A model for the integrity assessment of ageing repairable systems

The failure rate of mechanical repairable systems that deteriorate with time due to ageing can usually be visualized by a bathtub curve. This study shows that an equation that is valid in other respects for describing creep curves can easily be deduced from functional forms of the failure rate of mechanical repairable systems. Creeping pieces can be considered repairable systems that evolve under an applied load, as combining positive and negative feedback loops. More generally, this can be extended to mechanical repairable systems, the negative feedback loops corresponding to repair and overhaul operations. The equation describing creep curves reflects the ageing of mechanical repairable systems. A critical time at which the system can no longer be restored to full performance, in spite of repair and/or replacement of subparts, can then be predicted. An application example is given using published failure data corresponding to a submarine main-propulsion diesel engine. The proposed model should apply every time that mechanical system ageing is expressed by a bathtub curve     

Linear-spline approximation for semi-parametric modeling of failuredata with proportional hazards

Modeling of failure processes using proportional hazards involves estimating both a baseline failure intensity as well as the parameters of the proportional failure intensity. In the absence of any information regarding the baseline failure intensity, a nonparametric form is typically assumed. This paper proposes a linear-spline function to approximate this baseline failure intensity, and develops such a spline function appropriate to bad-as-old failure data generated from a repairable system. Field data from an industrial setting demonstrate an improved approximation using such a spline function as compared to other procedures in the literature     

Economic cost modeling of environmental-stress-screening andburn-in

The electronics industry is highly competitive. The introduction of new advanced integrated circuits continues. The manufacturer must develop a product with adequate life cycle, high quality and low failure rate in the specified time period. Environmental stress screening (ESS) is widely used in the electronics industry to assist in eliminating early or patent failures. The bathtub curve is a common model for the failure rate of a population. The classic bathtub curve has three parts: infant mortality; useful life; and wearout. However, some electronic products have latent defects which are not detected by conventional inspection or functional testing, and occur during the product useful life. Most electronic systems have finite useful life. As a result, technology obsolescence is of concern and should be considered, This paper uses a new interpretation and development of the bathtub curve that integrates: (1) latent failures; and (2) the concept of obsolescence, by constructing an integrated cost model used to determine both optimal burn-in and ESS times in the same environment     

Spurious exponentiality observed when incorrectly fitting adistribution to nonstationary data

Failure data for a repairable system can be represented either by a set of chronologically ordered arrival times at which the system failed, or by a set of interarrival times defined as the times observed between successive failures (ignoring repair times in both cases). The two representations are mathematically equivalent if the chronological order of the interarrival times is maintained. Methods aimed at describing the distribution of the observed interarrival times are meaningful only if the interarrival times are identically distributed. In all other cases, such analyses are meaningless and often result in maximally misleading impressions about the system behavior, as demonstrated here by several examples. That is, when the information in the chronological order of interarrival times is ignored, they often appear spuriously exponential, leading to the impression that the system can be modeled using a homogeneous Poisson process. Misunderstandings of this nature can be avoided by applying an appropriate test for trend before attempting to fit a distribution to the interarrival times. If evidence of trend is determined, then a nonstationary model such as the nonhomogeneous Poisson process should be fitted using the chronologically ordered data     

Statistical Analysis of a Compound Power-Law Model for Repairable Systems

A compound (mixed) Poisson distribution is sometimes used as an alternative to the Poisson distribution for count data. Such a compound distribution, which has a negative binomial form, occurs when the population consists of Poisson distributed individuals, but with intensities which have a gamma distribution. A similar situation can occur with a repairable system when failure intensities of each system are different. A more general situation is considered where the system failures are distributed according to nonhomogeneous Poisson processes having Power Law intensity functions with gamma distributed intensity parameter. If the failures of each system in a population of repairable systems are distributed according to a Power Law process, but with different intensities, then a compound Power Law process provides a suitable model. A test, based on the ratio of the sample variance to the sample mean of count data from s-independent systems, provides a convenient way to determine if a compound model is appropriate. When a compound Power Law model is indicated, the maximum likelihood estimates of the shape parameters of the individual systems can be computed and homogeneity can be tested. If equality of the shape parameters is indicated, then it is possible to test whether the systems are homogeneous Poisson processes versus a nonhomogeneous alternative. If deterioration within systems is suspected, then the alternative in which the shape parameter exceeds unity would be appropriate, while if systems are undergoing reliability growth the alternative would be that the shape parameter is less than unity.     

