-

70 IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 1, 1988 APRIL

A Direct Algorithm for Computing Reliability of a Consecutive-k Cycle

D. Z. Du

F. K. Hwang, Affiliate Member IEEE Massachusetts Institute of Technology, Cambridge

AT&T Bell Laboratories, Murray Hill

Key Wom3 - Consecutivek-out-ofn system, Recursive equation.

RoadcrAidk - Purpose: Report a derivation. Special math needed for explanations: Combinatorics. Special math needed to use results: None. Results useful to: Reliability analysts and theoreticians.

Abstract - A consecutive4 cycle is a circular system such that the system fails if and only if any k consecutive components all fail. Reliabillties for consecutive4 cycles are usually computed by recursive equations. However, most recursive equations pro- posed so far for the cycle involve reliabiuties for consecutive4 lines. Thus we have to run two passes where the first pass com- putes only the line reliabilities. We propose a recursive equation involving cycles only. It is simpler in form but much harder to understand on intuitive grounds. Another advantage is that the new cycle recursion has the same form as a line recursion previously proposed. Thus a uniform treatment of lines and cycles is possible. We use this uniform approach to obtain some explicit solutions of both line and cycle reliabilities for 2 < A d 4.

1. INTRODUCTION

A consecutive-k cycle is a circular system of n com- ponents such that the system fails if and only if any k con- secutive components all fail. Under the i.i.d. model the components are assumed to be statistically independent and each component has probability p of working and probability q = 1 - p of not working. Several papers [l-31 have proposed recursive algorithms to compute the reliabilities of consecutive-k cycles. The fastest one [ 11, re- quiring O(n) time, is the following (for n 2 k + 2):

R&, k, n) = PRL@~ k, n - 1) + qR&, k, n - 1)

- kp2qkRL@, k, n - k -2). (1)

where Rc denotes reliability for cycle and RL denotes reliability for line. Recently [4,5] , the closed-form solution for the recursive algorithms has been obtained (for n 2 k):

)] - 4". n - k(i + 1) - 1

While the closed-form solution is nice to have for many other purposes, it does not automatically qualify as the best way to compute the reliabilities. Granted, there are at most n/k nonzero terms in the sum; but each binomial coefficient involves n/k multiplications. Furthermore, the closed-form formula computes only the reliability for one fixed n while the recursive algorithm computes O(n) reliabilities, ie, for n' from k + 2 to n.

Most recursive algorithms proposed so far compute the cycle reliabilities in terms of line reliabilities. This is bothersome in concept and also in practice since we have to run two passes for the computation-the first pass just to compute the line reliabilities (using recursive algorithms for the line). In this paper we propose a recursive algorithm involving the cycle reliabilities only. The new recursion takes the same form as a line recursion previous- ly proposed, though its validity depends on very different and much more subtle reasons. Thus a uniform treatment for the line case and the cycle case is now possible. We use this uniform approach to obtain explicit solutions for both line and cycle reliabilities for 2 6 k 6 4.

2. NOTATION

n k minimum number of consecutive components

P

RL@,k,n) reliability of a consecutive-k-out-of-n line. R&k,n) reliability of a consecutive-k-out-of-n cycle.

number of components in the system.

whose failures cause system failure. probability that a component works.

4 1 - p .

3. THE MAIN RESULTS

We define R&, k, n) G RL(p, k, n) z 1 for 0 6 n <

We quote three results from the literature. k.

Lemma 1 [6]:

R&, k, k) = 1 - qk. R&, k, n) = 1 - 4" - npqk,

f o r k + 1 6 n d 2k.

Lemma 2.

k- 1

RL@, k, n) = pqi RL@, k, n - i - 1) for n 2 k. i=o

(Easily obtained by substituting pi = p in [3, (la)].

0018-9529/88/O400-0070$01 .@IO1988 IEEE

DU/HWANG: A DIRECT ALGORITHM FOR COMPUTING RELIABILITY OF A CONSECUTIVE-K CYCLE

~

71

Lemma 3 [2].

