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Published on January 16, 2008

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Linear consecutive-k-out-of-n systems:  Linear consecutive-k-out-of-n systems Variant optimal design problem Malgorzata O’Reilly University of Adelaide Nomenclature:  Nomenclature A linear consecutive-k-out-of-n:F system is an ordered sequence of n components such that the system fails if and only if at least k consecutive components fail. A linear consecutive-k-out-of-n:G system is an ordered sequence of n components such that the system works if and only if at least k consecutive components work. A particular arrangement of components in a system is referred to as a design. Assumptions:  Assumptions The system is either in a failing or a working state. Each component is either in a failing or a working state. The failures of the components are independent. Component reliabilities are distinct and within (0,1). The fourth assumption is made for the clarity of presentation, without loss of generality. Cases that include reliabilities 0 and 1 can be viewed as limits of other cases. Some of the proven strict inequalities will become nonstrict when these cases are included. Examples of linear consecutive-k-out-of-n:F systems:  Examples of linear consecutive-k-out-of-n:F systems A telecommunication system with n relay stations (satellites or ground stations) which fails when at least 2 consecutive stations fail, An oil pipeline system with n pump stations which fails when at least 2 consecutive pump stations are down. Examples of linear consecutive-k-out-of-n:G systems:  Examples of linear consecutive-k-out-of-n:G systems Consider n parallel-parking spaces on a street, with each space being suitable for one car. The problem is to find a probability that a bus, which takes 2 consecutive spaces, can park on this street. A bridge with n cables, where a minimum k cables are necessary to support the bridge. Applications of linear consecutive-k-out-of-n systems.:  Applications of linear consecutive-k-out-of-n systems. Vacum systems in accelerators Computer ring networks Systems from the field of integrated circuits Belt conveyors in open-cast mining Exploration of distant stars by spacecraft Generalizations of consecutive-k-out-of-n systems:  Generalizations of consecutive-k-out-of-n systems Consecutively connected systems Linearly connected systems Consecutive-k-out-of-m-from-n:F systems Consecutive-weighed-k-out-of-n:F systems m-consecutive-k-out-of-n:F systems 2-dimensional consecutive-k-out-of-n:F systems Connected-X-out-of-(m,n):F lattice systems Connected-(r,s)-out-of-(m,n):F lattice systems k-within-(r,s)-out-of-(m,n):F lattice systems Consecutively connected systems with multistate components Studies of reliability of consecutive-k-out-of-n systems:  Studies of reliability of consecutive-k-out-of-n systems Reliability formulae Algorithms to calculate reliability Approximating reliability by its upper and lower bounds Limiting the reliability or distributions associated with the systems Optimal design problem:  Optimal design problem Consider n components, each with different unreliability. Then, for a given linear consecutive-k-out-of-n system, what is the best arrangement of components? In other words, which design is optimal i.e. maximizes system reliability? Optimal designs have been classified into two types: invariant and variant. Invariant optimal designs are optimal always, subject only to the ordering of the numerical values of component reliabilities. The optimality of variant optimal designs depends on the numerical values of components reliabilities. Invariant optimal designs:  Invariant optimal designs Invariant optimal design for linear consecutive-k-out-of-n:F systems exist only for k  {1,2,n-2,n-1,n}. Invariant optimal design for linear consecutive-k-out-of-n:G systems exist only for k  {1,n-2,n-1,n} and for n/2  k < n-2. The theory of invariant optimal designs is now complete. Invariant optimal designs of linear consecutive-k-out-of-n:F systems:  Invariant optimal designs of linear consecutive-k-out-of-n:F systems For k = 2: (1,n,3,n-2,…,n-3,4,n-1,2) For k = n-2: (1,4,,3,2) For k = n-1: (1, ,2) For k  {1,n}: () Symbol  represents any possible arrangement. The assumed order of component