ) b E n,S ( X , τ , ( Y )) (b) u ( x ) = x . n ˆ E n,S ( X , Y )ˆ E n,S ( X , τ , ( Y )) (c) u ( x ) = (1 − e − x ) / n ˆ E n,S ( X , Y )ˆ E n,S ( X , τ , ( Y )) (d) u ( x ) = log x Figure 1: Plots of the estimators b E n,S ( X , Y ) and b E n,S ( X , τ , ( Y )). Example 3.3
Assume the lifetime vector X = ( X , X ) is assembled with Clayton [survival]copula with generator φ ( t ) = ( t +1) − , and X and X have exponential distributions with hazard ates λ = 0 . and λ = 0 . , respectively. Let Y and Y be two independent exponential randomvariables with respective hazard rates µ = 0 . and µ = 0 . . Clearly, X ≤ lr X and Y ≥ lr Y .Besides, it is easy to check that X is LWSAI [RWSAI]. For any increasing function u , we denote E u,S ( X , Y ) := E [ u (min { X + Y , X + Y } )] and E u,S ( X , τ , ( Y )) := E [ u (min { X + Y , X + Y } )] . From the population ( X , X , Y , Y ) , we generate i.i.d. samples ( X , , X , , Y , , Y , ) , . . . , ( X ,n , X ,n , Y ,n , Y ,n ) , where the generation of ( X ,i , X ,i ) , for i = 1 , . . . , n , are based on the method in Subsection 2.9of Nelsen [32]. Then, E u,S ( X , Y ) and E u,S ( X , τ , ( Y )) can be approximated by b E n,S ( X , Y ) = 1 n n X i =1 u (min { X ,i + Y ,i , X ,i + Y ,i } ) and b E n,S ( X , τ , ( Y )) = 1 n n X i =1 u (min { X ,i + Y ,i , X ,i + Y ,i } ) , respectively. Consider four different utility functions u ( x ) = x . , u ( x ) = x . , u ( x ) = (1 − e − x ) / , and u ( x ) = log x , for x ∈ R + . As observed in Figure 1, b E n,S ( X , Y ) ≥ b E n,S ( X , τ , ( Y )) for n = 1000 , , . . . , and all four utility functions. By law of large numbers, we have E u,S ( X , Y ) ≥ E u,S ( X , τ , ( Y )) , and thus the result of Theorem 3.2 is illustrated. As the next example shows, the LWSAI [RWSAI] condition in Theorem 3.2 may not bediscarded.
Example 3.4
Assume the lifetime vector X = ( X , X , X ) is assembled with Gumbel-Barnettcopula with generator φ ( t ) = e − t / , and X has distribution function ( e x − / ( e − for x ∈ [0 , , X has beta distribution with two shape parameters (3 , , and X has distributionfunction ( e x − / ( e − for x ∈ [0 , . Let Y , Y , Y be three independent beta randomvariables with shape parameters (3 , , (2 , and (1 , , respectively. One can easily check that Y ≥ lr Y ≥ lr Y . As can be seen in Figure 2(a), the distribution function of X and X crosseseach other, invalidating the existence of the usual stochastic order between these two randomvariables. By the implication in (1) , we know that X is neither LWSAI nor RWSAI. Forincreasing concave functions u ( x ) = x γ , we denote E u,S ( X , Y ) := E [(min { X + Y , X + Y , X + Y } ) γ ] and E u,S ( X , τ , ( Y )) := E [(min { X + Y , X + Y , X + Y } ) γ ] . . . . . . . X X (a) Distribution functions of X and X − − − − + − − g (b) b E u,S ( X , Y ) − b E u,S ( X , τ , ( Y )) Figure 2: Plots of Example 3.4.
From the population ( X , X , X , Y , Y , Y ) , we generate i.i.d. samples of one million observa-tions ( X , , X , , X , , Y , , Y , , Y , ) , . . . , ( X , , X , , X , , Y , , Y , , Y , ) . Then, for each γ = 0 . , . , · · · , . , E u,S ( X , Y ) and E u,S ( X , τ , ( Y )) can be approximatedby b E u,S ( X , Y ) = 110 X i =1 (min { X ,i + Y ,i , X ,i + Y ,i , X ,i + Y ,i } ) γ and b E u,S ( X , τ , ( Y )) = 110 X i =1 (min { X ,i + Y ,i , X ,i + Y ,i , X ,i + Y ,i } ) γ , respectively. As observed from Figure 2(b), the difference between b E u,S ( X , Y ) and b E u,S ( X , τ , ( Y )) is not always positive or negative, negating the conclusion of Theorem 3.2. Let Y be the lifetime of one single cold-standby redundancy R . Denote by S ( r ) ( X ; Y ) = min( X , . . . , X r − , X r + Y, X r +1 , . . . , X n )the resulting lifetime of the series system with R allocated to C r , for r = 1 , , . . . , n . Thefollowing corollary can be obtained from Theorem 3.2, which extends Proposition 3.1(i) of Singhand Misra [36], Theorem 5 of Li and Hu [28], and Theorem 2.1 of Misra et al. [31] to the caseof dependent components. Corollary 3.5 (i) If X is LWSAI, we have S (1) ( X ; Y ) ≥ icv S (2) ( X ; Y ) ≥ icv · · · ≥ icv S ( n ) ( X ; Y ) .(ii) If X is RWSAI, we have S (1) ( X ; Y ) ≥ st S (2) ( X ; Y ) ≥ st · · · ≥ st S ( n ) ( X ; Y ) . R , R have lifetimes Y , Y such that Y ≥ lr Y .Let S (1 , ( X ; Y , Y ) [ S (2 , ( X ; Y , Y )] be the lifetime of the series system with C allocatedby R [ R ] and C allocated by R [ R ]. The following result states that the allocation policy S (1 , ( X ; Y , Y ) is better than S (2 , ( X ; Y , Y ), which generalizes Theorem 2.2(i) of Li et al.[29] and Theorems 3.1 and 3.2 of Misra et al. [31]. Corollary 3.6 (i) If X is LWSAI, we have S (1 , ( X ; Y , Y ) ≥ icv S (2 , ( X ; Y , Y ) .(ii) If X is RWSAI, we have S (1 , ( X ; Y , Y ) ≥ st S (2 , ( X ; Y , Y ) . In this subsection, optimal allocation strategies of matched heterogeneous and independent cold-standby redundancies are pinpointed for parallel systems comprised of dependent components.
