Problem In Reliability Theory Research Articles

Economic Criteria for Optimizing the Assigned Service Life of Machinery and Equipment

The machine is used in a continuous production process in the enterprise that is a market participant. During its use, it can be subject to random failure after which losses occur due to interruption of the process. After failure, the machine is decommissioned and disposed of. Over time, the failure rate increases. Under these conditions, it turns out to be beneficial for the enterprise to set an assigned service life for the machine, after which (if no failure has occurred) it must be disposed of. We solve the problem of optimizing the assigned service life. Usually, to solve this problem, various optimality criteria are used, for example, average costs per unit of time (including the time to eliminate the consequences of a failure), reliability indicators, or other indicators that do not fully reflect the commercial interests of the enterprise owning the machine. Using the principles and methods of asset valuation, we build a mathematical model that allows us to calculate the optimal assigned life of the machine and at the same time assess the market value of the work performed by it. At the same time, in this problem, the optimality criterion is the ratio of the expected discounted costs to the expected discounted volume of work performed by the machine (in units of operating time). Using this criterion, the market value of the enterprise owning the machine will be the highest. The proposed model allows us to estimate the market value of the serviceable machine knowing its age. We compare the proposed and alternative optimality criteria. We give an example in which machine failures have a Rayleigh distribution. The proposed general approach can be used both in solving other optimization problems of reliability theory and for practical valuation of some types of machinery and equipment.

On a Surface Associated with Pascal’s Triangle

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries.

The income approach to the problems of optimizing the repair policy

The article is devoted to the application of approaches and methods used in the valuation of assets to the problems of reliability theory – the assignment of service life and the timing of preventive repairs to degrading item that are subject to failures. In the works on reliability, changes in the characteristics of the item after repair are described by Kijima models and their modifications, in which the state of item is characterized by a virtual / effective age. A similar indicator has been widely used in appraisal activity since the middle of the last century. Meanwhile, it turns out that indicators of this type do not allow us to adequately describe the state of degraded repairable items. It is proposed to describe their condition by two indicators – the age (operating time) at the beginning of the current inter-repair cycle and the same in this cycle. In combination with Kijima's ideas, this allows us to offer a more adequate model of changing the item’s characteristics after repair. To assign service life and terms of preventive maintenance of items, a well-grounded criterion of optimality is required.It is shown that the usually accepted criterion of the minimum expected maintenance and repair costs per unit of time does not meet the interests of enterprises participating in the market. We propose to use the so-called income approach to valuation, focused on maximizing the market value of the enterprise (in the simplest case, this approach leads to the criterion of the expected discounted costs per unit of work performed by the item). This makes it possible to assess the market value of the work performed by the item necessary for calculations, and the corresponding assessment method can be attributed to the cost approach.We construct and analyze the corresponding mathematical models, provide an algorithm for solving them and a numerical example demonstrating the irrationality of commonly used repair policies.

Central Limit Theorem for a Wavelet Estimator of a Probability Density with a Given Weight

The problem of estimating a probability density with a given weight is considered. Probability densities of this type arise in different cases, e.g., analyzing order statistics and studying random-size samples in problems of reliability theory, insurance, and other areas. When constructing an estimator, expansion is used with respect to a wavelet basis based on wavelet functions with bounded spectrum. It is proved that the considered estimator is asymptotically normal when the number of terms of the expansion is fixed and growing.

Fault Prediction Based on the Kernel Function for Ribbon Wireless Sensor Networks

There exist several applications of wireless sensor networks in which the reliable operation can be crucial. Fault prediction is a critical problem in reliability theory for ribbon wireless sensor networks (RWSNs). Accurate fault prediction can effectively improve the availability of the WSNs system. In this paper, we evaluated the network performance for RWSNs, studied the basic theory of kernel functions, proposed a new failure prediction method based on kernel function, and selected the radial basis function as kernel function failure prediction models from two aspects of node hardware failures and network failures for fault prediction. Theoretical evidence and experimental results have shown that the proposed algorithmic prediction method has higher accuracy of 12 and 15% than that of GRNN and PNN respectively. Finally, we provided extensive numerical results to demonstrate the usage and efficiency of the proposed algorithms and complement our theoretical analysis.

Multilevel Monte Carlo for Reliability Theory

As the size of engineered systems grows, problems in reliability theory can become computationally challenging, often due to the combinatorial growth in the number of cut sets. In this paper we demonstrate how Multilevel Monte Carlo (MLMC) — a simulation approach which is typically used for stochastic differential equation models — can be applied in reliability problems by carefully controlling the bias-variance tradeoff in approximating large system behaviour. In this first exposition of MLMC methods in reliability problems we address the canonical problem of estimating the expectation of a functional of system lifetime for non-repairable and repairable components, demonstrating the computational advantages compared to classical Monte Carlo methods. The difference in computational complexity can be orders of magnitude for very large or complicated system structures, or where the desired precision is lower.

