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.

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.

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.

scholarly journals Computational method for importance measure of the k-out-of-n system based on stress–strength interference

2019 ◽  
Vol 234 (1) ◽  
pp. 27-40
Author(s):  
Dong Lyu ◽  
Shubin Si ◽  
Zhiqiang Cai ◽  
Liyang Xie
Keyword(s):  
System Reliability ◽  
Strength Distribution ◽  
Computational Method ◽  
Risk Assessments ◽  
System Level ◽  
Importance Measure ◽  
Relative Significance ◽  
Numerical Examples ◽  
Component Importance ◽  
Distribution Parameters

Importance measures, which are used to evaluate the relative significance of various components to system reliability, have been widely applied in system reliability designs and risk assessments. This article deals with the importance measure for the k-out-of- n system of which components are loaded by common stress. Based on system-level stress–strength interference model, a new computational method for the Birnbaum importance measure is proposed for the k-out-of- n system. Then, two numerical examples are presented to further illustrate the proposed method and some key contents are discussed particularly as follows: (1) the importance measures for the system with s-identical components and nonidentical components are developed, (2) component importance changes as its own strength distribution parameters change and (3) the new method corrects the errors caused by ignoring the failure dependency.

OPTIMAL ALLOCATION OF MULTISTATE ELEMENTS IN LINEAR CONSECUTIVELY-CONNECTED SYSTEMS WITH DELAYS

2002 ◽  
Vol 09 (01) ◽  
pp. 89-108 ◽  
Author(s):  
GREGORY LEVITIN
Keyword(s):  
Genetic Algorithm ◽  
Generating Function ◽  
System Reliability ◽  
Function Method ◽  
Optimal Allocation ◽  
Allocation Problem ◽  
Time Period ◽  
Different Characteristics ◽  
Universal Generating Function

A linear consecutively-connected system consists of N + 2 linear ordered positions. The first position contains a source of a signal and the last one contains a receiver. M statistically independent multistate elements (retransmitters) with different characteristics are to be allocated at the N intermediate positions. The elements provide retransmission of the received signal to the next few positions. Each element can have different states determined by a number of positions that are reached by the signal generated by this element. The probability of each state for any given element depends on the position where it is allocated. The signal retransmission process is associated with delays. The system fails if the signal generated by the source can not reach the receiver within a specified time period. A problem of finding an allocation of the multistate elements that provides the maximal system reliability is formulated. An algorithm based on the universal generating function method is suggested for the system reliability determination. This algorithm can handle cases where any number of multistate elements are allocated in the same position while some positions remain empty. It is shown that such an uneven allocation can provide greater system reliability than an even one. A genetic algorithm is used as an optimization tool in order to solve the optimal element allocation problem.

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.

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