Theorem 3.7
Suppose that Y ≥ lr Y ≥ lr · · · ≥ lr Y n . For any ≤ i < j ≤ n ,(i) if X is LWSAI, then max( X + Y ) ≤ st max( X + τ i,j ( Y )) ;(ii) if X is RWSAI, then max( X + Y ) ≤ icx max( X + τ i,j ( Y )) . Proof.
By adopting the proof of Theorem 3.2, it suffices to show the non-positivity of E [ u (max( X + Y ))] − E [ u (max( X + τ i,j ( Y )))]= Z · · · Z R n − n Y k = i,j p k ( y k )d y k Z Z y i ≤ y j [ E [ u (max( X + y ))] − E [ u (max( X + τ i,j ( y )))]] × [ p i ( y i ) p j ( y j ) − p i ( y j ) p j ( y i )]d y i d y j , (8)where u is increasing for case (i), and increasing convex for case (ii). In view of (3), it is enoughto prove that E [ u (max( X + y ))] ≥ E [ u (max( X + τ i,j ( y )))] , y i ≤ y j . (9)Proof of (i): X is LWSAI. For any given X { i,j } = x { i,j } , we define g ( x i , x j ) = u (max( x + y )) = u (max { x i + y i , x j + y j , max { ( x + y ) { i,j } }} )and g ( x i , x j ) = u (max( x + τ i,j ( y ))) = u (max { x i + y j , x j + y i , max { ( x + y ) { i,j } }} ) . Then, for any x i ≤ x j and y i ≤ y j , it can be seen that∆ ( x i , x j ) = g ( x i , x j ) − g ( x i , x j )= u (max { x j + y j , max { ( x + y ) { i,j } }} ) − u (max { x i + y j , x j + y i , max { ( x + y ) { i,j } }} )12s always decreasing in x i ≤ x j for any increasing u . On the other hand, for any x i ≤ x j , g ( x i , x j ) + g ( x j , x i ) = g ( x i , x j ) + g ( x j , x i ) . (10)Thus, we have E [ u (max( X + y )) | X { i,j } = x { i,j } ] ≥ E [ u (max( X + τ i,j ( y ))) | X { i,j } = x { i,j } ] . Then, the proof is completed by applying Lemma 3.1 and iterated expectation formula.Proof of (ii): X is RWSAI. In light of the proof in (i), the desired result boils down toshowing that ∆ ( x i , x j ) is increasing in x j ≥ x i for any y i ≤ y j and increasing convex u .Case 1: x i + y j ≥ x j + y i . For this case, the function∆ ( x i , x j ) = u (max { x j + y j , max { ( x + y ) { i,j } }} ) − u (max { x i + y j , max { ( x + y ) { i,j } }} )is always increasing in x j ≥ x i by the increasing property of u .Case 2: x i + y j < x j + y i . For this case, we have∆ ( x i , x j ) = u (max { x j + y j , max { ( x + y ) { i,j } }} ) − u (max { x j + y i , max { ( x + y ) { i,j } }} ) . Note that x j + y j ≥ x j + y i , the following three situations are considered.Subcase 1: max { ( x + y ) { i,j } } ≥ x j + y j . Clearly, ∆ ( x i , x j ) = 0, due to which the proof istrivial.Subcase 2: x j + y j > max { ( x + y ) { i,j } } ≥ x j + y i . Note that∆ ( x i , x j ) = u ( x j + y j ) − u (max { ( x + y ) { i,j } } ) , which is obviously increasing in x j ≥ x i by the increasing property of u .Subcase 3: x j + y i > max { ( x + y ) { i,j } } . Observe that∆ ( x i , x j ) = u ( x j + y j ) − u ( x j + y i ) . For any x j ≥ x ′ j ≥ x i , the increasing convexity of u implies that∆ ( x i , x j ) = u ( x j + y j ) − u ( x j + y i ) ≥ u ( x ′ j + y j ) − u ( x ′ j + y i ) = ∆ ( x i , x ′ j ) , which means that ∆ ( x i , x j ) is increasing in x j ≥ x i .To conclude, we have shown that ∆ ( x i , x j ) is increasing in x j ≥ x i for any y i ≤ y j andincreasing convex u . Now, upon using Lemma 3.1 and iterated expectation formula, the proofis finished.For the parallel system with components C , . . . , C n having LWSAI [RWSAI] lifetimes, The-orem 3.7 implies that the redundancy R i should be put in standby with C i , i = 1 , . . . , n , ac-cording to the usual stochastic [increasing convex] ordering if the redundancies lifetimes satisfy Y ≥ lr · · · ≥ lr Y n . 13or one single cold-standby redundancy R with lifetime Y , let T ( r ) ( X ; Y ) = max( X , . . . , X r − , X r + Y, X r +1 , . . . , X n )be the resulting lifetime of the parallel system with R allocated to C r , for r = 1 , , . . . , n . Thefollowing corollary can be obtained from Theorem 3.7, extending Proposition 3.1(ii) of Singhand Misra [36], Theorem 4 of Li and Hu [28], and Theorem 2.2 of Misra et al. [31] to the caseof dependent components. Corollary 3.8 (i) If X is LWSAI, we have T (1) ( X ; Y ) ≤ st T (2) ( X ; Y ) ≤ st · · · ≤ st T ( n ) ( X ; Y ) .