Linear consecutive-k-out-of-n systems

Linear consecutive-k-out-of-nsystems Variant optimal design problem Malgorzata O’Reilly University of Adelaide

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 • 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 linearconsecutive-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 linearconsecutive-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 linearconsecutive-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 • 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 ofconsecutive-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 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 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 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 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 • 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 • Heuristic method (sub-optimal design) • Randomization method (sub-optimal design) • Binary search method (exact optimal design)

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, psfor 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 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 • 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 Systems with 2k  n  3k Malgorzata O’Reilly University of Adelaide

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. Illustration: 5-out-of-15 system

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 • 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 n  {2k,(2k+1)} is for it to be nonsingular.

Procedures to improve designs, based on necessary conditions Procedure A. In order to improve a nonsingular design of a linear consecutive-k-out-of-n:G system with 2k  n  2k+1, k>1, interchange symmetrical components so that design becomes singular.  Illustration: 7-out-of-15 system For a given nonsingular design, the number of possible singular designs produced in this manner is 2 (The second improved design is the reversed version of the first).

Procedure B. In order to improve a singular design of a linear consecutive-k-out-of-n:F system with 2k  n  2k+1, k>1, • Select an arbitrary nonempty set of up to (k-1) pairs of symmetrical components, and then • Interchange the two components in each selected pair.  • Illustration: 7-out-of-15 system The number of possible choices in Step 1 is (2k - 2). Consequently, the best improvement can be chosen, or if the number of choices is too large to consider all options, the procedure can be repeated as required.

Objectives of this research • To explore whether the necessary conditions for variant optimal design of systems with n  {2k,(2k+1)} can be extended to other cases • To establish necessary conditions for variant optimal design of those extended cases • To develop procedures of improving designs not satisfying those necessary conditions

RESULTS

Cases explored are n  2k, k>1 • Variant optimal designs exist only for F systems with 2 < k < n-2 and G systems with 2  k < n/2. • The case n  2k for F systems can be limited to n = 2k due to the following result: • Theorem. A design X is optimal for a linear consecutive-k-out-of-n:F system with n < 2k if and only if • The (2k-n) best components are placed from positions (n-k+1) to k in any order, and the design (q1,…,qn-k,qk+1,…,qn) • is optimal for a linear consecutive (n-k)-out-of-2(n-k):F system. 

Definition of X* Suppose X = (q1,…,q2k+m), k > 1, m  2. Let  be an arbitrary nonempty subset of {q(m),…,q(k)}. We denote by X* the design obtained from X by interchanging every component listed in with its symmetrical component. Illustration: 6-out-of-15 system (k = 6, m = 3)

Main results - F systems Theorem 1. Let X = (q1,…,q2k+m) be singular, 2  m  k. Then X* is nonsingular and is a better design of a linear consecutive-k-out-of-(2k+m):F system for any chosen X*.  Corollary 1. A necessary condition for the optimal design of a linear consecutive-k-out-of-(2k+m):F system with 2  m  k is for it to be nonsingular.  Although the above necessary condition corresponds to the case n  {2k,(2k+1)}, proof is more complicated and the results do not mirror exactly those earlier results.

Main results - G systems Theorem 2. Let X = (q1,…,q2k+m) be singular, 2  m  k. Then X* is nonsingular and X is a better design of a linear consecutive-k-out-of-(2k+m):G system for any chosen X*. Corollary 2. 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. 

Procedure 1 - F systems In order to improve a singular design of a linear consecutive-k-out-of-(2k+m):F system with 2  m  k, Step 1. Select an arbitrary nonempty set of pairs of symmetrical components, excluding (m-1) components on the left-hand side of the system, (m-1) components on the right-hand side of the system, and m components in the middle of the system, and then Step 2. Interchange the two components in each selected pair.  Illustration: 6-out-of-15 system (k = 6, m = 3) The number of possible choices in Step 1 is 2(k-m+1)-1. Consequently, the best improvement can be chosen or, if the number of possible choices is too large to consider all options, the procedure can be repeated as required.

