Pooja Mohan

GERT Analysis of

An m-consecutive-k-out-of-n:F system, introduced by W.S. Griffith, consists of an ordered linear sequence of n i.i.d. components that fails iff there are at least m non-overlapping runs of k consecutive failed components. However, a situation may occur in which an ordered linear sequence of n i.i.d. components fails iff there are at least m non-overlapping runs of at least k consecutive failed components. We call such a system an m-consecutive-at least-k-out-of-n:F system. This paper presents a graphical evaluation and review technique (GERT) analysis of both types of systems providing closed form explicit formulae for reliability evaluation in a unified manner. GERT, besides providing a visual picture of the system, helps to analyse the system in a less inductive manner. Numerical examples for each system are studied in detail by computing the reliability for various combinations of sets of values of the parameters involved. It is observed that m-consecutive-at least-k-out-of-n:F systems are more reliable than m-consecutive-k-out-of-n:F systems as the number of possible state combinations leading to systems failure are larger in the latter. Mathematica is used for systematic computations. Numerical investigations illustrate the efficiency of GERT in reliability analysis of such systems. In comparison with the existing formulae of m-consecutive-k-out-of-n:F systems for i.i.d. components, the formula obtained by GERT analysis, to be referred to as GERT-F, is much more efficient owing to its significantly low computational time, and easy implementation

m

This paper proposes a new model, a combined m-consecutive-k-out-of-n : F & consecutive-kc-out-of-n : F system, consisting of n components ordered in a line such that the system fails iff there exist at least kc consecutive failed components, or at least m non-overlapping runs of k consecutive failed components, where kc < mk. The components are assumed to be i.i.d. An algorithm for system reliability evaluation, along with an illustrative numerical example, is provided based on the analysis of the system using graphical evaluation and review technique (GERT). The algorithm is also evaluated in terms of the order of computational time. This model has applications to various complex systems such as infrared (IR) detecting and signal processing, and bank automatic payment systems.

-Consecutive-

An m-consecutive-k-out-of-n:F system is a system of n linearly ordered components which fails if and only if at least m non-overlapping sequences of k consecutive components fail. When m = 1, we have the classic consecutive-k-out-of-n:F system about which there is an extensive literature. In this paper, we study the situation in which a system consisting of n linearly ordered sequence of components fails if and only if there are at least m overlapping runs of k consecutive failed components. Graphical Evaluation and Review Technique (GERT) analysis is used for reliability evaluation of the system for both, i.i.d. components and (k–1)-step Markov dependent components, in a unified manner. Software Mathematica is used for systematic computation. Illustrative numerical examples are presented to substantiate the theory.

k

During the last quarter century, a mass of research has been investigated on consecutive-k-out-of-n:F systems, and its various generalizations, addressing problems of reliability computation. Besides, several new reliability models by combining the k-out-of- n model with other consecutive-k-out-of-n models have also been proposed. When the number of working components between the two failures is at maximum d, then the two failed components are called consecutive failures with sparse d, Zhao et al. [8]. In this paper, Graphical Evaluation and Review Technique (GERT) has been applied to study consecutive-k-out-of-n-out-of-:F systems with sparse d, m-consecutive-k-out-of-n:F systems with sparse d, and (n, f, k):F systems with sparse d.

-Out-of-

We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results.

n

This paper provides reliability analysis of linear strict consecutive-k-out-of-n:F system, introduced and studied by Bollinger in 1985, using Graphical Evaluation and Review Technique (GERT). The components are assumed to be i.i.d. One of the strengths of the GERT network is the graphical representation, which is intuitive and easy to understand. By means of Mason’s rule a simple formula for failure probability is obtained. Mathematica Software is used for systematic computations for different sets of values of n, k and p. Numerical examples illustrate the time efficiency of GERT Analysis.