(ii) If X is RWSAI, we have T (1) ( X ; Y ) ≤ icx T (2) ( X ; Y ) ≤ icx · · · ≤ icx T ( n ) ( X ; Y ) . Suppose there are two cold-standby redundancies R , R having lifetimes Y , Y such that Y ≥ lr Y . Let T (1 , ( X ; Y , Y ) [ T (2 , ( X ; Y , Y )] be the resulting lifetime of the parallel systemwith C allocated by R [ R ] and C allocated by R [ R ]. The following result implies that theallocation policy T (2 , ( X ; Y , Y ) is better than T (1 , ( X ; Y , Y ), which generalizes Lemma 3.2of Shaked and Shanthikumar [34], Theorem 2.2(ii) of Li et al. [29], and Theorem 3.3 of Misraet al. [31]. Corollary 3.9 (i) If X is LWSAI, we have T (1 , ( X ; Y , Y ) ≤ st T (2 , ( X ; Y , Y ) .(ii) If X is RWSAI, we have T (1 , ( X ; Y , Y ) ≤ icx T (2 , ( X ; Y , Y ) . To close this section, we present one example to illustrate Theorem 3.7.
Example 3.10
Under the setting of Example 3.3, we denote E u,P ( X , Y ) := E [ u (max { X + Y , X + Y } )] and E u,P ( X , τ , ( Y )) := E [ u (max { X + Y , X + Y } )] . Then, the function E u,P ( X , Y ) and E u,P ( X , τ , ( Y )) can be approximated by b E n,P ( X , Y ) = 1 n n X i =1 u (max { X ,i + Y ,i , X ,i + Y ,i } ) and b E n,P ( X , τ , ( Y )) = 1 n n X i =1 u (max { X ,i + Y ,i , X ,i + Y ,i } ) , respectively, where ( X ,i , X ,i , Y ,i , Y ,i ) ’s are independent copies of ( X , X , Y , Y ) . We considerfour different utility functions u ( x ) = x . , u ( x ) = x . , u ( x ) = 10(1 − e − . x ) , and u ( x ) =log x . Figure 3 shows that b E n,P ( X , Y ) ≤ b E n,P ( X , τ , ( Y )) for n = 1000 , , . . . , andall four types of utility functions. By law of large numbers, it must hold that E u,P ( X , Y ) ≤ E u,P ( X , τ , ( Y )) , which validates the effectiveness of Theorem 3.7. b E n,P ( X , Y ) b E n,P ( X , τ , ( Y )) (a) u ( x ) = x . b E n,P ( X , Y ) b E n,P ( X , τ , ( Y )) (b) u ( x ) = x . n ˆ E n,P ( X , Y )ˆ E n,P ( X , τ , ( Y )) (c) u ( x ) = 10(1 − e − . x ) n ˆ E n,P ( X , Y )ˆ E n,P ( X , τ , ( Y )) (d) u ( x ) = log x Figure 3: Plots of the estimators b E n,P ( X , Y ) and b E n,P ( X , τ , ( Y )).15 xample 3.11 Under the setting of Example 3.4, we denote E u,P ( X , Y ) := E [(max { X + Y , X + Y , X + Y } ) γ ] and E u,P ( X , τ , ( Y )) := E [(max { X + Y , X + Y , X + Y } ) γ ] . Then, the function E u,P ( X , Y ) and E u,P ( X , τ , ( Y )) can be approximated by b E u,P ( X , Y ) = 110 X i =1 (max { X ,i + Y ,i , X ,i + Y ,i , X ,i + Y ,i } ) γ and b E u,P ( X , τ , ( Y )) = 110 X i =1 (max { X ,i + Y ,i , X ,i + Y ,i , X ,i + Y ,i } ) γ , respectively. Figure 4 shows the difference between b E u,P ( X , Y ) and b E u,P ( X , τ , ( Y )) for γ =1 , . , · · · , . As can be seen from the plot, the difference curve is neither positive nor negativefor all the γ , which invalidates the effectiveness of Theorem 3.7. − . . . . . . g Figure 4: Difference between b E u,P ( X , Y ) and b E u,P ( X , τ , ( Y )).One natural interesting problem would be studying optimal cold-standby redundancies allo-cations for general k -out-of- n systems. The following example indicates that there is no certainanswer for this problem. Example 3.12
