A Multi-Level Trend-Renewal Process for Modeling Systems With Recurrence Data

A repairable system is a system that can be restored to an operational state after a repair event. The system may experience multiple events over time that are called recurrent events. To model the recurrent event data, the renewal process (RP), the nonhomogenous Poisson process (NHPP), and the trend-renewal process (TRP) are often used. Compared to the RP and NHPP, the TRP is more flexible for modeling, because it includes both RP and NHPP as special cases. However, for a multi-level system (e.g., system, subsystem, and component levels), the original TRP model may not be adequate if the repair is effected by a subsystem replacement and if subsystem-level replacement events affect the rate of occurrence of the component-level replacement events. In this article, we propose a general class of models to describe replacement events in a multi-level repairable system by extending the TRP model. We also develop procedures for parameter estimation and the prediction of future events based on historical data. The proposed model and method are validated by simulation studies and are illustrated by an industrial application. This article has online supplementary materials.     

Trend analysis of the power law process using Expectation-Maximization algorithm for data censored by inspection intervals

Trend analysis is a common statistical method used to investigate the operation and changes of a repairable system over time. This method takes historical failure data of a system or a group of similar systems and determines whether the recurrent failures exhibit an increasing or decreasing trend. Most trend analysis methods proposed in the literature assume that the failure times are known, so the failure data is statistically complete; however, in many situations, such as hidden failures, failure times are subject to censoring. In this paper we assume that the failure process of a group of similar independent repairable units follows a non-homogenous Poisson process with a power law intensity function. Moreover, the failure data are subject to left, interval and right censoring. The paper proposes using the likelihood ratio test to check for trends in the failure data. It uses the Expectation-Maximization (EM) algorithm to find the parameters, which maximize the data likelihood in the case of null and alternative hypotheses. A recursive procedure is used to solve the main technical problem of calculating the expected values in the Expectation step. The proposed method is applied to a hospital's maintenance data for trend analysis of the components of a general infusion pump.     

On Reliability of a Multi‐Socket Repairable System

Consider a set of the so‐called sibling components in a multi‐socket repairable system. In the case of an automobile, for example, these siblings would be spark plugs, light bulbs, tires, that is, identical components that are coded with the same part number. When field data are analyzed, a dilemma arises as to how to interpret a recurrent replacement of a sibling component: as a secondary failure of the component that has already been replaced once, or as the first failure of the component's sibling(s)? From the stand point of root‐cause analysis, the task is to understand whether recurrent failures are related to (i) a particular sibling, which might be operating in inauspicious conditions relative to other siblings, or (ii) to all siblings on the vehicle. One could attribute Scenario 1 to a system‐level (e.g. system interaction) problem, and Scenario 2 to a component‐level (supplier quality) problem. We first review a statistical procedure that solves the above‐mentioned dilemma in the framework of ordinary renewal process (ORP) and then extend the discussion to the non‐homogeneous Poisson process (NHPP) and the g‐renewal process (GRP). We also propose advanced Monte Carlo procedure for estimating GRP in this context. Copyright © 2016 John Wiley & Sons, Ltd.     

An Inequality on the Mean Time of Faultless Operation of a Parallel System I

The problem of estimating the reliability characteristics of parallel systems with spares and/or repairable elements is investigated. A parallel system is such that the failure of all units causes failure of the entire system. Under very general assumptions, we have found, among others, an upper as well as a lower bound on the mean time before failure of a parallel system with n independent units. An example is given in detail to illustrate how the theorems proved in section 2 can be applied and the main theorem is stated in section 4. Some bounds on the higher moments of the time to failure are treated in [l].     

A supplemental algorithm for the repairable system in the GO methodology

The GO methodology is an effective method of system reliability analysis and it can be used in the repairable system as well. The calculation formulas of the ordinary operators for the repairable components and the quantification algorithm of the steady characteristics of the repairable system have been presented in Ref. [6]. This paper provides a calculation formula of the failure rate for the repairable system with the independent components, and the calculation procedure of the equivalent failure rate and the average operation probability for the repairable system with the shared signals at complex case are presented. This supplemental algorithm is quite universal for the complex repairable system and quite convenient for calculation by computer. An example demonstrates the calculation process and the result shows that the algorithm is correct and available.     