reliabilities is p1 < p2 <…< pn . Invariant optimal designs of linear consecutive-k-out-of-n:G systems:  Invariant optimal designs of linear consecutive-k-out-of-n:G systems For n/2  k  n-1: (1,3,…,2(n-k)-1,,2(n-k),…,2) For k  {1,n}: () Symbol  represents any possible arrangement. The assumed order of component reliabilities is p1 < p2 <…< pn . Variant optimal designs:  Variant optimal designs Linear consecutive-k-out-of-n systems have variant optimal designs for all F systems with 2 < k < n-2 and all G systems with 2  k < n/2. The information about the order of component reliabilities is not sufficient to find the optimal design. One needs to know the exact value of component reliabilities. Different sets of component reliabilities produce different optimal designs, so that for a given linear consecutive-k-out-of-n system there is more than one possible optimal design. Methods in dealing with the variant optimal design problem:  Methods in dealing with the variant optimal design problem Heuristic method (sub-optimal design) Randomization method (sub-optimal design) Binary search method (exact optimal design) Heuristic method:  Heuristic method The heuristic method is based on the concept of Birnbaum reliability importance defined by the following formula, where R stands for reliability of a system, ps for the reliability of a component s where 1  s  n, 1 and 0 represent working and failing states of a component i. I(i) = R(System/i works) - R(System/i fails) = R(p1,...,pi-1,1,pi+1,...,pn) - R(p1,...,pi-1,0,pi+1,...,pn). The heuristic method implements the idea that a component with a higher reliability should be placed in a position with a higher Birnbaum importance. Randomization method:  Randomization method Compares a limited number of randomly chosen design and obtains the best amongst them. It is based on general necessary conditions for the optimal design. Binary search method:  Binary search method Has been applied to linear consecutive-k-out-of-n:F systems with n/2  k  n and is based upon the following general necessary conditions for the optimal design. Components from positions 1 to min{k,(n-k+1)} are arranged in non-decreasing order of component reliability; Components from positions n to max{k,(n-k+1)} are arranged in non-decreasing order of component reliability; The (2k-n) most reliable components are arranged from positions (n-k+1) to k in any order if n<2k. Necessary conditions for the variant optimal design of linear consecutive-k-out-of-n systems:  Necessary conditions for the variant optimal design of linear consecutive-k-out-of-n systems The (k+1)-th component Malgorzata O’Reilly University of Adelaide General necessary conditions for the variant optimal design:  General necessary conditions for the variant optimal design Components from positions 1 to k are arranged in non-decreasing order of component reliability, Components from positions n to (n-k+1) are arranged in non-decreasing order of component reliability. Definition of singularity:  Definition of singularity We define a design X = (q1,q2,...,qn) to be singular if either qi > qn+1-i for all 1  i  [n/2] (integer part of n/2) or qi < qn+1-i for all 1  i  [n/2]. Otherwise it is nonsingular. Components qi and qn+1-i are referred to as symmetrical.  Illustration: 7-out-of-15 system Other necessary conditions for the variant optimal design:  Other necessary conditions for the variant optimal design A necessary condition for the optimal design of a linear consecutive-k-out-of-n:G system with n  {2k,(2k+1)} is for it to be singular. A necessary condition for the optimal design of a linear consecutive-k-out-of-n:F system with 2k  n  3k is for it to be nonsingular. Let X= (q1,…,q2k+m) be the optimal design of a linear consecutive-k-out-of-(2k+m):G system with 2  m  k. If (q1,…,qm-1,qk+1,…,qk+m,q2k+2,…,q2k+m) is singular, then X must be singular too. Remark:  Remark The existing general necessary conditions for the variant optimal design of linear consecutive systems provide comparisons between reliabilities of components restricted to positions from 1 to k and positions from n to (n-k+1). Objectives of this research:  Objectives of this research To examine relationships between components at some other positions, including the (k+1)-th component To establish necessary conditions for the optimal design of linear consecutive systems based on those comparisons. To provide procedures to improve designs not satisfying those necessary conditions RESULTS:  RESULTS Definitions:  Definitions Definition 1. We define Xi;j to be a design obtained from X by interchanging components i and j.  