Consider a -out-of- system comprising three independent components with X , X and X having hazard rates λ = 0 . , λ = 0 . and λ = 0 . , respectively. Supposethe cold-standby redundancy has exponential lifetime Y with hazard rate λ . Let us considerthree possible allocation policies (i) P : allocated to X ; (ii) P : allocated to X ; and (iii) P :allocated to X . By taking u ( x ) = u . and λ = 0 . or λ = 2 . , Figure 5 presents the empiricalvalues (denoted by b E ( P i ) ) of the expected function of the resulting lifetime under policies P i ,for i = 1 , , . As observed from the plots, the optimal allocation policy (in the sense of theincreasing convex order) may depend on the reliability performance of the redundancy. × P P P (a) λ = 0 . n × P P P (b) λ = 2 . Figure 5: Plots of b E ( P i ) for different allocation policies P i , i = 1 , , In this section, we investigate optimal allocations of m i.i.d. cold-standby redundancies withrandom lifetimes Y = ( Y , . . . , Y m ) to a series or parallel system comprised of n dependent com-ponents with lifetimes X = ( X , . . . , X n ). Let r ∈ A := { ( r , . . . , r n ) | P ni =1 r i = m, r i ∈ N , i =1 , . . . , n } be the allocation policy with r i cold-standby redundancies allocated to component C i , i = 1 , . . . , n . Denote by S ( X + Y ; r ) [ T ( X + Y ; r )] the lifetime of the series [parallel] systemwith allocation policy r ∈ A .To begin with, let us review the notion of totally positive of order 2 ( T P ). A function h ( x, y ) is said to be T P in ( x, y ), if h ( x, y ) ≥ h ( x , y ) h ( x , y ) ≥ h ( x , y ) h ( x , y ),whenever x ≤ x and y ≤ y . Interested readers are referred to Karlin and Rinott [27] for acomprehensive study on the properties and applications of T P . Theorem 4.1
Suppose the redundancies lifetimes Y , . . . , Y m have common log-concave densityfunctions. For any ≤ i < j ≤ n ,(i) if X is LWSAI, then S ( X + Y ; r ) ≥ icv S ( X + Y ; τ i,j ( r )) whenever r i ≥ r j ;(ii) if X is RWSAI, then S ( X + Y ; r ) ≥ st S ( X + Y ; τ i,j ( r )) whenever r i ≥ r j . Proof.
Define Z l = Y P l − i =1 r i +1 + · · · + Y P li =1 r i , for l = 1 , , . . . , n , where r ≡
0. Let f ( r l ) ( z l )be the the density function of Z l , which are convolutions of r l copies of Y for l = 1 , , . . . , n . Forany integrable function u , we have E [ u ( S ( X + Y ; r ))] − E [ u ( S ( X + Y ; τ i,j ( r )))]= E [ u (min( X + Z ))] − E [ u (min( X + τ i,j ( Z )))]17 Z · · · Z R n + E [ u (min( X + z ))] f ( r i ) ( z i ) f ( r j ) ( z j ) n Y l = i,j f ( r l ) ( z l )d z · · · d z n − Z · · · Z R n + E [ u (min( X + z ))] f ( r i ) ( z j ) f ( r j ) ( z i ) n Y l = i,j f ( r l ) ( z l )d z · · · d z n = Z · · · Z R n − n Y l = i,j f ( r l ) ( z l )d z l Z Z z i ≤ z j (cid:26)(cid:2) E [ u (min( X + z ))] − E [ u (min( X + τ i,j ( z )))] (cid:3) × [ f ( r i ) ( z i ) f ( r j ) ( z j ) − f ( r i ) ( z j ) f ( r j ) ( z i )] (cid:27) d z i d z j . (11)The desired result is equivalent to proving that (11) is non-negative.On the one hand, according to the proof of Theorem 3.2 we have shown under cases (i) and(ii) that E [ u (min( X + z ))] ≤ E [ u (min( X + τ i,j ( z )))] , z i ≤ z j . (12)On the other hand, from the log-concavity of the density function of Y , we can conclude that f ( r l ) ( y ) is T P in ( r l , y ) ∈ { , , . . . , m } × R + , where f ( r l ) ( y ) = f ( y ) ∗ f ( y ) ∗ · · · ∗ f r l ( y )denotes the density function of convolutions of r l copies of Y , l = 1 , , . . . , n . Thus, it can beobtained that f ( r i ) ( z i ) f ( r j ) ( z j ) − f ( r i ) ( z j ) f ( r j ) ( z i ) ≤ , z i ≤ z j and r i ≥ r j . (13)Upon combining (12) with (13), the non-negativity of (11) is established. Hence, the theoremfollows.Many lifetime distributions have log-concave densities, to name a few, the Beta distributionwith both parameters greater than 1, the Gamma distribution with shape parameter greater than1 and scale parameter equaling to 1, the Weibull distribution with shape parameter greater than1 and scale parameter equaling to 1, and so on. Based on Theorem 4.1, it can be figured out thatmore redundancies should be allocated to the weaker components in the sense of the increasingconcave [usual stochastic] ordering when the components lifetimes are LWSAI [RWSAI]. It mightbe of great interest to investigate whether the log-concavity property of the density for the coldstandbys is a must or not in Theorem 4.1. Numerical examples show that this assumption mightbe relaxed or discarded, for which we cannot prove so far and thus leave it as an open problem.The following numerical example shows the validity of Theorem 4.1. Example 4.2
