Reliability and Importance Measures for Combined m-Consecutive-k-Out-of-n: F and Consecutive-kb-Out-of-n: F Systems with Non-Homogeneous Markov-Dependent Components

2018 ◽  
Vol 25 (05) ◽  
pp. 1850022 ◽  
Author(s):  
Mahmoud Boushaba ◽  
Azzedine Benyahia
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
System Reliability ◽  
Probability Generating Function ◽  
Importance Measure ◽  
Joint Reliability ◽  
Independent Case ◽  
New Formula ◽  
Marginal Reliability ◽  
F System

A Combined [Formula: see text]-Consecutive-[Formula: see text]-out-of-[Formula: see text] and Consecutive-[Formula: see text]-out-of-[Formula: see text]: F System consists of [Formula: see text] components ordered in a line such that the system fails iff there exist at least [Formula: see text] consecutive failed components, or at least [Formula: see text] nonoverlapping runs of [Formula: see text] consecutive failed components, where [Formula: see text]. This system was been introduced by Mohan et al. [P. Mohan, M. Agrawal and K. Sen, Combined [Formula: see text]-consecutive-[Formula: see text]-out-of-[Formula: see text]: F and consecutive-[Formula: see text]-out-of-[Formula: see text]: F systems, IEEE Trans. Reliab. 58 (2009) 328–337] where they propose an algorithm to evaluate system reliability by using the (GERT) technique, in the independent case. In this paper, we propose a new formula of the reliability of this system for nonhomogeneous Markov-dependent components. For a Combined [Formula: see text]-Consecutive-[Formula: see text]-out-of-[Formula: see text] and Consecutive-[Formula: see text]-out-of-[Formula: see text]: F System with nonhomogeneous Markov-dependent components, we derive closed-form formulas for the marginal reliability importance measure of a single component, and the joint reliability importance measure of two or more than two components using probability generating function (pgf) and conditional pgf methods.

scholarly journals A linear m-consecutive-k-out-of-n system with sparse d of non-homogeneous Markov-dependent components

2018 ◽  
Vol 233 (3) ◽  
pp. 328-337 ◽  
Author(s):  
Xiaoyan Zhu ◽  
Mahmoud Boushaba ◽  
Abdelmoumene Boulahia ◽  
Xian Zhao
Keyword(s):  
Closed Form ◽  
Conditional Probability ◽  
Generating Function ◽  
System Reliability ◽  
Function Method ◽  
Probability Generating Function ◽  
Importance Measure ◽  
Numerical Examples ◽  
Joint Reliability ◽  
Marginal Reliability

Consider non-homogeneous Markov-dependent components in an m-consecutive- k-out-of- n:F (G) system with sparse [Formula: see text], which consists of [Formula: see text] linearly ordered components. Two failed components are consecutive with sparse [Formula: see text] if and if there are at most [Formula: see text] working components between the two failed components, and the m-consecutive- k-out-of- n:F system with sparse [Formula: see text] fails if and if there exist at least [Formula: see text] non-overlapping runs of [Formula: see text] consecutive failed components with sparse [Formula: see text] for [Formula: see text]. We use conditional probability generating function method to derive uniform closed-form formulas for system reliability, marginal reliability importance measure, and joint reliability importance measure for such the F system and the corresponding G system. We present numerical examples to demonstrate the use of the formulas. Along with the work in this article, we summarize the work on consecutive- k systems of Markov-dependent components in terms of system reliability, marginal reliability importance, and joint reliability importance.

COMBINATORIAL APPROACH TO COMPUTING COMPONENT IMPORTANCE INDEXES IN COHERENT SYSTEMS

2011 ◽  
Vol 26 (1) ◽  
pp. 117-128 ◽  
Author(s):  
Ilya B. Gertsbakh ◽  
Yoseph Shpungin
Keyword(s):  
System Reliability ◽  
Simple Formula ◽  
Analytic Formula ◽  
Importance Measure ◽  
Coherent Systems ◽  
Joint Reliability ◽  
Component Importance ◽  
Definition Of ◽  
Combinatorial Parameter ◽  
Combinatorial Representation

We consider binary coherent systems with independent binary components having equal failure probability q. The system DOWN probability is expressed via its signature's combinatorial analogue, the so-called D-spectrum. Using the definition of the Birnbaum importance measure (BIM), we introduce for each component a new combinatorial parameter, so-called BIM-spectrum, and develop a simple formula expressing component BIM via the component BIM-spectrum. Further extension of this approach allows obtaining a combinatorial representation for the joint reliability importance (JRI) of two components. To estimate component BIMs and JRIs, there is no need to know the analytic formula for system reliability. We demonstrate how our method works using the Monte Carlo approach. We present several examples of estimating component importance measures in a network when the DOWN state is defined as the loss of terminal connectivity.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

Coupled dynamics of populations supported by discrete sites and their continuum limit

2010 ◽  
Vol 466 (2121) ◽  
pp. 2561-2586 ◽  
Author(s):  
Timothy R. Field ◽  
Robert J. A. Tough
Keyword(s):  
Evolution Equation ◽  
Closed Form ◽  
Generating Function ◽  
Rate Equation ◽  
Continuum Limit ◽  
Infinite Chain ◽  
Migration Process ◽  
The Mean ◽  
The Continuum ◽  
Birth Death

The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

Importance measures for degrading components based on cooperative game theory

2019 ◽  
Vol 37 (2) ◽  
pp. 189-206
Author(s):  
Yingsai Cao ◽  
Sifeng Liu ◽  
Zhigeng Fang
Keyword(s):  
Cooperative Game ◽  
Shapley Value ◽  
System Reliability ◽  
Design Methodology ◽  
System Structure ◽  
Importance Measure ◽  
Engineering Systems ◽  
Content Type ◽  
Timely Feedback ◽  
Degradation Models

Purpose The purpose of this paper is to propose new importance measures for degrading components based on Shapley value, which can provide answers about how important players are to the whole cooperative game and what payoff each player can reasonably expect. Design/methodology/approach The proposed importance measure characterizes how a specific degrading component contributes to the degradation of system reliability by using Shapley value. Degradation models are also introduced to assess the reliability of degrading components. The reliability of system consisting independent degrading components is obtained by using structure functions, while reliability of system comprising correlated degrading components is evaluated with a multivariate distribution. Findings The ranking of degrading components according to the newly developed importance measure depends on the degradation parameters of components, system structure and parameters characterizing the association of components. Originality/value Considering the fact that reliability degradation of engineering systems and equipment are often attributed to the degradation of a particular or set of components that are characterized by degrading features. This paper proposes new importance measures for degrading components based on Shapley value to reflect the responsibility of each degrading component for the deterioration of system reliability. The results are also able to give timely feedback of the expected contribution of each degrading component to system reliability degradation.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

System Reliability Allocation and Optimization Based on Generalized Birnbaum Importance Measure

2019 ◽  
Vol 68 (3) ◽  
pp. 831-843 ◽  
Author(s):  
Shubin Si ◽  
Mingli Liu ◽  
Zhongyu Jiang ◽  
Tongdan Jin ◽  
Zhiqiang Cai
Keyword(s):  
System Reliability ◽  
Importance Measure ◽  
Reliability Allocation

Interconnected population processes

1973 ◽  
Vol 10 (01) ◽  
pp. 1-14 ◽  
Author(s):  
E. Renshaw
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Approximate Solutions ◽  
Death Process ◽  
Birth And Death Process ◽  
Population Processes

This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.

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