Reliability lower bounds for two-dimensional consecutive-k-out-of-n:F systems

Abstract

The two-dimensional consecutive-k-out-of-n:F system has received extensive research interest due to the main fact that it can be applied into various areas, e.g., safety monitoring systems, design of electronic devices, disease diagnosis, and pattern recognition. As known, recursive algorithms can be used to derive the exact system reliability. However, the computing time complexity is exponential and it is infeasible for larger systems. In this paper, with the use of artificial perfect components, we convert the two-dimensional consecutive-k-out-of-n:F system into a general one-dimensional consecutive-k-out-of-n:F system. Then, based upon the exact formula for the reliability of one-dimensional consecutive-k-out-of-n:F system, a simple formula is presented for the reliability lower bound of the two-dimensional consecutive-k-out-of- n:F system. Numerical results show the performance of the presented approach. In addition, it is shown that the proposed approach can be easily extended to t-within-connected-(r,s)-out-of-(m,n):F systems. An example is illustrated to derive the system reliability lower bounds.

Introduction

High reliability is very important to all engineering systems and equipments, for example, atomic power plants, aircrafts, and computers and automobiles, etc. It is well known that the reliability of series system is not high, especially of a large series system, and the reliability of parallel system is high, but tends to be very expensive (Chao et al. [1]). A new system, namely consecutive-k-out-of-n:F system (or (one-dimensional) C(k,n:F) system), has caught much attention since 1980 due to the followings (Chao et al. [1]).

• C(k,n:F) system usually has much higher probability than the series systems, and

• C(k,n:F) system is less expensive than the parallel systems.

The C(k,n:F) system consists of n linearly connected components, and it fails if and only if there are consecutive k or more than k components failed. Since 1980, much research has been devoted to the derivation of exact formulae for the reliability of C(k,n:F) system. In the early years, most of the proposed formulae were based upon the recursive equations and assumed that all the components are s-independent and have the same probability. In 1984, Chao and Lin [2] first observed that the general C(k,n:F) system can be imbedded in a Markov chain with 2^k states. However, only systems with small k can be manipulated. Later, Fu [3] successfully reduced the Markov chain into k+1 states and considerably simplified the probability structure of C(k,n:F) system. Subsequently, Fu and Hu [4], and Chao and Fu [5], [6] developed the following simple and well-known exact formula for the reliability of C(k,n:F) system.

$$\text{R(k,n,p:F)=π}{}_0\text{∏}\text{i=1}\text{n}\text{M}{}_{\mathrm{i}}\text{U}{}^{\mathrm{<mtext>T</mtext>}}\text{,}$$

where

$$\text{M}{}_{\mathrm{i}}\text{=}\text{p}{}_{\mathrm{i}}\text{1−p}{}_{\mathrm{i}}\text{0}\text{…}\text{0}\text{0}\text{p}{}_{\mathrm{i}}\text{0}\text{1−p}{}_{\mathrm{i}}\text{…}\text{0}\text{0}\text{:}\text{:}\text{:}\text{:}\text{:}\text{:}\text{p}{}_{\mathrm{i}}\text{0}\text{0}\text{…}\text{0}\text{1−p}{}_{\mathrm{i}}\text{0}\text{0}\text{0}\text{…}\text{0}\text{1}\text{∈R}{}^{\mathrm{(k+1)\times (k+1)}}\text{.}$$

π₀=(1,0,…,0)∈R^1×(k+1), U=(1,1,…,1,0)∈R^1×(k+1), p_i is the reliability of component i.

Interested readers are referred to the excellent survey paper by Chao et al. [1] for more details of C(k,n:F) systems and their relevant research.

The two-dimensional C(k,n:F) system was first introduced by Salvia and Lasher [7]. The system consists of n² components in a square grid of side n and it fails if and only if there is at least one square of side $\text{k}\text{(2⩽k⩽n−1)}$ that contains all failed components. For example, the system in Fig. 1 is a two-dimensional C(k,n:F) system, and when k=2, it fails if components (2,2), (2,3), (3,2), and (3,3) are failed.