Assume X = ( X , X ) is assembled with Clayton [survival] copula with generator φ ( t ) = ( t + 1) − and X and X have exponential distribution with hazard rates λ = 0 . and λ = 0 . , respectively. Let Y , . . . , Y be independent Weibull random variables with common ˆ E n,S ( r )ˆ E n,S ( τ , ( r )) (a) u ( x ) = x . n ˆ E n,S ( r )ˆ E n,S ( τ , ( r )) (b) u ( x ) = x . n ˆ E n,S ( r )ˆ E n,S ( τ , ( r )) (c) u ( x ) = (1 − e − x ) / n ˆ E n,S ( r )ˆ E n,S ( τ , ( r )) (d) u ( x ) = log x Figure 6: Plots of the estimators b E n,S ( r ) and b E n,S ( τ , ( r )).19 cale parameter µ = 1 and shape parameter β = 1 . . It is easy to check that X is LWSAI[RWSAI], and Y has log-concave density function. Let r = (3 , . For any increasing function u , we denote E u,S ( r ) := E [ u (min { X + Y + Y + Y , X + Y + Y } )] and E u,S ( τ , ( r )) := E [ u (min { X + Y + Y , X + Y + Y + Y } )] . From the population ( X , X , Y , . . . , Y ) , we generate an i.i.d. sample ( X , , X , , Y , , . . . , Y , ) , . . . , ( X ,n , X ,n , Y ,n , . . . , Y ,n ) , where the samples ( X ,i , X ,i ) ’s are generated via the method in Subsection 2.9 of Nelsen [32].Then, the functions E u,S ( r ) and E u,S ( τ , ( r )) can be approximated by b E n,S ( r ) = 1 n n X i =1 u (min { X ,i + Y ,i + Y ,i + Y ,i , X ,i + Y ,i + Y ,i } ) and b E n,S ( τ , ( r )) = 1 n n X i =1 u (min { X ,i + Y ,i + Y ,i , X ,i + Y ,i + Y ,i + Y ,i } ) , respectively. As observed in Figure 6, for utility functions u ( x ) = x . , u ( x ) = x . , u ( x ) =(1 − e − x ) / and u ( x ) = log x , it holds that b E n,S ( r ) ≥ b E n,S ( τ , ( r )) for n = 1000 , , . . . , .By law of large numbers, we have E u,S ( r ) ≥ E u,S ( τ , ( r )) , which supports the result of Theorem4.1. The following example sheds light on the optimal allocation strategies for series systems,which cannot be proven so far due to technicality difficulty and is left as an open problem.
Example 4.3
Under the setup of Example 4.2, we plot the values of b E n,S ( r ) in Figure 7 forthree allocation policies r = (3 , , r = (4 , , and r = (5 , . These numerical simulationssuggest that the optimal allocation policy for a series system might be such that (i) more cold-standby redundancies should be allocated to weaker components, and (ii) the number of sparesgiven in each node should be as close as possible. The following corollary can be obtained from Theorem 4.1, which partially generalizes The-orem 4.2 of Zhuang and Li [52] to the case of dependent components.
Corollary 4.4
Suppose the redundancies lifetimes Y , . . . , Y m and have common log-concavedensity functions. For any ≤ i < j ≤ n ,(i) if X has an Archimedean copula with log-convex generator and such that X ≤ rh · · · ≤ rh X n , then S ( X + Y ; r ) ≥ icv S ( X + Y ; τ i,j ( r )) whenever r i ≥ r j ;(ii) if X has an Archimedean survival copula with log-convex generator and such that X ≤ hr · · · ≤ hr X n , then S ( X + Y ; r ) ≥ st S ( X + Y ; τ i,j ( r )) whenever r i ≥ r j . ˆ E n,S ( r )ˆ E n,S ( r )ˆ E n,S ( r ) (a) u ( x ) = x . n ˆ E n,S ( r )ˆ E n,S ( r )ˆ E n,S ( r ) (b) u ( x ) = x . Figure 7: Plots of the estimators b E n,S ( r ) for different allocation policies r . In this subsection, we present optimal allocation strategies of i.i.d. cold-standby redundancies toa parallel system comprised of dependent components having LWSAI or RWSAI joint lifetimes.
Theorem 4.5
Suppose the redundancies lifetimes Y , . . . , Y m have common log-concave densityfunctions. For any ≤ i < j ≤ n ,(i) if X is LWSAI, then T ( X + Y ; r ) ≤ st T ( X + Y ; τ i,j ( r )) whenever r i ≥ r j ;(ii) if X is RWSAI, then T ( X + Y ; r ) ≤ icx T ( X + Y ; τ i,j ( r )) whenever r i ≥ r j . Proof.
