Journal of the Operati~ns Research Society of Japan Vol. 39, No. 3, September 1996

RELIABILITY OF A CIRCULAR CONNECTED-(r,s)-OUT-OF-(m,n): F LATTICE SYSTEM
Hisashi Yamamoto Masami Miyakawa

Tezkyo Unzverszty of Sczence and Technolog9

Unzverszty of Tokyo

(Received November 11, 1994; Final August 25, 1995)

A circular connected-(r>s)-out-of-(m,n):Flattice system has m n compoaen%s is consisted of n and and m circles. The system fails if and only if a connected (r,s)-matrix of components fail. Fixskly, this paper proposes a recursive algorithm for the reliability of $he system>which requires O(rnn2sm) computing time. In the statisticaIly independent identically distributed case7the c o m p u t i ~ ~ g is able to be reduced. time Secondly, the upper a ~ lower bounds for the system reliability are obtained for the circular connectedd (r,s)-out-of-(m,n):l? lattice system. FinalIy, it is proved that the reliability of the large system tmds to where p 1A and 7 are exp[-pArs] as n = , m Q - I 2 n -+ m if every component has failure probability ~ n - ' / / ~ , constant, ,LA, A > Ol 7 > r , ri rays

Abstract

I . INTRODUCTION The linear (circular) consecutive-k-out-of-n:F system has n llnewly (circularly) ordered components. Each compnent either functkms or fails. The systems fail if and only if k consecutive compnents fail (see Figure 1.1). Defining the rmdom variable Zi by if component i functions zi = if component i Fdils, for i = 1 2 , .*, n, the reliability of the linear consect~tive-k-out-of-n:F system can be expressed as

PI-{ n{zi= 0, a s i s a + k - 1)') and the circular consecutive-k-out-of-n:F system as a=l n-k+l

pro{^^ =Q a s i sa+k-I)'), a=l n

whereZi z Z i " n , i = n+l, n+2,e**y for ni-k-1. Such systems

have been studied by many reseachers including Bolfingermd S d ~ ~ i a &maz*Liebenjzaz ad [3], Rom [4], Ftt [5]and Hwafig