Competing risk models in reliability systems, a weibull distribution model with bayesian analysis approach

In reliability theory, the most important problem is to determine the reliability of a complex system from the reliability of its components. The weakness of most reliability theories is that the systems are described and explained as simply functioning or failed. In many real situations, the failures may be from many causes depending upon the age and the environment of the system and its components. Another problem in reliability theory is one of estimating the parameters of the assumed failure models. The estimation may be based on data collected over censored or uncensored life tests. In many reliability problems, the failure data are simply quantitatively inadequate, especially in engineering design and maintenance system. The Bayesian analyses are more beneficial than the classical one in such cases. The Bayesian estimation analyses allow us to combine past knowledge or experience in the form of an apriori distribution with life test data to make inferences of the parameter of interest. In this paper, we have investigated the application of the Bayesian estimation analyses to competing risk systems. The cases are limited to the models with independent causes of failure by using the Weibull distribution as our model. A simulation is conducted for this distribution with the objectives of verifying the models and the estimators and investigating the performance of the estimators for varying sample size. The simulation data are analyzed by using Bayesian and the maximum likelihood analyses. The simulation results show that the change of the true of parameter relatively to another will change the value of standard deviation in an opposite direction. For a perfect information on the prior distribution, the estimation methods of the Bayesian analyses are better than those of the maximum likelihood. The sensitivity analyses show some amount of sensitivity over the shifts of the prior locations. They also show the robustness of the Bayesian analysis within the range between the true value and the maximum likelihood estimated value lines.

Modeling Dependence Between Two Multi-State Components via Copulas

Modeling statistical dependence between two systems or components is an important problem in reliability theory. Such a problem has been well studied for binary systems and components. In the present paper, we provide a way for modeling s-dependence between two multi-state components. Our method is based on the use of copulas which are very popular for modeling s-dependence. We obtain expressions for the joint state probabilities of the two components, and illustrate the results for the case when the degradation in both components follows a Markov process.

Multi-Mode Failure Models for Attribute Test Data in Reliability Systems, a Bayesian Analysis Approach Using Multi-Nomial Distribution Model

This paper describes the extending model of Multi-mode Failure Models by using the Weibull and Gamma distribution models presented in a conference [1,2]. Different than the models in the previous papers which are for variable test data, in this paper we will describe the use of attribute test data for our model. In reliability theory, the most important problem is to determine the reliability of a complex system from the reliability of its components. The weakness of most reliability theories is that the systems are described and explained as simply functioning or failed. In many real situations, the failures may be from many causes depending upon the age and the environment of the system and its components. Another problem in reliability theory is one of estimating the parameters of the assumed failure models. The estimation may be based on data collected over censored or uncensored life tests. In many reliability problems, the failure data are simply quantitatively inadequate. The Bayesian analyses are more beneficial than classical analyses in such cases. The Bayesian estimation analyses allow us to combine past knowledge or experience in the form of an apriori distribution with life test data to make inferences of the parameter of interest. In this paper, we have investigated the application of the Bayesian estimation analyses to multi-mode failure systems for attribute test data. The cases are limited to the models with independent causes of failure. We select our investigation by using the Multi-nomial distribution as our model. This distribution is widely used in reliability analysis for attribute test data. This model describes the likelihood function and follows with the description of the posterior function. A Beta prior is used in our analysis for each model and it is followed by the estimation of the point, interval, and reliability estimations.

A Note On Beta Inverse-Weibull Distribution

The inverse Weibull distribution is one of the widely applied distribution for problems in reliability theory. In this article, we introduce a generalization—referred to as the Beta Inverse-Weibull distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the Beta Inverse-Weibull distribution. The shapes of the corresponding probability density function and the hazard rate function have been obtained and graphical illustrations have been given. The distribution is found to be unimodal. Results for the non central moments are obtained. The relationship between the parameters and the mean, variance, skewness, and kurtosis are provided. The method of maximum likelihood is proposed for estimating the model parameters. We hope that this generalization will attract wider applicability to the problems in reliability theory and mechanical engineering.

FUZZY MODELS OF MARKOV PROCESSES IN RELIABILITY THEORY

It was researched the applicability of fuzzy Markov processes in problems of reliability theory.

Holistic reliability analysis of weighted voting systems from a multi-state perspective

The computation of the reliability of weighted voting systems is an important problem in reliability theory due to its potential application in security, target identification, safety and monitoring areas. Voting systems are used in a wide variety of applications where an acceptance or rejection decision has to be made about a binary proposition presented to the system. For these systems, it is of interest to obtain the probability so that based on the vote of decision-making units, the system aggregates these votes into the right decision when presented with such a proposition. This paper presents a holistic work on weighted voting system reliability by presenting modeling, computation, estimation and optimization techniques. The modeling part takes advantage of the structure of weighted voting systems to present a model of its reliability as a multi-state system. Next, based on the multi-state view of the system, an exact computational approach based on multi-state minimal cut and path vectors is introduced. The paper then acknowledges the computational complexity of the problem and provides a Monte Carlo simulation approach that estimates system reliability accurately and in an efficient computational time. Finally, an optimization heuristic that generates quasi-optimal solutions is presented that is able to solve the problem of maximizing the reliability of a weighted voting system based on a specified number of decision-making units with known reliability characteristics.