In light of the proof of Theorem 4.1, one can see that E [ u ( T ( X + Y ; r ))] − E [ u ( T ( X + Y ; τ i,j ( r )))]= E [ u (max( X + Z ))] − E [ u (max( X + τ i,j ( Z )))]= Z · · · Z R n − n Y l = i,j f ( r l ) ( z l )d z l Z Z z i ≤ z j (cid:26)(cid:2) E [ u (max( X + z ))] − E [ u (max( X + τ i,j ( z )))] (cid:3) × [ f ( r i ) ( z i ) f ( r j ) ( z j ) − f ( r i ) ( z j ) f ( r j ) ( z i )] (cid:27) d z i d z j . (14)Then, the non-positivity of (14) can be established under cases (i) and (ii) by using (13) andthe proof method as in Theorem 3.7.It can inferred from Theorem 4.5 that more redundancies should be allocated to bettercomponents in order to reach a resulting parallel system with higher lifetime in the sense of theusual stochastic [increasing convex] ordering when the joint lifetimes of the original componentsare LWSAI [RWSAI]. It remains open to investigate whether the log-concavity assumption onthe common density function of the cold standbys could be relaxed (or removed) or not.21he following example illustrates Theorem 4.5. n ˆ E n,P ( r )ˆ E n,P ( τ , ( r )) (a) u ( x ) = x . n ˆ E n,P ( r )ˆ E n,P ( τ , ( r )) (b) u ( x ) = x . n ˆ E n,P ( r )ˆ E n,P ( τ , ( r )) (c) u ( x ) = 10(1 − e − . x ) n ˆ E n,P ( r )ˆ E n,P ( τ , ( r )) (d) u ( x ) = log x Figure 8: Plots of the estimators b E n,P ( r ) and b E n,P ( τ , ( r )). Example 4.6
Under the setup of Example 4.2, for any increasing function u , we denote E u,P ( r ) := E [ u (max { X + Y + Y + Y , X + Y + Y } )] and E u,P ( τ , ( r )) := E [ u (max { X + Y + Y , X + Y + Y + Y } )] . The function E u,P ( r ) and E u,P ( τ , ( r )) can be approximated by b E n,P ( r ) = 1 n n X i =1 u (max { X ,i + Y ,i + Y ,i + Y ,i , X ,i + Y ,i + Y ,i } ) and b E n,P ( τ , ( r )) = 1 n n X i =1 u (max { X ,i + Y ,i + Y ,i , X ,i + Y ,i + Y ,i + Y ,i } ) , espectively. Figure 8 plots the estimators b E n,P ( r ) and b E n,P ( τ , ( r )) for n = 1000 , , . . . , under utility functions u ( x ) = x . , u ( x ) = x . , u ( x ) = 10(1 − e − . x ) , and u ( x ) = log x . Then,by law of large numbers it holds that E u,P ( r ) ≤ E u,P ( τ , ( r )) , which validates Theorem 4.5. n ˆ E n,P (˜ r )ˆ E n,P (˜ r )ˆ E n,P (˜ r ) (a) α = 0 . n ˆ E n,P (˜ r )ˆ E n,P (˜ r )ˆ E n,P (˜ r ) (b) α = 1 . Figure 9: Plots of the estimators b E n,P ( ˜ r ) for different allocation policies ˜ r .However, the explicit configuration of the optimal allocation strategy remains to be deter-mined for parallel systems. The following example conjectures on the optimal allocation, whichcannot be proven so far due to technical difficulty and is thus left as an open problem. Example 4.7
The values of b E n,P ( ˜ r ) are displayed in Figure 9 for three policies ˜ r = (2 , , ˜ r = (1 , and ˜ r = (0 , under the setting of Example 4.6. These plots suggest that, for aparallel system, all redundancies might be allocated to the component with the best performance,which agrees with intuition. Similar with Corollary 4.4, the following result can be obtained from Theorem 4.5.
Corollary 4.8
Suppose the redundancies lifetimes Y , . . . , Y m have common log-concave densityfunctions. For any ≤ i < j ≤ n ,(i) if X has an Archimedean copula with log-convex generator and such that X ≤ rh · · · ≤ rh X n , then T ( X + Y ; r ) ≥ st T ( X + Y ; τ i,j ( r )) whenever r i ≥ r j ;(ii) if X has an Archimedean survival copula with log-convex generator and such that X ≤ hr · · · ≤ hr X n , then T ( X + Y ; r ) ≥ icx T ( X + Y ; τ i,j ( r )) whenever r i ≥ r j . In reliability theory and engineering practice, it is an important research issue to seek for optimalredundancies allocation strategies for coherent systems. In this article, we investigate optimal23llocation policies of cold-standby redundancies in series and parallel systems comprised ofdependent components having LWSAI or RWSAI joint lifetimes. Under the assumption that thecold-standby redundancies are independent of the original components, optimal allocations arepinpointed both for series and parallel systems under matching allocation when the cold-standbyspares are independent and ordered via the likelihood ratio ordering. For the homogeneous cold-standby redundancies, optimal allocation policies are also derived for both series and parallelsystems. The optimal allocation strategies for series systems are opposite to those for parallelsystems, which are consistent with the findings in Boland et al. [6].You et al. [40] established the optimal allocations of hot-standby redundancies for k -out-of- n systems with components having LTPD lifetimes. They applied the allocation strategies indistributing new wires to a set of cables in order to increase the strength of the cables. Cableswith great tensile strength are commonly demanded for designing a high voltage electricitytransmission network. The strength of a cable composed of several wires can be viewed as a k -out-of- n system, where the wires may be regarded as components. Consider the selected wireslabeled as 1, 2 and 8 in the data set tested and reported in Hald [20] and embodied in Handet al. [21]. Denote the tensile strength of wires 1, 2, 8 by X , X and X , respectively. Byshowing that Fr´echet distribution fits the strength of these wires well, and