Analysis of equivalent dynamic reliability with repairs under partial information

For repairable system it is usual to evaluate the effectiveness of repair by considering the life extended between failures. Traditional definition of dynamic reliability is not suitable for the evaluation due to that the population diminishes gradually as failed events may induce non-repairable situations. More often the failure record presents sometimes only the sequence of failure number, it does not specify the system in a repairable or non-repairable condition as it fails. This kind of information is dim in describing the whole picture of failures after repair that should be accounted for estimating the repaired system dynamic reliability. In this paper the cumulative failure data set with repairs, which depicts the failure number in the successive operating ranges, is constructed from such incomplete information. It is developed by fuzzy consideration which (1) distinguishes repairable vs. non-repairable cases in the failure sequence data, and (2) identifies the system failed again after repair in the next time failure sequence data. The membership values in the fuzzy treatment about the data are incorporated with physical sense of cumulative damage. Thus, an equivalent dynamic reliability with repairs (EDRWR) is obtained in comparison with that by jumps representation at the time taking repairs. Finally, an example with different times of repairs for about 191 bus motors is used to demonstrate the suggested methodology. The fittings of EDRWR are quite well accepted in Weibull distribution.     

Estimation Techniques for Common Cause Failure Data With Different System Sizes

Modeling of system lifetimes becomes more complicated when external events can cause the simultaneous failure of two or more system components. Models that ignore these common cause failures lead to methods of analysis that overestimate system reliability. Typical data consist of observed frequencies in which i out of m (identical) components in a system failed simultaneously, i = 1,…, m. Because this attribute data is inherently dependent on the number of components in the system, procedures for interpretation of data from different groups with more or fewer components than the system under study are not straightforward. This is a recurrent problem in reliability applications in which component configurations change from one system to the next. For instance, in the analysis of a large power-supply system that has three stand-by diesel generators in case of power loss, statistical tests and estimates of system reliability cannot be derived easily from data pertaining to different plants for which only one or two diesel generators were used to reinforce the main power source. This article presents, discusses, and analyzes methods to use generic attribute reliability data efficiently for systems of varying size.     

A bivariate optimal replacement policy for a multistate repairable system

In this paper, a deteriorating simple repairable system with k+1 states, including k failure states and one working state, is studied. It is assumed that the system after repair is not “as good as new” and the deterioration of the system is stochastic. We consider a bivariate replacement policy, denoted by (T,N), in which the system is replaced when its working age has reached T or the number of failures it has experienced has reached N, whichever occurs first. The objective is to determine the optimal replacement policy (T,N)^* such that the long-run expected profit per unit time is maximized. The explicit expression of the long-run expected profit per unit time is derived and the corresponding optimal replacement policy can be determined analytically or numerically. We prove that the optimal policy (T,N)^* is better than the optimal policy N^* for a multistate simple repairable system. We also show that a general monotone process model for a multistate simple repairable system is equivalent to a geometric process model for a two-state simple repairable system in the sense that they have the same structure for the long-run expected profit (or cost) per unit time and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results.     

Reliability Modelling of Multiple Repairable Units

This paper proposes a model selection framework for analysing the failure data of multiple repairable units when they are working in different operational and environmental conditions. The paper provides an approach for splitting the non‐homogeneous failure data set into homogeneous groups, based on their failure patterns and statistical trend tests. In addition, when the population includes units with an inadequate amount of failure data, the analysts tend to exclude those units from the analysis. A procedure is presented for modelling the reliability of a multiple repairable units under the influence of such a group to prevent parameter estimation error. We illustrate the implementation of the proposed model by applying it on 12 frequency converters in the Swedish railway system. The results of the case study show that the reliability model of multiple repairable units within a large fleet may consist of a mixture of different stochastic models, that is, the homogeneous Poisson process/renewal process, trend renewal process, non‐homogeneous Poisson process and branching Poisson processes. Therefore, relying only on a single model to represent the behaviour of the whole fleet may not be valid and may lead to wrong parameter estimation. Copyright © 2015 John Wiley & Sons, Ltd.     

Life cycle cost modelling using marked point processes

This article presents a life cycle cost (LCC) model of a repairable system. The model is based on a marked point process and allows for non-constant failure intensity as well as stochastic nature of costs associated with system's failures. The model is applied to failure data from computer numerically controlled (CNC) machines.     