R&, k, n) = p2 k- 1

(i + l)q'RL@, k, n - i - 2) i=O

f o r n 2 k + 1.

We are now ready to prove our main results.

k-1

Theorem 1. R&, k, n) = pqi R&, k, n - i - l ) , i=O

for n 2 2k.

Proof. For n 2 2k + 1 the proof is rather straightfor- ward.

k-1

R&, k, n) = p2 0' + 1)d RL@, k, n - j - 2) j=0

by lemma 3 k-1 k-1

= P 2 c j=0 t i + 1 ) q ' C p q i R L @ , k , n - j - i - 3 ) i=O

by lemma 2 k- 1

( j + l)@RL@, k, n - j - i - i=0

k- 1

= pqi R&, k, n - i - 1 ) by lemma 3. i=0

For n = 2k the proof is more involved and appears in the appendix. QED

Theorem 1 and lemma 2 (for n 2 2k)show that the cy- cle and the line have the same recursive equation, though the initial conditions are different as shown by lemmas 1 and 2 (for k d n d 2k - 1) . However, the same recursive equation fits the two cases for very different reasons. For the line case term i in the sum is simply the joint probabili- ty of the system in a working state, and component i + 1 is the first (or last) working component; but not so for the cy- cle case. Also note that theorem 1 is not true for n = 2k - 1. For example:

R&, 2,3) = p3 + 3p2q = 3p2 - 2p3

+ PR&, 2, 2) + P9R&, 2 , l )

= p(l - q2) + pq = p + p2 - p3.

For n 2 2k + 1 a variation of theorem 1 has been previously given [5] without proof. We now state the result and give a proof.

Theorem 2. R&, k, n) = R&, k, n - 1 )

- pqkR&, k, n - k - 1 ) for n 2 2k + 1.

Pro0 f.

R&, k, n) = k- 1

pqi R&, k, n - i - 1 ) by theorem 1

= p R&, k, n - 1) + q i=0 cpqiR&, k, n - i - 2)

i=O

k- 1

- pqk R&, k, n - k - 1 )

- p q k R & , k , n - k - 1 ) by theorem 1

= R&, k, n - 1 ) - pqk R&, k, n - k - 1).

The recursion in theorem 2 is in the same form as a line recursion in [6]. Again, it is difficult to find a direct argument for theorem 2 not using line results. Also note that theorem 2 is not true for n = 2k. For example:

= 3p2 - 2p3 - pq2 = - p + 5p2 - 3p3.

4. EXPLICIT SOLUTIONS FOR SMALL k

The closed-form solution for the cycle recursive equa- tions as exhibited in (2) involves a sum over O(n) terms. By an explicit solution we mean an expression without a sum or a product over O(n) terms though n is allowed as an ex- ponent. For k = 2 such an explicit solution was given in ~41:

(note typo errors in [4])

by using a combinatorial identity. We now give a systematic way of obtaining such explicit solutions for 2 d k d 4 .

Consider the equation:

Then (3) has k nonzero roots which are also the roots of:

k- 1

(4)

Let xi, ..., xk denote the k roots. Then a general solution for the recursive equation of theorem 1 is:

k

R&, k, n) = aiX: ( 5 ) i= 1

72 IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 1,1988 APRIL

where al, ..., (Yk are determined by the initial conditions given in lemma 1 . Note that for 2 < k < 4 we can always solve (4) by the theory of polynomial equations. Hence we can always obtain explicit solutions for R&, k, n ) in the form of (5 ) for 2 < k < 4. A similar argument holds for the line case. An interesting observation is that though the recursive equation of theorem 2 is more appropriate for computing the reliability, it is not as good as the recursive equation of theorem 1 for obtaining explicit solutions, since the corresponding polynomial equation has degree k + 1 .