that the dependencestructure is fitted by Gumbel-Barnett copula statistically well, You et al. [40] shows that thevector ( X , X , X ) can be modelled by some absolutely continuous LTPD distribution, whichis equivalent to LSWAI. Therefore, if these three wires are assembled in series or parallel, theallocation strategies derived in Sections 3 and 4 can be applied in the cable to increase thestrength. For example, assume that a cable is made from these three wires, and its strengthis measured as the weakest of the wires. Consider the repairing strategy such that a brokenwire is replaced by a new one. Ignoring the replacing time, the lifetime of the cable can beapproximated by that of a series system under some cold-standby allocation strategy. If thereare three extra wires with independent strengths ordered in the sense of the likelihood ratioorder, each of which is to replace one broken wire, then the best replacing strategy is to put thestrongest one to wire 2, the moderate one to wire 1, and the weakest one to wire 8.Since both LWSAI and RWSAI are positive dependence notions, it is of natural interest tostudy whether the optimal allocation policies established here still hold for the case of negativelydependent components. Besides, another possible extension might be seeking for the best allo-cation policies when the original components have WSAI (c.f. Cai and Wei, [9]) joint lifetimes.We leave these as open problems. Acknowledgments
The authors are very grateful for the valuable comments from two anonymous reviewers, whichhave greatly improved the presentation of this paper. Yiying Zhang thanks the start-up grant24t Nankai University.
References [1] Ardakan, M. A. and Hamadani, A. Z. (2014). Reliability-redundancy allocation problem withcold-standby redundancy strategy.
Simulation Modelling Practice and Theory , , 107-118.[2] Barlow, R. and Proschan, R. (1981). Statistical Theory of Reliability and Life Testing . SilverSpring, MD: Madison.[3] Belzunce, F., Ma´rtinez-Puertas, H. and Ruiz, J. (2011). On optimal allocation of redundantcomponents for series and parallel systems of two dependent components.
Journal of StatisticalPlanning and Inference , , 3094-3104.[4] Belzunce, F., Ma´rtinez-Puertas, H. and Ruiz, J. (2013). On allocation of redundant com-ponents for systems with dependent components. European Journal of Operational Research , , 573-580.[5] Boland, P. J., El-Neweihi, E. and Proschan, F. (1988). Active redundancy allocation incoherent systems. Probability in the Engineering and Informational Sciences , , 343-353.[6] Boland, P. J., El-Neweihi, E. and Proschan, F. (1992). Stochastic order for redundancyallocation in series and parallel systems. Advances in Applied Probability , , 161-171.[7] Brito, G., Zequeira, R. I. and Vald´es, J. E. (2011). On the hazard rate and reversed haz-ard rate orderings in two-component series systems with active redundancies. Statistics & Probability Letters , , 201-206.[8] Cai, J. and Wei, W. (2014). Some new notions of dependence with applications in optimalallocation problems. Insurance: Mathematics and Economics , , 200-209.[9] Cai, J. and Wei, W. (2015). Notions of multivariate dependence and their applications inoptimal portfolio selections with dependent risks. Journal of Multivariate Analysis , , 156-169.[10] Chen, J., Zhang, Y., Zhao, P. and Zhou, S. (2018). Allocation strategies of standby re-dundancies in series/parallel system. Communications in Statistics-Theory and Methods , ,708-724.[11] da Costa Bueno, V. (2005). Minimal standby redundancy allocation in a k -out-of- n : F system of dependent components. European Journal of Operational Research , , 786-793.[12] Da, G. and Ding, W. (2016). Component level versus system level k -out-of- n assemblysystems. IEEE Transactions on Reliability , , 425-433.2513] Ding, W., Zhao, P. and Zhou, S. (2017). On optimal allocation of active redundancies tomulti-state r -out-of- n systems. Operations Research Letters , , 508-512.[14] Doostparast, M. (2017). Redundancy schemes for engineering coherent systems via asignature-based approach. arXiv:1708.07059. [15] Eryilmaz, S. (2017). The effectiveness of adding cold standby redundancy to a coherentsystem at system and component levels. Reliability Engineering & System Safety , , 331-335.[16] Eryilmaz, S. and Erkan, T. E. (2018). Coherent system with standby components. AppliedStochastic Models in Business and Industry , , 395-406.[17] Finkelstein, M., Hazra, N. K. and Cha, J. H. (2018). On optimal operational sequence ofcomponents in a warm standby system. Journal of Applied Probability , , 1014-1024.[18] Gholinezhad, H. and Hamadani, A. Z. (2017). A new model for the redundancy allocationproblem with component mixing and mixed redundancy strategy. Reliability Engineering & System Safety , , 66-73.[19] Hadipour, H., Amiri, M. and Sharifi, M. (2018). Redundancy allocation in series-parallelsystems under warm standby and active components in repairable subsystems. ReliabilityEngineering & System Safety , https://doi.org/10.1016/j.ress.2018.01.007 [20] Hald, A. (1952). Statistical Theory with Engineering Applications . John Wiley & Sons: NewYork.[21] Hand, D. J., Daly, F., McConway, K., Lunn, D. and Ostrowski, E. (1994).