Some current problems in reliability theory

Reliability work occupies an increasingly important place in engineering practice. Although the details differ depending on whether mechanical, electrical, chemical, or other systems are under analysis, the reliability concepts and the mathematical foundations cut across the specific fields of application. Over the past 50 years thousands of papers and dozens of books on mathematical models of reliability have been published. A comprehensive survey alone on the current developments in the mathematical theory of reliability would fill a voluminous book. Based on importance both for theory and application and taking into account the interests of the author, current investigations in four important branches of reliability theory are considered: coherent systems, stochastic networks, software reliability, and maintenance theory.

Probabilistic Reliability Engineering

Fundamentals. Reliability Indexes. Unrepairable Systems. Load-Strength Reliability Models. Distributions with Monotone Intensity Functions. Repairable Systems. Repairable Duplicated Systems. Analysis of Performance Effectiveness. Two-Pole Networks. Optimal Redundancy. Optimal Technical Diagnosis. Additional Optimization Problems in Reliability Theory. Heuristic Methods in Reliability. Index.

Compound poisson approximation in systems reliability

The compound Poisson “local” formulation of the Stein-Chen method is applied to problems in reliability theory. Bounds for the accuracy of the approximation of the reliability by an appropriate compound Poisson distribution are derived under fairly general conditions, and are applied to consecutive-2 and connected-s systems, and the 2-dimensional consecutive-k-out-ofn system, together with a pipeline model. The approximations are usually better than the Poisson “local” approach would give. © 1996 John Wiley & Sons, Inc.

On redundancy allocations in systems

Allocation of a redundant component in a system in order to optimize, in some sense, the lifetime of the system is an important problem in reliability theory, having practical applications. Consider a series system consisting of two components (say C 1 and C 2), having independent random lifetimes X 1 and X 2, and suppose a component C having random lifetime X (independent of X 1 and X 2) is available for active redundancy with one of the components. Let U 1 = min(max(X 1, X), X 2) and U 2 = min(X 1, max(X 2, X)), so that U 1 (U 2) denote the lifetime of a system obtained by allocating C to C 1 (C 2). We consider the criterion where C 1 is preferred to C 2 for redundancy allocation if . Here we investigate the problem of allocating C to C 1 or C 2, with respect to the above criterion. We also consider the standby redundancy for series and parallel systems with respect to the above criterion. The problem of allocating an active redundant component in order that the resulting system has the smallest failure rate function is also considered and it is observed that unlike stochastic optimization, here the lifetime distribution of the redundant component also plays a role, making the problem of even active redundancy allocation more complex.

On redundancy allocations in systems

Allocation of a redundant component in a system in order to optimize, in some sense, the lifetime of the system is an important problem in reliability theory, having practical applications. Consider a series system consisting of two components (say C1 and C2), having independent random lifetimes X1 and X2, and suppose a component C having random lifetime X (independent of X1 and X2) is available for active redundancy with one of the components. Let U1 = min(max(X1, X), X2) and U2 = min(X1, max(X2, X)), so that U1 (U2) denote the lifetime of a system obtained by allocating C to C1 (C2). We consider the criterion where C1 is preferred to C2 for redundancy allocation if . Here we investigate the problem of allocating C to C1 or C2, with respect to the above criterion. We also consider the standby redundancy for series and parallel systems with respect to the above criterion. The problem of allocating an active redundant component in order that the resulting system has the smallest failure rate function is also considered and it is observed that unlike stochastic optimization, here the lifetime distribution of the redundant component also plays a role, making the problem of even active redundancy allocation more complex.

Simulation of stochastic processes with known correlation functions

Abstract A procedure for the simulation of normal stochastic processes with specified correlation functions is developed and described in detail. A known function is fitted to the given correlation by least-squares optimisation and is then employed to determine the form of a random number generator, which is the source of the random shift parameters in a set of broken line processes with ordinates determined from zero mean normal distributions. For a given number of such processes, the domain of their correlation functions and the variances of the normal distributions are also determined by the initial optimisation. The sum of the resulting broken line processes is the required simulated process. Any point on this process can be individually simulated, without the need to store large numbers of generated process values, so that the method is suitable for use on microcomputers. Large numbers of points can be generated quickly, as most process parameters are determined before the simulation run. The procedure is therefore suitable for the construction of other processes in the simulation of outcrossing problems in reliability theory.

On the computational complexity of reliability redundancy allocation in a series system

Finding the optimal redundancy that maximizes the system reliability is one of the important problems in reliability theory. A good deal of effort has been done in this field. In this paper, we prove that some reliability redundancy optimization problems are NP-hard. We also derive alternative proofs for the NP-hardness of some special cases of the knapsack problem.

Stochastic order for redundancy allocations in series and parallel systems

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.