Procedure 2 - G systems Suppose a design of a linear consecutive-k-out-of-(2k+m):G system with 2  m  k is nonsingular. Consider its subsystem composed of (m-1) components on the left-hand side, (m-1) components on the right-hand side, and m components in the middle, in order as in the design. If such subsystem is singular, then in order to improve the design, interchange all required components so that the design becomes singular. Illustration: 6-out-of-15 system (k = 6, m = 3)

Randomization method 1. Generate a random design of a linear consecutive-k- out-of-n system, 2k  n  3k. 2. Apply Procedures A-B or Procedures 1-2 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 Example. Consider a linear consecutive-4-out-of-10 system and assume q1>q2>…>q10. Let Y=(1,4,5,7,9,10,8,6,2,3) and Z=(1,2,5,7,9,10,8,6,4,3). Then by Theorem 1, a nonsingular Y is a better design than a singular Z. This is despite the fact that Z satisfies general necessary conditions for the optimal design, while Y does not. 

Singular and nonsingular optimal designs for G systems exist Example. (1,5,7,9,8,6,4,3,2) is a nonsingular design of a linear consecutive-3-out-of-9:G system. It is optimal for q1=0.151860, q2=0.212439, q3=0.304657, q4=0.337662, q5=0.387477, q6=0.600855, q7=0.608716, q8=0.643610, q9=0.885895. (1,3,4,5,7,9,8,6,2) is a singular design of a linear consecutive-3-out-of-9:G system. It is optimal for q1=0.0155828, q2=0.1593690, q3=0.3186930, q4=0.3533360, q5=0.3964650, q6=0.4465830, q7=0.5840900, q8=0.8404850, q9=0.8864280. 

METHOD

Definitions Definition 1. Let X=(q1,…,q2k+m), 2 m  k, with either m=2T+1 for some T>0 or m=2T+2 for some T0. We define: W(0)_X = F(X), W(t )_X= F(q1,…qk,1,….,1,qk+t+1,…,qk+m-t, 1,…,1,qk+m+1,…,q2k+m) for 1  t  T, W(T+1)_X= F(q1,…qk,1,…,1, qk+m+1,…,q2k+m).

Notation: [a  b] = a+b-ab Definition 2. Let X=(q1,…,q2k+m), 2 m  k, with either m=2T+1 for some T>0 or m=2T+2 for some T0. We define: M(t)_X= [qt+1...qk  qk+m+1... q2k+m-t] for 0  t  T.

Definition 3. Let X=(q1,…,q2k+m), 2 m  k. If m=2T+2 for some T0, we define R(T)_X= pk+T+1qk+m-T{[qT+1…qk qk+m+1…q2k+m-T] }+ qk+T+1pk+m-T{[qk+m+1…q2k+m-T qT+2…qk] }. If m=2T+1 for some T0, then for 0 t  T-1 we define R(t)_X= pk+T+1qk+m-T{ [qt+1…qk F(qk+t+2,…,qk+m-t-1,qk+m+1,…,q2k+m-t-1]}+ qk+T+1pk+m-T{ [qk+m+1…q2k+m-t  F(qt+2,…,qk,qk+t+2,…,qk+m-t-1]}.

Used formulas If m=2T+1, then W(T)_X= pk+T+1M(T)_X + qk+T+1W(T+1)_X. If m=2T+2, then W(T)_X= pk+T+1pk+m-TM(T)_X + pk+T+1qk+m-TW(T+1)_X + R(T)_X.  W(t)_X= pk+t+1pk+m-tM(t)_X + pk+t+1qk+m-tW(t+1)_X + R(t)_X.

Propositions Let X=(q1,…,q2k+m), 2 m  k, with either m=2T+1 for some T>0 or m=2T+2 for some T0. Then Proposition 1. M(t)_X > M(t)_ X*for 0  t  T.  Proposition 2. R(T)_X > R(T)_ X*.  Proposition 3. R(t)_X > R(t)_ X* for 0  t  T. 

Proof of Theorem 1 - outline Theorem 1. Let X = (q1,…,q2k+m) be singular, 2  m  k. Then X* is nonsingular and is a better design of a linear consecutive-k-out-of-(2k+m):F system for any chosen X*.  Proof Step 1. W(T+1)_X  W(T+1)_X* Step 2. W(T)_X > W(T)_X* Step 3. If W(t+1)_X > W(t+1)_ X* then W(t)_X > W(t)_X* From Steps 1-3 and mathematical induction we have W(0)_X > W(0)_ X* that is F(X) > F(X*), and so X* is a better design. 

Linear consecutive-k-out-of-n systems