Generalized reliability bounds for coherent structures

In this article we introduce generalizations of several well-known reliability bounds. These bounds are based on arbitrary partitions of the family of minimal path or cut sets of the system and can be used for approximating the reliability of any coherent structure with i.i.d. components. An illustration is also given of how the general results can be applied for a specific reliability structure (two-dimensional consecutive-k1 × k2-out-of-n1 × n2 system) along with extensive numerical calculations revealing that, in most cases, the generalized bounds perform better than other available bounds in the literature for this system. © 2000 Applied Probability Trust

On a Markov chain approach for the study of reliability structures

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description

Consecutives-k, r-out-of-n:DFM systems

A dual failure mode (DFM) system is a system whose components are subject to two different kinds of failure (three-state units). In this paper, a new system (the consecutive-k, r-out-of-n:DFM system) is introduced by considering an extension of the well-known and much studied consecutive-k-out-of-n:F system to a dual failure mode environment. Recursive formulae are provided for the evaluation of the reliability of a consecutive-k, r-out-of-n:DFM system with unequal component failure probabilities. Finally, some simple upper and lower reliability bounds are established which provide quite adequate approximations, at least for highly reliable systems. Copyright © 1996 Elsevier Science Ltd

Non-parametric randomness tests based on success runs of fixed length

Let Xn be a random variable enumerating the number of appearances of a specific pattern in a sequence of n Bernoulli trials. A new method is presented for obtaining the conditional distribution of Xn given the number of successes in the n trials. The method is applied to three fixed-length run statistics and the results are used for establishing and investigating certain non-parametric tests of randomness

Bounds for Reliability of Consecutive k-within-m-out-of-n:F Systems

This paper provides upper & lower bounds for the reliability of a (linear or circular) consecutive K-within-m-out-of-n:F system with unequal component-failure probabilities. Numerical calculations indicate that, for systems with components of good enough reliability, these bounds quite adequately estimate system reliability. The estimate is easy to calculate, having computational complexity O(m2 • n). For identically distributed components, a Weibull limit theorem for system time-to-failure is proved. © 1993 IEE

A general class of nonparametric control charts

In this article, we introduce a new general class of nonparametric Shewhart-type control charts using the lengths of runs of test sample observations between successive observations of a reference sample. Several control charts that have appeared in the literature are members of the new family. In addition, 3 new nonparametric control charts that belong to the class are introduced and studied. Numerical results depict that the proposed charts attain competitive in-control and out-of-control performance as compared with existing nonparametric charts. Copyright © 2018 John Wiley & Sons, Ltd

Sooner waiting time problems in a sequence of trinary trials

Let X1, X1,⋯ be a (linear or circular) sequence of trials with three possible outcomes (say S, S* or F) in each trial. In this paper, the waiting time for the first appearance of an S-run of length k or an S*-run of length r is systematically investigated. Exact formulae and Chen-Stein approximations are derived for the distribution of the waiting times in both linear and circular problems and their asymptotic behaviour is illustrated. Probability generating functions are also obtained when the trials are identical. Finally, practical applications of these results are discussed in some detail

Distribution theory of runs: A Markov chain approach

The statistics of the number of success runs in a sequence of Bernoulli trials have been used in many statistical areas. For almost a century, even in the simplest case of independent and identically distributed Bernoulli trials, the exact distributions of many run statistics still remain unknown. Departing from the traditional combinatorial approach, in this article we present a simple unified approach for the distribution theory of runs based on a finite Markov chain imbedding technique. Our results cover not only the identical Bernoulli trials, but also the nonidentical Bernoulli trials. As a byproduct, our results also yield the exact distribution of the waiting time for the mth occurrence of a specific run. © 1994 Taylor & Francis Group, LLC

Application of the stein‐chen method for bounds and limit theorems in the reliability of coherent structures

The Stein‐Chen method for establishing Poisson convergence is used to approximate the reliability of coherent systems with exponential‐type distribution functions. These bounds lead to quite general limit theorems for the lifetime distribution of large coherent systems. © 1993 John Wiley & Sons, Inc. Copyright © 1993 Wiley Periodicals, Inc., A Wiley Compan

Reliability bounds for coherent structures with independent components

In this article a minimal path upper bound and a minimal cut lower bound on the reliability of a coherent system are derived for the case of independent (but not necessarily identical) components. Coupling these bounds with the classical Esary and Proschan's (1963) bounds, some limit theorems are established for the reliability of large coherent systems under quite general conditions. © 1995