Residual Life Distribution Research Articles

Variance residual life function in discrete random ageing

The random variable Xt= X − t¦X ≥ t, which is called the residual life random variable, has gathered the attention of most researchers in reliability. The mean and the variance of this variable in continuous distribution have been studied by several authors. But, in discrete case, only in recent years, some studies have been done for the mean of this variable. In this paper, we define and study the properties of variance of Tk= T − k¦T ≥ k where T is a discrete random variable. Besides similar results for discrete and continuous lifetime distributions, relationships with its mean, monotonicity and the associated ageing classes of distributions are obtained for discrete cases. Furthermore, some characterization results about the class of increasing (decreasing) variance residual life distributions based on mean residual life and residual coefficient of variation, are presented and the lower and upper bound for them are achieved.

L-moments of residual life

The mean, variance and coefficient of variation of residual life which are based on the usual moments of the residual life distribution are extensively used in reliability analysis. It has been established in various theoretical and empirical studies that the L-moments have some advantages over the usual moments in many situations. Accordingly in the present paper we study the properties of L-moments of residual life in the context of modeling lifetime data, characterizing life distributions and other applications. The role of certain quantile functions and quantile-based concepts in reliability analysis are also investigated.

Prognostics-Based Identification of the Top-$k$ Units in a Fleet

This paper considers a fleet of identical units where each unit consists of the same critical components. The degradation state of each critical component is assumed to be monitored by an on-board sensor. The paper presents a methodology for identifying the top-k (the k most reliable) units in a fleet using sensor-based prognostic information. Specifically, we develop a prognostics-based ranking (PBR) algorithm that combines stochastic degradation models with computer science database ranking algorithms. The stochastic degradation modeling framework is used to compute and update, in real-time, residual life distributions (RLDs) of the critical components of each unit. Using a base case exponential degradation model, we identify conditions necessary to establish stochastic ordering among the RLDs of similar components. A preference relationship, consistent with the stochastic ordering results, is then used to sort the units of the fleet based on the RLDs of their respective components. A database ranking algorithm, known as the threshold algorithm (TA), is then used to identify the top-k units without necessarily computing all the RLDs. The paper concludes with an illustrative example.

Predictive Maintenance Management Using Sensor-Based Degradation Models

This paper presents a sensory-updated degradation-based predictive maintenance policy (herein referred to as the SUDM policy). The proposed maintenance policy utilizes contemporary degradation models that combine component-specific real-time degradation signals, acquired during operation, with degradation and reliability characteristics of the component's population to predict and update the residual life distribution (RLD). By capturing the latest degradation state of the component being monitored, the updating process provides a more accurate of the remaining life. With the aid of a stopping rule, maintenance routines are scheduled based on the most recently updated RLD. The performance of the proposed maintenance policy is evaluated using a simulation model of a simple manufacturing cell. Frequency of unexpected failures and overall maintenance costs are computed and compared with two other benchmark maintenance policies: a reliability-based and a conventional degradation-based maintenance policy (without any sensor-based updating).

Residual Life Predictions in the Absence of Prior Degradation Knowledge

Recent developments in degradation modeling have been targeted towards utilizing degradation-based sensory signals to predict residual life distributions. Typically, these models consist of stochastic parameters that are estimated with the aid of an historical database of degradation signals. In many applications, building a degradation database, where components are run-to-failure, may be very expensive and time consuming, as in the case of generators or jet engines. The degradation modeling framework presented herein addresses this challenge by utilizing failure time data, which are easier to obtain, and readily available (relative to sensor-based degradation signals) from historical maintenance/repair records. Failure time values are first fitted to a Bernstein distribution whose parameters are then used to estimate the prior distributions of the stochastic parameters of an initial degradation model. Once a complete realization of a degradation signal is observed, the assumptions of the initial degradation model are revised and improved for future predictions. This approach is validated using real world vibration-based degradation information from a rotating machinery application.