5 . CONCLUSIONS We have given an O(n) time recursive equation to

compute the reliabilities of consecutive-k-out-of-n cycles. This recursive equation involves cycles only and hence can be computed in one pass. Furthermore, it has the same form as a recursive equation for lines. Thus a uniform treatment can be taken. We have also used this uniform approach to obtain explicit solutions for both lines and cycles for 2 < k < 4.

APPENDIX Proof of Theorem 1 for n = 2k

k-1

R&, k, 2k) = p’ (j + 1)q‘ RL@, k, 2k - j - 2) j=O

by lemma 3 k-2 k- 1

= p ’ (j + 1)d C p q i R L @ , k, 2k - j - i - 3) j=0 i=O

+ p2kqk-’RL@, k, k - 1 ) by lemma 2 k- 1 k-2

= i=O c P d [P’ j=0 c 0’ + 1)q‘ 7

RL@, k, 2k - j - i - 3) + p2kqk’ RL@, k, k - 1 )

= i=O c p q i [ p ’ j = O (j + l)q‘RL@, k, 2k - j - i - 3 )

- p2kqk-’] + pqk-$’

1 k-2 k- 1

k-2

(j + l)#RL@, k, k - j - 2) j=O

+ pZkqk-’RL(P, k, k - 1)

k-2

= pqi Rc(p, k, 2k - i - 1) i=O

+ p3qk’(l - kqkl + (k - l)qk)/(l - q)’ + p2kqk-’

k-2

pqi R&, k, 2k - i - 1 ) + pqk-’(l - qk) i=O

k- 1

= pqi RC(p, k, 2k - i - 1 ) . i=0

ACKNOWLEDGMENT

We thank the referees and the Editor for helpful com- ments. We also thank Prof. Papastavridis for informing us that theorem 2 has appeared in [ 5 ] .

REFERENCES

[l] I. Antonopoulou, S. Papastavridis, “Fast recursive algorithm to evaluate the reliability of a circular consecutive-k-out-of-n:F system,” IEEE Trans. Reliability, vol R-36, 1987 Apr, pp 83-84.

[2] C. Derman, G. Lieberman, S. Ross, “On the consecutive-k-out-of- n:F system,” IEEE Trans. Reliability, vol R-31, 1982 Apr, pp 57-63.

[3] F. K. Hwang, “Fast solutions for consecutive-k-out-of-n:F system,” IEEE Trans. Reliability, vol R-31, 1982 Dec, pp 4 4 7 4 8 .

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

[5] M. Lambiris, S. Papastavridis, “Exact probability formulas for linear and circular consecutive-k-out-of-n:F systems,” IEEE Trans. Reliability, vol R-34, 1985 Jun, pp 124-126. J. G. Shanthikumar, “Recursive algorithm to evaluate the reliability of a consecutive-k-out-of-n:F system,” IEEE Trans. Reliability, vol

[6]

R-31, 1982 D ~ c , pp 442443.

AUTHORS

Dr. F. K. Hwang; AT&T Bell Laboratories; 600 Mountain Avenue; Murray Hill, New Jersey 07974 USA.

F. K. Hwang received the BA (1960) from National Taiwan Universi- ty, MBA (1964) from City University of New York and PhD (]%E) in statistics from North Carolina State University. He has been a member of the technical staff at Bell Laboratories since 1967. He is the author of 200 papers (some in Chinese) in combinatorics, optimization, and applied mathematics.

Dr. D. Z. Du; Department of Mathematics; Massachusetts Institute of Technology; Cambridge, Massachusetts 02139 USA.

D. Z. Du received the MS (1980) from the Institute of Mathematics, Academia Sinica and PhD (1985) in Computer Science from University of California, Santa Barbara. He spent a year at the Mathematical Sciences Research Institute at Berkeley before he joined the Department of Mathematics at MIT. Du has published 50 papers in mathematical pro- gramming, computer science and combinatorics.

Manuscript TR87-002 received 1987 January 8; revised 1987 May 29.

- p3kqk-’ ( 1 - qk-’)/(1 - 9) IEEE Log Number 16739 4 TR W