Definition 2. We define Xi(1),..,i(r);j(1),…,j(r) to be a design obtained from X by interchanging components i(s) and j(s) for all 1  s  r.  General result for n > 2k, k  2:  General result for n > 2k, k  2 Theorem 1. Let X be a design of a linear consecutive-k-out-of-n:F system or linear consecutive-k-out-of-n:G system n > 2k, k  2. If q1 < qk then X1;k is a better design.  Corollary 1. If X be a design of a linear consecutive-k-out-of-n:F system or linear consecutive-k-out-of-n:G system n > 2k, k  2. If X is optimal, then q1 > qk and qn < qn-k.  F systems with n = 2k+1, k > 2:  F systems with n = 2k+1, k > 2 Theorem 2. Let X be a design for a linear consecutive-k-out-of-(2k+1):F system, k > 2. Let X satisfies general necessary conditions for the optimal design. Assume qk+1  qk. If q1…qk-1  qk+2…q2k, then Xk;k+1 is a better design, otherwise X1,…,k-1;2k,…,k+2 is a better design and (q2k+1,q1,…,q2k) is a better design.  Corollary 2. Let X be a design for a linear consecutive-k-out-of-(2k+1):F system, k > 2. If X is optimal, then min{q1,q2k+1} > qk+1 > max{qk,qk+2}.  F systems with n = 2k+2, k>2:  F systems with n = 2k+2, k>2 Theorem 3. Let X be a design for a linear consecutive-k-out-of-(2k+2):F system, k > 2. Assume qk+1 < qk+2. If qk+3…q2k+1  q2…qk, then Xk+1;k+2 is a better design, while if qk+3…q2k+1  q2…qk, then X1;2k+2 is a better design.  Corollary 3. Let X be a design for a linear consecutive-k-out-of-(2k+2):F system, k > 2. If X is optimal, then (q1,qk+1,qk+2,q2k+2) is singular, and (q1,…,qk,qk+3,…,q2k+2) is nonsingular.  Necessary conditions - Summary:  Necessary conditions - Summary q1 > qk and qn < qn-k for F and G systems, n > 2k, k  2, min{q1,q2k+1} > qk+1 > max{qk,qk+2} for F systems with n = 2k+1, (q1,qk+1,qk+2,q2k+2) is singular, and (q1,…,qk,qk+3,…,q2k+2) is nonsingular for F systems with n = 2k+2. Procedure 1 - F and G systems with n > 2k, k  2:  Procedure 1 - F and G systems with n > 2k, k  2 Let X be a design for a linear consecutive-k-out-of-n:F or a linear consecutive-k-out-of-n:G system with n > 2k, k  2. In order to improve the design: If q1 < qk+1, interchange the components q1 and qk+1, Next, if qn < qn-k, interchange components qn and qn-k.  Illustration: 4-out-of-15 system Procedure 2 :  Procedure 2 Let X be a design for a linear consecutive-k-out-of-(2k+1):F, k  2 In order to improve deign, rearrange components to satisfy general necessary conditions for the optimal design, and If qk+1 < qk 1. Interchange components qk+1 and qk when q1…qk-1  qk+2…q2k; otherwise Take q2k+1 component, put it on the left hand side of the system, next to the q1 component (position 0), Illustration: 7-out-of-15 system Slide32:  2. In such obtained design, rearrange components on positions from 1 to k, and then components on positions from (2k+1) to (k+2) so that general necessary conditions are satisfied. Illustration: 7-out-of-15 system 3. If required, repeat steps 1-3 to further improve this design or until the condition qk+1 > qk is satisfied.  If qk+1 < qk , reverse the order of components and apply steps 1-3 to such rearranged design.  Procedure 3:  Procedure 3 Let X be a design for a linear consecutive-k-out-of-(2k+2):F system, k  2. In order to improve deign: If q1 > q2k+2 and qk+1 < qk+2, interchange components qk+1 and qk+2 when qk+3…q2k+1  q2…qk, or q1 and q2k+2 when qk+3…q2k+1  q2…qk. If q1 < q2k+2 and qk+1 > qk+2, interchange components qk+1 and qk+2 when qk+3…q2k+1  q2…qk, or q1 and q2k+2 when qk+3…q2k+1  q2…qk.  Illustration: 7-out-of-16 system Binary search method:  Binary search method The following necessary conditions for the optimal design of linear consecutive-k-out-of-n:F systems with n  {2k+1,2k+2} are applied: Two worst components are placed on positions 1 and n. (Assume the worst is on position 1 wlog), qk+1 > max{qk,qk+2} if n = 2k+1, qk+1 > qk+2 if n = 2k+2, The design is nonsingular, Components from positions 1 to k as well as components on positions from n to (n-k+1) are arranged in non-decreasing order of component reliability. Randomization method:  Randomization method 1. Generate a random design of a linear consecutive-k- out-of-n system, n > 2k, k  2. 