IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 5 , 1991 DECEMBER 553

An Explicit Solution for the Number of Minimal p-Cutsequences in a Consecutive-k-out-of-n: F System

F. K. Hwang AT&T Bell Laboratories, Murray Hill

Key Words - Consecutive-k-out-of-n:F system, Failure-time distribution, Minimal cutset, Minimal cutsequence.

Reader Aids - Purpose: Derive exact closed-form solution Special math needed for explanations: Elementary combinatorics

Special math needed to use results: None Results useful to: Reliability analysts and theoreticians

and probability

such that {cl, . . . , ck} is also a cutsequence. Bollinger & Salvia used the minimal p-cutsequences (they called them “failure paths of length p”) to derive a formula for FT when GT is exponen- tial. They observed that while the formula is still tedious for computing FT, since it involves convolution of GT, it is pretty handy to compute moments. They set up recursive equations to compute the numbers of minimal p-cutsequences which are needed in the formula. This paper gives an explicit solution for these numbers.

2. NOTATION & ASSUMPTIONS

Abstract - B o w e r & Salvia proposed an approach to study the failure-time distribution of a consecutive-k-out-of-n:F system by studying the number of minimal p-cutsequences. They gave recursive equations to compute this number for independent, iden- t i d y distributed components. This paper shows that the recur- sive equations have an explicit solution.

Notation

k,n from the definition of k-out-of-n:F system gilb(n/k) number of minimal p-cutsequences in a consecutive- rp’k k-out-of-n:F system

Ck( n - p + 1, p) number of s-cutsets in a consecutive-k-out- of-n:F system

FT( 0 ) failure-time Cdf of a consecutive-k-out-of-n:F system GT(*) failure-time Cdf of a component 1. INTRODUCTION

There are two approaches to study the failure-time distribu- tion FT of a consecutive-k-out-of-n:F system with iid com- ponents whose Cdf is G T ( t ) .

1. Compute snapshots of FT at given moments T = t . The advantage is that FT(t) is easily obtained by substituting G(t) for the component reliability p in an algorithm for the reliability R of a consecutive-k-out-of-n:F system. The disad- vantage is that a continuous Cdf FT cannot be composed from a finite set of snapshots. This difficulty was observed by Shanthikumar [7] who used the recursive-equations algorithm of R to compute FT( t ) . The composition problem is resolved, however, if F T ( t ) has an explicit solution which is continuous in t. The first such solution was demonstrated by Chen & Hwang [3] who used the “generating function of first system failure” algorithm of R to derive FT( t ) as a continuous function of GT( t) . The practicality of this solution is marred by the fact that the explicit solution contains too many terms. Later, a much more succinct explicit solution of R was independently discovered by Lambiris & Papastavridis [5], and Hwang [4]. Papastavridis & Hadjichristos [6] used this solution to: a) derive a more computable expression of FT( t) , and b) give explicit solutions for first and second moments of FT when GT is Weibull.

2 . Derive FT in its general form, as proposed by Boll- inger & Salvia [2], not as a composition of FT(t ) . Define a p-cutsequence,as a permutation of a p-cutset. An m-cutsequence {c1, . . . , cp> is minimal if there does not exist a 1 I k < p

Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

Assumptions

1. The system is consecutive-k-out-of-n:F. 2. At time zero, all components are operating. 3. Failure times of the components are iid.

3. MAIN RESULT

Bollinger & Salvia gave the following recursive equations to compute rp,k for k + 1 5 p < n:

rp,k = [number of p-sequences]

- [number of p-cutsequences which are not minimal]

- [number of p-sequences which are not cutsequences]

= n!/(n - p ) ! - p-1 p-1

i = l p i

r i + n (n-j)p!Ck(n-p+l,p)

rk,k = k!(n-k+l).

Values of Ck(n-p+ 1, p) have been tabulated for k = 3, s I 9, n 5 18 in Bollinger [l].

001 8-9529/91/ 1200-0553$01 .00@199 1 IEEE

554 IEEE TRANSACTIONS ON RELIABILITY, VOL. 40, NO. 5, 1991 DECEMBER

A closed-form solution of Ck(n - p + 1 , p) [4,5] is:

Lemma. Ck(n-p+l ,p)

Theorem.

Y rp,k = ( n - p + l ) ( p - l ) ! ( - 1 ) ;

i = O

n - p + l (;-;;I> -P! i=O ( i )(:I:) for n l p z k z 1.

Proof: See the appendix.

One may also argue directly for the theorem. A sequence is a cutsequence iff the corresponding set is a cutset. However, each p-set generates p! p-sequences. Therefore,

1 - [Pr{a random (i- 1)-set is not a

1 cutset} - Pr{a random i-set is not a cutset}

- p!Ck(n-p+ 1,p)

n! - 5 Pr{a random i-set is a minimal i = k

n! -- - Pr{a random (p - 1)-set is not a cutset} ( n - p ) !

is the number of (p - 1)-sequences which are not cutsequences. Multiplying by the (n - p + 1 ) choices of failed component p, one obtains the number of p-sequences whose first p - 1 com- ponents do not constitute a cutsequence. Subtracting from this the number of p-sequences which are not cutsequences, gives the number of minimal p-cutsequences.

APPENDIX: PROOF OF THEOREM

Proof: For p = k, the r.h.s. in the theorem is:

- k + l ) ( k - l ) ! ( n - k + l ) - k ! [( n - k )-(n- t l ) ]

We prove the general p 1 k+ 1 case by induction on p,

The theorem follows directly from the lemma. Q.E.D.

REFERENCES

[I J R. C. Bollinger, “Direct computation for consecutive-k-out-of-n:F systems”, IEEE Trans. Reliability, vol R-31, 1982 Dec, pp 444-446.

[2] R. C. Bollinger, A. A. Salvia, “Consecutive-k-out-of-n:F system with se- quential failures”, IEEE Trans. Reliability, vol R-34, 1985 Apr, pp 43-45.

[3] R. W. Chen, F. K. Hwang, “Failure distributions of consecutive-k-out- of-n:F systems”, IEEE T r m . Reliability, vol, R-34, 1985 Oct, pp 338-341.

[4] F. K. Hwang, “Simplified reliabilities for consecutive-k-out-of-n systems”, SIAM J. Algebraic Discrete Methods, vol 7, 1986 Apr, pp 258-264.

[5] M. Lambiris, S. Papastavridis, “Exact reliability formulas for linear and circular consecutive-k-out-of-n:F systems”, IEEE Trans. Reliability, vol R-34, 1985 Jun, pp 124-126.

[6] S. Papastavridis, J . Hadjichristos, “Mean time to failure for a consecutive- k-out-of-n:F system”, IEEE Trans. Reliability, vol R-36, 1987 Apr, pp

[7] J . G . Shanthikumar, “Lifetime distribution of consecutive-k-out-of-n:F systems with exchangeable lifetimes”, IEEE Trans. Reliability, vol R-34, 1985 Dec, pp 480-483.

85-86.

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