A Handbook ofSmall Data Sets . Chapman & Hall: London.[22] Hazra, N. K. and Nanda, A. K. (2014). Component redundancy versus system redundancyin different stochastic orderings.
IEEE Transactions on Reliability , , 567-582.[23] Hazra, N. K. and Nanda, A. K. (2015). A note on warm standby system. Statistics & Probability Letters , , 30-38.[24] Hollander, M., Proschan, F. and Sethuraman, J. (1977). Functions decreasing in transporta-tion and their applications in ranking problems. The Annals of Statistics , , 722-733.[25] Jeddi, H. and Doostparast, M. (2016). Optimal redundancy allocation problems in engi-neering systems with dependent component lifetimes. Applied Stochastic Models in Businessand Industry , , 199-208.[26] Joe, H. (2014). Dependence Modeling with Copulas . CRC Press, Boca Raton, FL.2627] Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlationinequalities I: multivariate totally positive distributions.
Journal of Multivariate Analysis , ,467-498.[28] Li, X. and Hu, X. (2008). Some new stochastic comparisons for redundancy allocations inseries and parallel systems. Statistics & Probability Letters , , 3388-3394.[29] Li, X., Yan, R. and Hu, X. (2011). On the allocation of redundancies in series and parallelsystems. Communications in Statistics-Theory and Methods , , 959-968.[30] Misra, N., Misra, A. K. and Dhariyal, I. D. (2011a). Active redundancy allocations in seriessystems. Probability in the Engineering and Informational Sciences , , 219-235.[31] Misra, N., Misra, A. K. and Dhariyal, I. D. (2011b). Standby redundancy allocations inseries and parallel systems. Journal of Applied Probability , , 43-45.[32] Nelsen, R. B. (2006). An Introduction to Copulas . Springer: New York.[33] Shaked, M. and Shanthikumar, J. G. (1992). Optimal allocation of resources to nodes ofparallel and series systems.
Advances in Applied Probability , , 894-914.[34] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders . New York: Springer-Verlag.[35] Shanthikumar, J. G. and Yao, D. D. (1991). Bivariate characterization of some stochasticorder relations.
Advances in Applied Probability , , 642-659.[36] Singh, H. and Misra, N. (1994). On redundancy allocation in systems. Journal of AppliedProbability , , 1004-1014.[37] Vald´es, J. E., Arango, G. and Zequeira, R. I. (2010). Some stochastic comparisons in seriessystems with active redundancy. Statistics & Probability Letters , , 945-949.[38] Vald´es, J. E. and Zequeira, R. I. (2006). On the optimal allocation of two active redundanciesin a two-component series systems. Operations Research Letters , , 49-52.[39] Wei, W. (2017). Joint stochastic orders of high degrees and their applications in portfolioselections. Insurance: Mathematics and Economics , , 141-148.[40] You, Y., Fang, R. and Li, X. (2016). Allocating active redundancies to k -out-of- n reliabil-ity systems with permutation monotone component lifetimes. Applied Stochastic Models inBusiness and Industry , , 607-620.[41] You, Y. and Li, X. (2014). On allocating redundancies to k -out-of- n reliability systems. Applied Stochastic Models in Business and Industry , , 361-371.2742] You, Y. and Li, X. (2015). Functional characterizations of bivariate weak SAI with anapplication. Insurance: Mathematics and Economics , , 225-231.[43] Yun, W. Y. and Cha, J. H. (2010). Optimal design of a general warm standby system. Reliability Engineering & System Safety , , 880-886.[44] Zhang, Y. (2018). Optimal allocation of active redundancies in weighted k -out-of- n systems. Statistics & Probability Letters , , 110-117.[45] Zhang, Y., Amini-Seresht, E. and Ding, W. (2017). Component and system active redundan-cies for coherent systems with dependent components. Applied Stochastic Models in Businessand Industry , , 409-421.[46] Zhang, Y., Li, X. and Cheung, K. C. (2018). On heterogeneity in the individual model withboth dependent claim occurrences and severities. ASTIN Bulletin , , 817-839.[47] Zhang, Y. and Zhao, P. (2015). Comparisons on aggregate risks from two sets of heteroge-neous portfolios. Insurance: Mathematics and Economics , , 124-135.[48] Zhao, P., Chan, P., Li, L. and Ng, H. K. T. (2013a). Allocation of two redundancies intwo-component series systems. Naval Research Logistics , , 588-598.[49] Zhao, P., Chan, P., Li, L. and Ng, H. K. T. (2013b). On allocation of redundancies intwo-component series systems. Operations Research Letters , , 690-693.[50] Zhao, P., Zhang, Y. and Li, L. (2015). Redundancy allocation at component level versussystem level. European Journal of Operational Research , , 402-411.[51] Zhao, P., Zhang, Y. and Chen, J. (2017). Optimal allocation policy of one redundancy in a n -component series system. European Journal of Operational Research , , 656-668.[52] Zhuang, J. and Li, X. (2015). Allocating redundancies to k -out-of- n systems with indepen-dent and heterogeneous components. Communications in Statistics-Theory and Methods ,44