Mixture Failure Rate Research Articles

Mixture reliability analysis of a product consisting of new and remanufactured components

PurposeThe rapid increase in importance of the remanufacturing operation in the present scenario is just because of its ability to retrieve the functional value of the End-of-Use or End-of-Life products which is as good as the original product. However, customers are still concerned about the reliability of the remanufactured product which is considered as one of the major problems in the area of remanufacturing. The purpose of this paper is to study and analyse the behavioural pattern of the mixture failure rate of a remanufactured product.Design/methodology/approachIn order to analyse the behavioural pattern of the mixture failure rate, different proportions of new and remanufactured products are mixed. In this paper, a two-parameter Weibull distribution is used to observe the mixture failure rate characteristics. Also, the mixture failure rate of the remanufactured product is evaluated under two conditions, that is when the shape parameter of new and remanufactured components is the same and when the shape parameter values are different.FindingsFrom the analysis, it is observed that the mixture failure rate is always decreasing in nature when the shape parameter values are same. In that case, the value of the mixture failure rate depends only on the proportion of the new components. When the shape parameter values are different, the mixture failure rate characteristics depend upon the shape parameter value of the remanufactured product.Originality/valueThe results of the research can be applied to any remanufactured automotive product. This study also shows the behavioural characteristics of the mixture failure rate of a remanufactured product at different mixture proportions.

On Some Properties of $$\alpha $$-Mixtures

The $$\alpha $$ -mixtures is a new, flexible family of distributions that includes many mixture models as special cases. This paper is mainly focused on relevant stochastic comparisons and ageing properties of $$\alpha $$ -mixtures of survival functions. In particular, we prove that ageing properties of $$\alpha $$ -mixtures for additive and multiplicative hazards models depend on the properties of the baseline failure rate functions and the corresponding conditional moments of mixing distributions. Partial orderings of the finite $$\alpha $$ -mixtures in the sense of the usual stochastic order and the hazard rate order are discussed. Finally, we extend some results on the shape of the mixture failure rate obtained in the literature for usual mixtures to the case of $$\alpha $$ -mixtures.

Some results on discrete mixture failure rates

Some properties of the discrete mixture failure rates are studied. Specifically, similar to the continuous case, it is shown that the population mixture failure rate is always smaller than the unconditional expectation in the family of subpopulations failure rates. The analog of the multiplicative and the additive frailty models is introduced via the corresponding survival function. Another approach via the alternative discrete failure rate is also discussed. Stochastic comparisons for two mixed distributions with equal and different mixing distributions are studied.

The failure rate dynamics in heterogeneous populations

Most populations encountered in real world are heterogeneous. In reliability applications, the mixture (observed) failure rate, obviously, can be considered as a measure of ‘average’ quality in these populations. However, in addition to this average measure, some variability characteristics for failure rates can be very helpful in describing the time-dependent changes in quality of heterogeneous populations. In this paper, we discuss variance and the coefficient of variation of the corresponding random failure rate as variability measures for items in heterogeneous populations. Furthermore, there is often a risk that items of poor quality are selected for important missions. Therefore, along with the ‘average quality’ of a population, more ‘conservative’ quality measures should be also defined and studied. For this purpose, we propose the percentile and the tail-mixture of the failure rates as the corresponding conservative measures. Some illustrative examples are given.

Understanding the shape of the mixture failure rate (with engineering and demographic applications)

AbstractMixtures of distributions are usually effectively used for modelling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behaviour as t→∞. Some demographic and engineering examples are considered. The corresponding inverse problem is discussed. Copyright © 2009 John Wiley & Sons, Ltd.

‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 2

‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 2

‘Understanding the shape of the mixture failure rate’ Rejoinder by Maxim Finkelstein

‘Understanding the shape of the mixture failure rate’ Rejoinder by Maxim Finkelstein

‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 1

‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 1

Reliability Equivalence Factors of Series System with Mixture Failure Rates

Reliability Equivalence Factors of Series System with Mixture Failure Rates

On asymptotic failure rates in bivariate frailty competing risks models

A bivariate competing risks problem is considered for a rather general class of survival models. The lifetime distribution of each component is indexed by a frailty parameter. Under the assumption of conditional independence of components the correlated frailty model is considered. The explicit asymptotic formula for the mixture failure rate of a system is derived. It is proved that asymptotically, as t → ∞ , the remaining lifetimes of components tend to be independent in the defined sense. Some simple examples are discussed.