Discounted warranty cost of minimally repaired series systems

Many factors should be considered in modeling DWC (discounted warranty cost) of repairable systems or products including system structure, components' failure processes, methods of discounting as well as the warranty policy itself. In this paper, we present DWC models for repairable series systems. In particular, a free repair warranty policy and a pro-rata warranty policy are studied. The impact of repair actions on components' failure times is assumed to be minimal, hence NHPPs are used to describe the failure processes. Two types of discounting methods are considered in this paper: a continuous discount function and a discrete discount function. Expressions for both the expected value and variance of DWC are derived. The applications of our findings can be seen in warranty design, warranty reserve determination and risk analysis. Our approach incorporates the information of system structure, the value of time and the impact of repair actions, which are of great importance to warranty cost prediction and evaluation, but have not been sufficiently studied in the literature of warranty analysis.     

Stochastic analysis of systems with primary and secondary failures

This paper presents the stochastic analysis of repairable systems involving primary as well as secondary failures. To this end, two models are considered. The first model represents a system with two identical warm standbys. The failure rates of units and the system are constant and independent while the repair times are arbitrarily distributed. The second system modeled consists of three repairable regions. The system operates normally if all three regions are operating, otherwise it operates at a derated level unless all three regions fail. The failure rates and repair times of the regions are constant and independent. The first model is analyzed using the supplemental variable technique while the second model is analyzed using the regenerative point technique in the Markov renewal process. Various expressions including system availability, system reliability and mean time to system failure are obtained.     

Generalized Linear Mixed Models for Reliability Analysis of Multi-Copy Repairable Systems

The power law process (PLP) is usually applied to failure data from a single repairable system. When a system has a number of copies for analysis, the usual approach is to assume homogeneity among all system copies, and then to pool data from these copies. In the real world, however, it may be more reasonable to assume heterogeneity among the system copies. Therefore, this paper proposes a new generalized linear mixed model (GLMM), called PLP-GLMM, to analyse failure data from multi-copy repairable systems. In the PLP-GLMM, the underlying model for each system copy is assumed to be a PLP at Stage 1, and parameters vary among copies at Stage 2. The PLP-GLMM can make inferences about both the population, and each system copy when accounting for copy-to-copy variance. A modified Anderson-Darling test is adapted to the goodness-of-fit test of the PLP-GLMM. A numerical application is given to show the effectiveness of the model     

Cost-benefit analysis of one-server two-unit imperfect switch system with degradation

This paper deals with the cost-benefit analysis of 1-server 2-unit system with imperfect switch where the items (units/switch) are not ‘as good as new’ after the repair. The successive repair times of each item are arbitrarily distributed while the successive failure rates are constants. If repair of a unit is not possible it is discarded incurring some salvage value. After a fixed number of failures the unit is scrapped without any salvage value. The switch is always repairable and is scrapped only when both the units are scrapped. The expected revenue, expected busy period due to different types of repairs of units/switch, expected salvage value and expected number of repair completions of the switch, all in [0, t], are obtained to carry out the cost-benefit analysis.     