The role of NHPP models in the practical analysis of maintenance failure data

The analysis of failure data is an important facet in the development of maintenance strategy for equipment. Only by properly understanding the mechanism of failure, through the modelling of failure data, can a proper maintenance plan be developed. This is normally done by means of probabilistic analysis of the failure data. From this, conclusions can be reached regarding the effectiveness and efficiency of preventive replacement (and overhaul) as well as that of predictive maintenance. The optimal frequency of maintenance can also be established by using well developed optimisation models. These optimise outputs, such as profit, cost and availability. The problem with this approach is that it assumes that all repairable systems are repaired to the ‘good-as-new’ condition at each repair occasion. Maintenance practice has learnt, however, that in many cases equipment slowly degrades even while being properly maintained (including part replacement and periodic overhaul). The result of this is that failure data sets often display degradation. This renders conventional probabilistic analysis useless. During the last two decades, a few researches applied themselves to the solution of this problem.This paper briefly examines the present state of the theoretical foundation of repairable systems analysis techniques and then develops two formats of the Non-Homogeneous Poisson Process model (NHPP model) for practical use by the maintenance analyst. This includes an identification framework, goodness-of-fit tests and optimisation modelling. The model is tested on two failure data sets from literature and one from industry.     

A quantification algorithm for a repairable system in the GO methodology

The GO methodology is an effective method of system reliability analysis. It has been applied to non-repairable systems. This paper discusses the application of the GO method to a repairable system which is described by a Markov model and presents the quantification algorithm of the steady characteristics of the repairable system. The calculation formulas of the ordinary operators and the logical gates are derived and the steady reliability parameters of the system such as average operation probability and average failure frequency can be directly computed by the GO method. The result of an example shows that the algorithm is correct. The algorithm will be useful for the safety analysis of most engineering repairable systems.     

Modeling imperfectly repaired system data via grey differential equations with unequal-gapped times

In this paper, we argue that grey differential equation models are useful in repairable system modeling. The arguments starts with the review on GM(1,1) model with equal- and unequal-spaced stopping time sequence. In terms of two-stage GM(1,1) filtering, system stopping time can be partitioned into system intrinsic function and repair effect. Furthermore, we propose an approach to use grey differential equation to specify a semi-statistical membership function for system intrinsic function times. Also, we engage an effort to use GM(1,N) model to model system stopping times and the associated operating covariates and propose an unequal-gapped GM(1,N) model for such analysis. Finally, we investigate the GM(1,1)-embed systematic grey equation system modeling of imperfectly repaired system operating data. Practical examples are given in step-by-step manner to illustrate the grey differential equation modeling of repairable system data.     

Optimal design for resilient load‐sharing systems with nonidentical components

For system maintenance, strategic component restoration planning is an important conceptual framework for load‐sharing k‐out‐of‐n: G system. A cost‐effective treatment of failure events is imperative with the purpose of reinstating the system ability. This paper presents a new optimal design method for load‐sharing repairable k‐out‐of‐n: G system, in which a flowgraph is used in conjunction with multiresponse optimization. By introducing the concept of modular design, the system is partitioned into scalable and repairable maintenance modules. The determination of the optimal design depends on the type of system components, the module‐based system structure, and the repair rule setting. An extended flowgraph model, which links covariates into transition branches, is used for modeling the system failure evolution. With consideration of various system performance measurements as responses, multiresponse optimization with weighted principal component analysis is used to achieve an optimal design of maintenance modules as well as repair policy. The methodology presented in this paper provides an efficient way to design a system having nonidentical components and arbitrary repair time distributions with consideration of the variety of maintenance policies as well as the diversity of system operating conditions.     

Different insights for improving part and system reliability obtained from exactly same DFOM “failure numbers”

Techniques for improving the reliability and maintainability of both nonrepairable and repairable items can be suggested by failure data analysis. It is shown that a given set of failure numbers leads to very different improvement strategies when the numbers are the times-between-successive-failures of one or more repairable items, rather than the times-to-failure of nonrepairable items. Since this should have been obvious more than 50 years ago, at the onset of formal reliability engineering activities, several reasons are proffered for the widespread and protracted misinterpretation of even the most basic—and simple!—conceptual and practical differences between nonrepairable and repairable items.     

