Birnbaum Reliability Importance Research Articles

RELIABILITY AND IMPORTANCE MEASURES OF WEIGHTED-k-OUT-OF-n SYSTEM

Recently many popular reliability systems are being extended to weighted systems where weight associated with each component refers to load/capacity of that component. The simplest k-out-of-n system extended under this concept is known as weighted-k-out-of-n system. A weighted-k-out-of-n: G(F) system consists of n components each one having a positive integer weight wi, i = 1, 2, …, n such that the total system weight is w and the system works (fails) if and only if the accumulated weight of working (failed) components is at least k. In this paper, we introduce the concept of Weighted Markov Bernoulli Trial (WMBT) {Xi, i ≥ 0} and study the distribution of [Formula: see text], the total weight of successes in the sequence of n WMBT and conditional distribution of [Formula: see text] given that uth trial results in failure/success. We refer the distribution of [Formula: see text] as Weighted Markov Binomial Distribution (WMBD). Further we discuss application of WMBD and conditional WMBD in evaluation of reliability, Birnbaum reliability importance (B-importance) and improvement potential importance of weighted-k-out-of-n: G/F system. The numerical work is included to demonstrate the computational simplicity of the developed results. Further we compare our study with the existing results in terms of efficiency and find that our results are efficient for large values of k.

Birnbaum Reliability Importance for (n,f,k) and ⟨n,k,f⟩ System

The ( n, f, k): F/⟨ n, k, f ⟩: F system is the combination of most popular consecutive-k-out-of-n and f-out-of-n system and its failure is caused by two different failure criteria. The ( n, f, k): F (⟨ n, k, f ⟩: F) system consists of n components ordered in a line or circle, while the system fails if and only if there exist at least f failed components or (and) at least k consecutive failed components. In this paper, we consider the sequence {Xu, u ⩾ 1} of {0, 1}-valued Markov Bernoulli trials (MBT) and study the Birnbaum reliability importance for components of ( n, f, k): F(G), and ⟨ n, k, f ⟩: F(G) system through the conditional joint distribution of , where and is the number of non overlapping occurrences of i-runs of length ki(i = 0, 1) in n MBT. The formula for evaluation of exact Birnbaum reliability importance is developed and the results are demonstrated numerically. We also bring out the important inter-relations between the reliability and reliability importance of four systems as f-out-of-n: F, consecutive-k-out-of-n: F, ( n, f, k): F and ⟨ n, k, f ⟩: F systems.

Computational Methods for Reliability and Importance Measures of Weighted-Consecutive-System

We consider the two reliability systems, namely, the weighted-consecutive- k-out-of- n:F (Cω(k,n:F)), and weighted-m-consecutive- k-out-of- n:F system (Cmω(k,n:F)). The weighted-systems have in general n components, each one having a positive integer weight ωi, i=1,2,...n such that total weight of all components of the system is ω=Σi=1nωi. The Cω(k,n:F) fails iff the total weight of failed consecutive components is at least k. We propose Cmω(k,n:F) as a generalization of Cω(k,n:F) which fails iff there are at least m non-overlapping groups of consecutive failed components with a total weight of at least k. Here we study the reliability, Birnbaum reliability importance, and improvement potential importance of a weighted-consecutive system based on the distribution of the failure run statistic in the sequence of weighted Bernoulli trials. We develop a simplified, efficient formula for the evaluation of reliability and importance measures of the systems under consideration, and demonstrate the results numerically.

BIRNBAUM IMPORTANCE FOR CONSECUTIVE-k SYSTEMS

Reliability importance of components of various systems is an important part in the reliability study. In this paper Birnbaum-reliability importance (B-importance) of components of three popular consecutive-type systems as consecutive-k-out-of-n: F-system, m-consecutive-k-out-of-n: F system and r-within-consecutive-k-out-of-n: F system is studied. The B-importance is studied when the survival probabilities of components of the system are Markov dependent. This study is based on conditional distribution of number of occurrences of failure runs of length k and number of occurrences of scanning windows of length k containing r failures in the sequence of Markov Bernoulli trials. Simplified formulae for calculation of B-importance of each component of the three consecutive-type systems are developed in terms of survival probabilities. To demonstrate the simplicity of the derived results, numerical study is also included.

A new approach to system reliability

Calculating system-reliability via the knowledge of structure function is not new. However, such attempts have had to compromise with the increasing complexity of a system. This paper overcomes this problem through a new representation of the structure function, and demonstrates that the well-known systems considered in the state-of-art follow this new representation. With this new representation, the important reliability calculations, such as Birnbaum reliability-importance, become simple. The Chaudhuri, et al. (1991) bounds which exploit the knowledge of structure function were implemented by our simple and easy-to-use algorithm for some s-coherent structures, viz, s-series, s-parallel, 2-out-of-3:G, bridge structure, and a fire-detector system. The Chaudhuri bounds are superior to the min-max and Barlow-Proschan bounds. This representation is useful in implementing the Chaudhuri bounds. With this representation of the structure function, the computation of important reliability measures such as the Birnbaum structural and reliability importance are easy. The use of Chaudhuri bounds is recommended for general use, especially when cost and/or time are critical. The C-H-A algorithm (in this paper) is simple and easy to use. It depends on the knowledge of the path sets of a given structure. Standard software packages are available to provide the minimal path sets of any s-coherent system. The C-H-A algorithm has been programmed in SAS, S-PLUS, and MATLAB.

Some further results on ranking the importance of system components

In this paper, we present two theorems concerning the importance of system components, which respectively characterize the cut-importance ranking introduced by Butler and the criticality ranking due to Boland, Proschan and Tong in terms of the well-known Birnbaum reliability importance measure. These results lead to a relationship between the cut-importance and the criticality rankings.

Comparing Criticality of Nodes via Minimal Cut (Path) Sets for Coherent Systems

In 1989, Boland, Proschan, and Tong [2] introduced the notion of criticality ranking among nodes and developed a procedure for obtaining an optimal assignment of components in coherent systems. In this article we obtain characterizations of the criticality ranking in terms of minimal cut (path) sets for coherent systems. Furthermore, utilizing the characterizations, it is shown that the criticality ranking defined by Boland et al. [2] is consistent with the cut-importance ranking introduced by Butler in 1979 [4]. A relationship between the criticality ranking and the well-known and widely used Birnbaum reliability importance measure is also derived.

Reliability and component importance of a consecutive-k-out-of-n system

Abstract This paper explores the duality between the consecutive- k -out-of- n F and G systems, studies the Birnbaum reliability importance of the components in an i.i.d. system, and reviews the research in uniform treatment of the linear and circular consecutive- k -out-of- n :F systems. Upper and lower bounds for the reliability of a linear consecutive- k -out-of- n system are developed by constructing k -fold series and parallel redundant systems from the linear system.

Multistate reliability models

In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance.

Multistate reliability models

In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance.

A complete importance ranking for components of binary coherent systems, with extensions to multi‐state systems

AbstractMeans of measuring and ranking a system's components relative to their importance to the system reliability have been developed by a number of authors. This paper investigates a new ranking that is based upon minimal cuts and compares it with existing definitions. The new ranking is shown to be easily calculated from readily obtainable information and to be most useful for systems composed of highly reliable components. The paper also discusses extensions of importance measures and rankings to systems in which both the system and its components may be in any of a finite number of states. Many of the results about importance measures and rankings for binary systems are shown to extend to the more sophisticated multi‐state systems. Also, the multi‐state importance measures and rankings are shown to be decomposable into a number of sub‐measures and rankings.