Bayes prediction for the number of failures of a repairable system

After observing a repairable system for some time, one may wish to predict the number of failures of the system in some fixed future interval. Such a prediction depends on the: (1) assumed model for the failure process; and (2) length of the interval. The authors use a Bayes approach to obtain point and interval predictions for the number of failures in a future interval. Two situations are discussed: (1) the power law process (PLP) governs failure times during the period of observation, but in the future interval the homogeneous Poisson Process (HPP) governs the failure times; and (2) the failure process is the PLP. A rationale and an example of each situation is presented. They discuss the use of informative and noninformative priors for the parameters of the failure process. The Bayes approach can incorporate both sources of uncertainty: (1) the number of failures in the future interval is random, so even if the parameters of the failure process are known, the number of failures that would occur in a future interval would still not predict with certainty; and (2) the parameters of the failure process are not known and must be estimated from the observed data     

Cost-Optimized Burn-In Duration for Repairable Electronic Systems

A mathematical model permits determining the duration of cost-optimized burn-in and evaluating the resultant saving for repairable electronics systems. Infant mortality failures occur according to a nonhomogeneous Poisson Process; repair actions restore the system to a bad-as-old condition. The s-expected costs associated with factory and field failures are traded-off with the costs of implementing a burn-in program. Under the constraints of the model, the optimum burn-in duration and consequent cost saving are independent of the eventual life of the system in the field. A numerical example illustrates these concepts.     

A flexible bathtub hazard model for non-repairable systems with uncensored data

This research effort has developed a mathematical model for bathtub shaped hazards (failure rates) for operating systems with uncensored data. The model will be used to predict the reliability of systems with such hazards. Early in the life-time of a system, there may be a relatively large number of failures due to initial weaknesses or defects in materials and manufacturing processes. This period is called the “infant mortality” period. During the middle period of an operating system fewer failures occur and are caused when the environmental stresses exceed the design strength of the system. It is difficult to predict the environmental stress amplitudes or the system strengths as deterministic functions of time, thus the middle-life failures are often called “random failures.” As the system ages, it deterioates and more failures occur. This region of failure is called the “wearout” period. Graphing these failure rates simultaneously will result in a bathtub shaped curve. The model developed for this bathtub pattern of failure takes into account all three failure regions simultaneously. The model has been validated for accuracy by using Halley's mortality table and is used to predict the reliability with both least squares and maximum likelihood estimators.     

An analysis of reliability in the early stages of photovoltaic systems in Japan

In order to disseminate Photovoltaic (PV) technologies into the energy network, the cost down is not only important, but also improving the performance of the PV system is significant issues. Long‐term reliability is one of the most important issues in terms of PV system performance. Previous researches were mainly focused on the reliability of PV modules, but the PV system is composed of a power conditioner, wiring, junction box, and so on. To improve the reliability of PV systems, it is important to accumulate trouble cases focused on all components of PV system. In this paper, we aim at evaluation of the reliability for the PV system on the early stages of PV system's lifetime by using large number of Japanese PV systems' data from the field Test in Japan. New Energy and Industrial Technology Development Organization has been running the “Field test project in Japan” from 1992. In this project, PV system users have cooperated with the collection of monitoring data and reported on the information of maintenance and certain failures of PV systems for 4 years after installation of PV system. Using those reports each year of installation, we evaluated reliability of PV systems by means of parameters such as Mean Time Between Failure, Mean Time To Repair, and the suspension time of PV system. As a result, the main trouble of PV systems was related power conditioner, and a few trouble of PV module was caused by typhoon. Moreover, the trend of the failure rate before FY 2000 of installation was demonstrated as the trend of initial failure in “bathtub curve;” however, the trend of its after FY 2001 of installation was indicated as the accidental failure in “bathtub curve.” Further, the operator simply forgot to restart the power conditioner after maintenance or suspensions of PV system in many trouble cases, and the user did not notice that it had been suspended for a while. These trouble cases can be avoidable easily through the effective alarm such as error message of power conditioner systems with monitoring systems. Thereby, monitoring with the evaluation method of PV systems is one of the important technologies due to the long‐term reliability and stable operation. Copyright © 2010 John Wiley & Sons, Ltd.     