On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.

Asymptotic behavior of a general class of mixture failure rates

Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, as t → ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

Asymptotic behavior of a general class of mixture failure rates

Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, ast→ ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

Bathtub Shaped Failure Rates From Mixtures: A Practical Point of View

We show that a bathtub shaped failure rate can be obtained from a mixture of two increasing failure rate (IFR) models. Specifically, we study the failure rate of the mixture of an exponential distribution, and a Weibull distribution with strictly increasing failure rate. Under some reasonable conditions, we show that, from a practical point of view, the mixture failure rate is bathtub. Similar results can be obtained from other mixtures.

On the behavior and shape of mixture failure rates from a family of IFR Weibull distributions

AbstractPopulations of many types of component are heterogeneous and often consist of a small number of different subpopulations. This is called a mixture and it arises in a number of situations. For example, a majority of products in industrial populations are mixtures of defective items with shorter lifetimes and standard items with longer lifetimes. It is a well‐known result that distributions with decreasing failure rates are closed under mixture. However, mixtures of distributions with increasing failure rates are not easily classifiable. If the subpopulations involved in the mixture have increasing failure rates, there might be some upward movement in the mixture and later a general downward pull towards the strongest component. Little work has been done in describing the shape of mixture failure rates when all subpopulations do not have decreasing failure rate. In this paper, we present general results that describe the shape and behavior of a failure rate of a mixture obtained from two Weibull subpopulations with strictly increasing failure rates. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004

Modeling lifetimes with unknown initial age

Abstract The failure rate of a distribution, which describes a lifetime random variable with unknown initial age, is studied. A specific model of mixing formalizes this setting. In accordance with this model, it turns out that under certain assumptions the operation of mixing distributions with increasing failure rates can result in the mixture failure rate with a bathtub shape: decreasing in the initial interval of time and then increasing, asymptotically approaching the failure rate of the baseline distribution. Two specific cases of the baseline distribution are considered: the Weibull and the truncated extreme value distributions. A possibility of mixing via the mean residual lifetime function is discussed.

MODELING A FAILURE RATE FOR A MIXTURE OF DISTRIBUTION FUNCTIONS

It is well known that mixtures of decreasing failure rate (DFR) distributions are always DFR. It turns out that, very often, mixtures of increasing failure rate (IFR) distributions can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. In this article, several types of underlying continuous IFR distribution are considered. Two models of mixing are studied: additive and multiplicative. The limiting behavior of a mixture failure rate function is analyzed. It is shown that the conditional characteristics (expectation and variance) of the mixing parameter are crucial for the limiting behavior. Several examples are presented and possible generalizations are discussed.

On an inverse problem in mixture failure rates modelling

AbstractMixtures of decreasing failure rate (DFR) distributions are always DFR. It turns out that very often mixtures of increasing failure rate distributions can decrease or show even more complicated patterns of dependence on time. For studying this and other relevant effects two simple models of mixing with additive and multiplicative failure rates are considered. It is shown that for these models an inverse problem can be solved, which means that given an arbitrary shape of the mixture failure rate and a mixing distribution, the failure rate for a governing distribution can be uniquely obtained. Some examples are considered where this operation can be performed explicitly. Possible generalizations are discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Why the mixture failure rate decreases

Abstract It is well known that mixtures of decreasing failure rate (DFR) distributions are always decreasing. It turns out that very often mixtures of increasing failure rate (IFR) distributions can decrease as well. In this paper several types of continuous mixtures of IFR distributions are considered. The corresponding limiting behavior of the mixture failure rate function is analyzed for the specific case of mixing which can be interpreted in terms of the proportional hazards model. It is shown that the conditional characteristics (expectation and variance) of the mixing parameter are crucial for the limiting behavior. Several examples are presented and possible generalizations are discussed.