2. Apply Procedures 1-3 to improve the design, if necessary. 3. Rearrange components on positions from 1 to k and then on positions from n to (n-k+1) in non-decreasing order of component reliability. 4. Compare this design with the previous design and keep the better one. 5. Repeat steps 1-4 as require (enough designs have been generated, or the improvements in step 4 becomes insignificant despite many repetitions). Significance of the results Remark 1:  Significance of the results Remark 1 Birnbaum reliability importance I(i) = R(System/i works) - R(System/i fails) = R(p1,...,pi-1,1,pi+1,...,pn) - R(p1,...,pi-1,0,pi+1,...,pn).  The intuition applied in many algorithms is that a component with a higher reliability should be placed in a position with a higher Birnbaum importance. Previous result. If we assume that all components of the system have the same reliability, then I(1)  I(k+1) for F and G systems with n  2k+1.  Theorem 1 and necessary condition stated in Corollary 1 give a stronger result, which also allows the component reliabilities to be distinct. Significance of the results Remark 2:  Significance of the results Remark 2 Suppose that X is a design of a linear consecutive system with n > 2k. Let i and j be the intermediate components satisfying k  i < j  n-k+1. According to previously published results such components are incomparable in a sense that the information qi > qj is insufficient in determining whether pairwise rearrangement of components i and j improves the system. According to the results established for systems with n=2k+1 (Theorem 2, Corollary 2) and n=2k+2 (Theorem 3, Corollary 3), it is possible however to establish, as a necessary condition, which of components i and j should be more reliable for the design to be optimal. Significance of the results Remark 3:  Significance of the results Remark 3 Although variant optimal designs depend upon the particular choices of component reliabilities, the necessary conditions for the optimal design established here rely only on the order of component reliabilities and not their exact values. Therefore they can be applied in the process of eliminating nonoptimal designs from the set of potential optimal designs when it is possible to compare component reliabilities, without necessarily knowing their exact values (e.g. based on the age of the components). METHOD:  METHOD Proof of Theorem 1 - outline:  Proof of Theorem 1 - outline Formula 1. F(X) - F(X1;k) = qk+2(qk+1 - q1)  W, where W = [F(q2,…,qk,1,1,qk+3,…,qn) - F(1,q2,…,qk,0,1,qk+3,…,qn)].  Definition. W(0) = F(1,1,qk+3,…,qn) - F(0,1, qk+3,…,qn), W(i) = F(11,…,1i,1,1, qk+3,…,qn) - F(11,…,1i,0,1, qk+3,…,qn) for 1  i  k-2, W(k-1) = F(11,…,1k-1,1,1, qk+3,…,qn) - F(1, 11,…,1k-1,0,1, qk+3,…,qn).  Formula 2. W = pkW(0) + [pk-1qkW(1) + pk-2qk-1qkW +…+ p2q3…qkW(k-2)] + q2…qkW(k-1).  Note that W(k-1) = 0 and W(i) > 0 for all 0  i  k-2. Hence W > 0, and so the theorem follows. Proof of Theorem 2 - outline:  Proof of Theorem 2 - outline Formula 1. F(X) - F(Xk;k+1) = (qk-qk+1){q1...qk-1 - qk+2...q2k + qk+2...q2k+1 - q1...qk-1qk+2...q2k+1}.  Formula 2. F(X) - F(X1,…,k-1;2k,…,k+2) = {qk+2...q2k - q1...qk-1}  {qk+1p2k+1 + q2k+1 - qk}.  Case q1… qk-1  qk+2... q2k follows directly from the Formula1. Let q1… qk-1 < qk+2... q2k. If q2k+1 < qk then by general necessary conditions q1 > q 2>…> qk > q2k+1 > q2k >…>qk+2, and then q1… qk-1 > qk+2... q2k, contrary to the assumption. Hence q2k+1> qk and so X1,…,k-1;2k,…,k+2 is a better design. Apply formula 1 to X1,…,k-1;2k,…,k+2 and the theorem follows.  Proof of Theorem 3 - outline:  Proof of Theorem 3 - outline Formula 1. F(X) - F(Xk+1;k+2) = (qk+2 - qk+1) [(p2k+2qk+3...q2k+1- p1q2...qk) - (p2k+2 - p1)q2...qkqk+3...q2k+1].  Formula 2. F(X) - F(X1;2k+2) = (q1 - q2k+2) [(pk+1q2...qk - pk+2qk+3...q2k+1) - (pk+1 -pk+2)q2...qkqk+3...q2k+1].  If qk+3...q2k+1  q2...qk then p2k+2qk+3...q2k+1 - p1q2...qk  (p2k+2 - p1)q2...qk > (p2k+2 - p1) q2...qkqk+3...q2k+1, otherwise pk+1q2...qk - pk+2qk+3...q2k+1  (pk+1 -pk+2) qk+3...q2k+1 > (pk+1 -pk+2)q2...qkqk+3...q2k+1. Hence Theorem 3 follows. 

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