Parametric Programming Approach for a Two-Unit Repairable System With Imperfect Coverage, Reboot and Fuzzy Parameters

This paper proposes a procedure for constructing the membership functions of the system characteristics of a two-unit repairable system in which the coverage factor for an operating unit failure is possibly considered. Times to failure, and times to repair of the operating units are assumed to follow fuzzified exponential distributions. In addition, the recovery times, and reboot times of the failed units also follow fuzzified exponential distributions. The $alpha$-cut approach is used to extract the fuzzy repairable system from a family of conventional crisp intervals for the desired system characteristics, determined with a set of parametric nonlinear programs using their membership functions. Two numerical examples illustrate the practicality of the proposed approach. Because the system characteristics are governed by the membership functions, more information is provided for use by designers and practitioners. The successful extension of the parameter spaces to fuzzy environments permits the repairable system to have wider practical applications.      

Mean residual life and its association with failure rate

The life time distributions having decreasing, increasing, or upside-down bathtub shaped MRL (mean residual life) are used as models in many applications. Mi (1995) has shown that if a component has a bathtub shape failure rate function, then the MRL is unimodal but the converse does not hold. This paper develops sufficient conditions for the unimodal MRL to imply that the failure rate function has a bathtub shape     

Reliability evaluation of a repairable network with limited capacity and structural redundancy

A repairable network with limited capacity is considered. Every link and node of a network can be presented by a structural component consisting of an individual element or a group of elements with different kinds of structural redundancy. Repairs of failed elements are carried out in the process of network operation. The number of elements that are serviced by a repair centre is usually greater than the number of repair units in this centre—so called limited repair. Therefore, a queue of failed elements may arise in the repair centre, which causes statistical interdependence of elements and components of the network. A method and models for computing steady-state availability, mean time to failure (MTTF) and mean time to repair (MTTR) of the repairable network with structural redundancy and limited capacity are presented. The proposed method and models are implemented in the application program package used for reliability evaluation of computer networks and distributed systems.     

A new method to compute reliability of repairable m-out-of-n systems by arbitrary distributions

In this paper it is presented a new method to compute reliability characteristics of repairable m-out-of-n systems by arbitrary distributed time to failure and time to repair. The method allows to consider the systems with large number of units and does not take much computational times.     

Comparison of mean time to first failure and mean up time

Two unit standby redundant repairable system, parallel systems and series systems are studied. The time between failures and the repair time are assumed exponentially distributed. Expressions for mean time to first failure (MTTFF) and mean up time (MUT) have been obtained. Comparisons have been made between MTTFF and MUT and the difference illustrated by numerical examples for different systems.     

Network reliability evaluation: Application of bathtub failure rate curve

This paper presents reliability evaluation of networks composed of two and three-state devices whose failure rate follows a bathtub curve. Networks reliability equations and (time dependent) curves are presented. Each network reliability time dependent curve is also plotted for the device constant failure rate.     

Load-capacity interference and the bathtub curve

Load-capacity (stress-strength) interference theory is used to derive a heuristic failure rate for an item subjected to repetitive loading which is Poisson distributed in time. Numerical calculations are performed using Gaussian distributions in load and capacity. Infant mortality, constant failure rate (Poisson failures), and aging are shown to be associated with capacity variability, load variability, and capacity deterioration, respectively. Bathtub-shaped failure rate curves are obtained when all three failure types are present. Changes in load or capacity distribution parameters often strongly affect the quantitative behavior of the failure-rate curves, but they do not affect the qualitative behavior of the bathtub curve. Neither is it likely that the qualitative behavior will be affected by the use of nonGaussian distributions. The numerical results, however, indicate that infant mortality and wear-out failures interact strongly with load variability. Thus bathtub curves arising from this model cannot be represented as simple superpositions of independent contributions from the three failure types. Only if the three failure types arise from independent failure mechanisms or in different components is it legitimate simply to sum the failure rate contributions