Recently, the two-dimensional C(k,n:F) system has received extensive research interest due to the main fact that it can be applied into various areas, e.g., safety monitoring systems, design of electronic devices, disease diagnosis, and pattern recognition (Salvia and Lasher [7]; Koutras et al. [8]). While much work has been devoted to the reliability formula of C(k,n:F) system, little research has been done on the reliability of two-dimensional C(k,n:F) system. As Salvia and Lasher [7] mentioned, the main reason is that it is very difficult to derive “simple” explicit formulas for the reliability of two-dimensional C(k,n:F) system. Therefore, the most research efforts on the development of reliability upper/lower bounds for the two-dimensional C(k,n:F) system.

Upper bounds: Salvia and Lasher [7] suggested an approach for computing the reliability upper bound of two-dimensional C(k,n:F) system, however, Ksir [9] pointed out that this bound was incorrect. Later, Koutras et al. [10], Fu and Koutras [11], [12], and Barbour et al. [13] proposed several upper bounds for the system.

Lower bounds: Salvia and Lasher [7] suggested a reliability lower bound by employing a “binomial type” argument. Using Chen-Stein method, Koutras et al. [10] and Barbour et al. [13] proposed reliability lower bounds for the two-dimensional C(k,n:F) system. But extensive numerical results showed that the classic lower bound of Esary and Proshcan [14] is in general performing quite well and in view of its simplicity is certainly preferable over all other lower bounds (Koutras et al. [10]).

Interested readers are referred to Koutras et al. [8], [10] and Salvia and Lasher [7] for more details.

Recently, a more general system called t-within-connected-(r,s)-out-of-(m,n):F system has been proposed and investigated, in which the system of size m×n is failed if and only if there are at least $\text{t}\text{(1⩽t⩽rs)}$ failed components within any submatrix of size r×s. Such a system generalizes the mentioned two-dimensional C(k,n:F) system when r=s(=k), m=n, and t=rs. For the case of t=rs, Yamamoto and Miyakawa [15] proposed a recursive algorithm for the exact system reliability of rs-within-connected-(r,s)-out-of-(m,n):F systems with computing time complexity O(s^m−rm²rn). Malinowski and Preuss [16] developed the lower and upper bounds for the system reliability of rs-within-connected-(r,s)-out-of-(m,n):F systems. Using improved Bonferroni inequalities, Makri and Psillakis derived the lower and upper bounds on the reliability for k-within two-dimensional consecutive-r-out-of-n:F systems [17] and for k-within-connected-(r,s)-out-of-(m,n):F systems [18], respectively. In 2001, Akiba and Yamamoto [19] proposed a more complicated recursive algorithm for the exact system reliability of t-within-connected-(r,s)-out-of-(m,n):F system. As expected, the computing time complexity, min{O(mt^r(n−s)),O(mt^s(m−r))}, is highly exponential. For example, it requires more than $\text{2}\text{h}$ of CPU time for deriving the exact system reliability when m=30, n=7, r=s=3, and t=6. Clearly, for larger systems the computing time increases drastically for their recursive algorithm. Therefore, it is important to develop simple algorithms for deriving the reliability lower/upper bounds for such systems.

The objectives of the present paper are multiple. Firstly, we propose a simple formula for the reliability lower bound of the two-dimensional C(k,n:F) system. Numerical results are provided to show the performance of the proposed approach and are compared with those in the literatures. Secondly, we show that the proposed approach can be easily extended to t-within-connected-(r,s)-out-of-(m,n):F systems. An example is shown step by step to derive the system reliability lower bounds.

This paper is organized as follows. Section 2 presents the process of converting a two-dimensional C(k,n:F) system into a one-dimensional C(k,n:F) system. Then, a simple formula, based upon the exact formula for the reliability of one-dimensional C(k,n:F) system, is presented for the reliability lower bound of the two-dimensional C(k,n:F) system. In Section 3, we report and discuss the numerical results for the test problems. In Section 4, we extend the proposed approach to t-within-connected-(r,s)-out-of-(m,n):F systems. An example is shown to develop the reliability lower bound for the system. Brief conclusions are summarized in Section 5.