EFFICIENT ALGORITHM FOR RELIABILITY OF A CIRCULAR CONSECUTIVE-K-OUT-OF-N-F SYSTEM

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IEEE TRANSACTIONS ON RELIABILITY, VOL. 42, NO:1, 1993 MARCH

~

163

Efficient Algorithm for Reliability of a

Circular Consecutive-k-out-of-n

:

F System

Jer-Shyan

Wu

Rong-Jaye Chen,

Member IEEE

National Chiao-Tung University, Hsinchu National Chiao-Tung University, Hsinchu

Key Words

-

Circular consecutive-k-out-of-n:F system, System reliability, Algorithm.

Reader Aids

-

Purpose: Report a new algorithm

Special math needed for explanations: Probability theory Special math needed to use results: Same

Results useful to: Reliability analysts and theoreticians

Summary & Conclusions - The time complexities of previously published algorithms for circular consecutive-k-out-of-n:F system are O(n.k2) and O(n.k). This paper proposes a method to improve upon the original O(n

.

k2) algorithm and hence derives an O(n k)

algorithm.

Notation n k pi, qi

RL (iJ) , Rc( i J ) reliability of [linear, circular] system number of components in a system

minimum number of consecutive failed components that causes system failure

probability that component i [functions, fails]; qi+pi

consisting of components i, i+ 1, i + 2 ,

. .

.

, j

Other, standard notation is given in “Information for Readers

& Authors” at the rear of each issue.

3. IMPROVEMENT

Hwang [4] proposed a recursive 0 (n) algorithm for linear consecutive-k-out-of-n:F systems and an 0 (n . k 2 ) algorithm for circular consecutive-k-out-of-n:F system. The 0 ( n - k 2 ) algorithm is:

1. INTRODUCTION

{ s - 1 A consecutive-k-out-of-n:F system consists of a sequence

of n ordered components where each component either functions or fails. The system fails if and only if at least

k

consecutive components fail. There are two topologies for this system: a line and a circle. The reliability analysis of such systems was first studied by Chiang & Niu [2], and later by Derman, Lieber- man, Ross [3], Hwang [4], Shanthikumar [5], Antonopoulou

& Papastavridis [ 11, and Wu & Chen [6]. Hwang [4] proposed a recursive 0 (n) algorithm for linear consecutive-k-out-of-n:F system and an O ( n . P ) algorithm for a circular consecutive- k-out-of-n:F system. Antonopoulou & Papastavridis [ 11 an- nounced that they had found an O(n.k) recursive algorithm for computing the reliability of a circular system. Wu & Chen [6] also found a new O ( n - k ) algorithm for such system. This paper proposes an improvement upon the Hwang [4] O ( n - k 2 ) algorithm and introduces a new O(n.k) algorithm.

2. MODEL Assumptions

tions or fails.

1. Each component, subsystem, and system either func- 2. All n component-states are mutually s-independent. 3. Components 1, 2, .. . , n are arranged as to form a cir- 4. The system or subsystem fails if and only if at least k cle in that order.

consecutive components fail.

n ,

Now, we derive the 0 ( n

.

k ) algorithm for circular consecutive- k-out-of-n:F systems by (1): k n n , m=j+2 J In (2), we need to generate: R ~ ( 2 , n

-

k) ,

. . .

, RL (2,n - 1 ) ; R ~ ( 3 , n - k + l ) ,

...,

RL(3,n-1);

...;

R L ( k + l , n - l ) 0018-9529/92$03.00 01992 IEEE ___-

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164 IEEE TRANSACTIONS ON RELIABILITY, VOL. 42, NO. 1, 1993 MARCH

1 k-1 [2] D. T. Chiang, S. C. Niu, “Reliability of consecutive-k-out-of-n:F system”,

ZEEE Trans. Reliability, vol R-30, 1981 Apr, pp 87-89.

[3] C. Derman, G. Lieberman, S. Ross, “On the consecutive-k-out-of-n:F

system”, ZEEE Trans. Reliability, vol R-31, 1982 Apr, pp 57-63.

[4] F. K. Hwang, “Fast solutions for consecutive-k-out-of-n:F system”, ZEEE Trans. Reliability, vol R-31, 1982 Dec, pp 447-448.

[5] J. G. Shanthiiumar, “Recursive algorithm to evaluate the reliability of a

consecutive-k-out-of-n:F system”, ZEEE Trans. Reliability, vol R-31, 1982 Dec, pp 442-443.

[6] J. S. Wu, R. J. Chen, “An O ( kn ) recursive algorithm for circular

consecutive-k-out-of-n:F system”, ZEEE Trans. Reliability, vol41, 1992 Jun, pp 303-305.

n

qi,

..-)

n

qi; i = l i = l

fi

qi> . . . $

fi

qi.

i = n - k + 2 r = n

The Hwang [4] recursive O ( n ) algorithm can be expressed as:

RL( 1,n) = RL( 19n-1)

-

R L ( 1 , n - k - l ) ‘ P n - k

*

fJ

4i.

i = n - k + l

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AUTHORS While computing RL ( 1 ,n) , we also get RL( 1 , l ) , RL ( 1,2), .

. .

,

RL( 1 ,n

-

1 ) ; that is important. Using this property, we can

evaluate: R L ( 2 , n - k ) ,

...,

R L ( 2 , n - l ) ; R L ( 3 , n - k + 1 ) ,

...,

R L ( 3 , n - 1 ) ;

...; RL(k+

1 , n - 1 ) with time complexity

O ( n - 2 )

+

O ( n - 3 )

+

...+

O ( n - k - 1 ) = O ( n . k ) . And we can compute

r ~ f , ~

qi,

...,

n,k,/qi;

n Y z n - k + 2 qi,

...,

IIY=,,

qi in O ( k ) . Store all derived values in memory. The sum in

( 2 ) contains Yzk ( k

+

1 ) terms, so the time complexity for com-

putingRc(l,n) is: ~ ( n - k )

+

~ ( k )

+

o ( P )

= ~ ( n - k ) , for n

>

k.

REFERENCES

Dr. Rong-Jaye Chen; Dept. of Computer Science and Information Engineer- ing; National Chiao-Tung University; Hsinchu 30050 TAIWAN ROC.

Rong-Jaye Chen (M’90) was born in Taiwan in 1952. He received his

BS (1977) in Mathematics from National Tsing-Hua University and the PhD (1987) in Computer Science from the University of Wisconsin, Madison. Dr. Chen is an Associate Professor in the Dept. of Computer Science and Informa- tion Engineering at National Chiao-Tung University. He is a Member of IEEE. His research interests include reliability theory & algorithm, mathematical pro- gramming, and computer networks.

Jer-Shyan Wu; Dept. of Computer Science and Information Engineering; Na- tional Chiao-Tung University; Hsinchu 30050 TAIWAN ROC.

Jer-Shyan Wu was born in Taipei, Taiwan in 1967. He received his BS (1989) in Computer Science from National Taiwan University, his MS (1991) in Computer Science from National Chiao-Tung University. His research interests include reliability theory, queuing theory, and algorithms.

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

Manuscript TR91-195 received 1991 October 31; revised 1992 April 2.

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