A method for designing power supply chain networks accounting for failure scenarios and preventive maintenance

A power grid is vulnerable and failures are inevitable. Failures decrease the power supply with an adverse effect on meeting the demand for electricity. Therefore, there is a need for a method to design power grid networks that result in the least possible disruption to the power supply when a failure occurs. In the literature, the focus has been on the design of the generation system without considering the transmission system or failures in the transmission system. Since power grids are integrated generation and transmission systems, each system will affect the other, so both generation and transmission systems need to be considered, as they are in this article. Methods developed for the structural modelling and analysis of supply chains are shown to be useful. The focus in this article is on describing a method using the supply chain construct for designing power grids that are relatively insensitive to failure in the integrated generation and transmission system. The efficacy of the method is illustrated using data from the Tehran Regional Electric Company. One of the findings is that targeted failures have a higher impact on decreasing the performance of the power grid than random failures. However, the focus is on the method rather than the results per se.     

Estimating system reliability with fully masked data under Brown-Proschan imperfect repair model

This article presents a statistical procedure for estimating the lifetime distribution of a repairable system based on consecutive inter-failure times of the system. The system under consideration is subject to the Brown-Proschan imperfect repair model. The model postulates that at failure the system is repaired to a condition as good as new with probability p, and is otherwise repaired to its condition just prior to failure. The estimation procedure is developed in a parametric framework for incomplete set of data where the repair modes are not recorded. The expectation-maximization principle is employed to handle the incomplete data problem. Under the assumption that the lifetime distribution belongs to the two-parameter Weibull family, we develop a specific algorithm for finding the maximum likelihood estimates of the reliability parameters, the probability of perfect repair (p), as well as the Weibull shape and scale parameters (α, β) The proposed algorithm is applicable to other parametric lifetime distributions with aging property and explicit form of the survival function, by just modifying the maximization step. We derive some lemmas which are essential to the estimation procedure. The lemmas characterize the dependency among consecutive lifetimes. A Monte Carlo study is also performed to show the consistency and good properties of the estimates. Since useful research is available regarding optimal maintenance policies based on the Brown-Proschan model, the estimation results will provide realistic solutions for maintaining real systems.     

Bayesian reliability assessment of repairable systems during multi-stage development programs

New repairable systems are generally subjected to development programs in order to improve system reliability before starting mass production. This paper proposes a Bayesian approach to analyze failure data from repairable systems undergoing a Test-Find-Test program. The system failure process in each testing stage is modeled using a Power-Law Process (PLP). Information on the effect of design modifications introduced into the system before starting a new testing stage is used, together with the posterior density of the PLP parameters at the current stage, to formalize the prior density at the beginning of the new stage. Contrary to the usual assumption, in this paper the PLP parameters are assumed to be dependent random variables. The system reliability is measured in terms of the number of failures that will occur in a batch of new units in a given time interval, for example the warranty period. A numerical example is presented to illustrate the proposed procedure.     

Model for Failure Point Process of a Repairable System and Application

A engineering system is usually repairable,and failure process of a repairable system is often described by a failure point process. The power law model is a commonly used approach to model the failure point process.This paper introduces the concept and model for the failure process of repairable system. The method of parameter estimation is developed,and failure observations are fitted into a power-law model by using the least square method. Two applications of the pressent model are discussed according to the practical failure data of the central cooling system of a nuclear power plant. One application is determining the optimal overhaul time,and the other is evaluating the quality of maintenance. This paper provides references for the overhaul decision making and maintenance quality evaluation in reality.     

A stochastic regime switching model for the failure process of a repairable system

This paper presents a stochastic model and estimation procedure for analyzing the failure process of a repairable system. We consider repairable systems whose successive interfailure times reveal a significant dependence while showing an insignificant trend. Neither the renewal process nor the non-homogeneous Poisson process are adequate for modeling such failure processes. Especially when the interfailure times show a cyclic pattern, we may consider a switching of the regimes (states) governing the lifetime distribution of the system. We propose a Markov switching model describing the failure process for such a case. The model postulates that a finite number of states governs the distinct lifetime distributions, and the state makes transitions according to a discrete-time Markov chain. Each of the distinct lifetime distributions represents a failure type that may change after successive repairs. Our model generalizes the mixture model by allowing the mixture probabilities to change during the transient period of the system. The model can capture the transient behavior of the system. The interfailure times constitute a set of incomplete data because the states are not explicitly identified. For the incomplete data, we propose a procedure for finding the maximum likelihood estimates of the model parameters by adopting the expectation and maximization principle. We also suggest a statistical method to determine the number of significant states. A Monte Carlo study is performed with two-parameter Weibull lifetime distributions. The results show consistency and good properties of the estimates. Some sets of Proschan's air conditioning unit data [Technometrics, 1963, 5′ 375–383] are also analyzed and the results are discussed with respect to the number of significant states and the performance of the prediction.