Residual-life estimation for components with non-symmetric priors

Condition monitoring uses sensory signals to assess the health of engineering systems. A degradation model is a mathematical characterization of the evolution of a condition signal. Our recent research focuses on using degradation models to compute residual-life distributions for degrading components. Residual-life distributions are important for providing probabilistic estimates of failure time for use in maintenance planning and spare parts inventory management. To obtain residual-life distributions, our earlier work assumed the degradation model's stochastic parameters to be normally distributed. This paper investigates the performance of these residual-life distributions when the underlying normality assumptions are not satisfied. The paper also develops methods for estimating residual-life when the stochastic parameters of the degradation model follow more general distributions.

A note on cause-specific residual life

In medical research, investigators often wish to characterize the distributions of remaining lifetimes. While nonparametric analyses of residual life distributions have been widely studied with independently right-censored data, residual life analysis has not been examined in the competing risks setting, with multiple, potentially dependent, failure types. We define the cause-specific residual life distribution as the residual cumulative incidence function conditionally on survival to a given time. Because of the improper form of the cause-specific distribution, the mean cause-specific residual lifetime does not exist, theoretically. We develop nonparametric inferences for the cause-specific residual life function and its corresponding quantiles, which may exist. Theoretical justification, including uniform consistency and weak convergence, is established. Simulation studies and a breast cancer data analysis demonstrate the practical utility of the methods.

Prognostic Degradation Models for Computing and Updating Residual Life Distributions in a Time-Varying Environment

This paper presents a degradation modeling framework for computing condition-based residual life distributions of partially degraded systems and/or components functioning under time-varying environmental and/or operational conditions. Our approach is to mathematically model degradation-based signals from a population of components using stochastic models that combine three main sources of information: real-time degradation characteristics of component obtained by observing the component's in-situ degradation signal, the degradation characteristics of the component's population, and the real-time status of the environmental conditions under which the component is operating. Prior degradation information is used to estimate the model coefficients. The resulting generalized stochastic degradation model is then used to predict an initial residual life distribution for the component being monitored. In-situ degradation signals, along with real-time information related to the environmental conditions, are then used to update the residual life distributions in real-time. Because these updated distributions capture current health information and the latest environmental conditions, they provide precise lifetime estimates. The performance of the proposed models is evaluated using real world vibration-based degradation signals from a rotating machinery application.

Preservation of some life length classes for age distributions associated with age-dependent branching processes

Under a Bellman–Harris age-dependent branching process, a parent lives a random length of time and at death produces a random number of offspring. The parent lifetimes have the common cumulative distribution G ( ⋅ ) and the number of offspring per parent has a common probability distribution. The current age distribution is known to converge to a limiting distribution A ( ⋅ ) , called the Lotka limit law, which depends only on G ( ⋅ ) and the mean number of offspring. Besides the age distribution, we also consider a residual life distribution and a total life distribution. We investigate the preservation of classes of life lengths for these distributions and their limits.

Moment inequalities for DVRL distributions, characterization and testing for exponentiality

Our goal in this paper is to establish inequalities for the moments of decreasing variance residual life (DVRL) distributions. As a consequence we derive a new characterization of exponentiality. Then we use two of these inequalities to construct new tests for exponentiality versus DVRL. Pitman’s asymptotic relative efficiency is employed to assess the performance of the tests. For some classes of life distributions our tests are better than, or well comparable with, other available tests. We carried out numerical simulations and produced a table for the critical values of one of the proposed tests.

A Neural Network Degradation Model for Computing and Updating Residual Life Distributions

The ability to accurately estimate the residual life of partially degraded components is arguably the most challenging problem in prognostic condition monitoring. This paper focuses on the development of a neural network-based degradation model that utilizes condition-based sensory signals to compute and continuously update residual life distributions of partially degraded components. Initial predicted failure times are estimated through trained neural networks using real-time sensory signals. These estimates are used to derive a prior failure time distribution for the component that is being monitored. Subsequent failure time estimates are then utilized to update the prior distributions using a Bayesian approach. The proposed methodology is tested using real world vibration-based degradation signals from rolling contact thrust bearings. The proposed methodology performed favorably when compared to other reliability-based and statistical-based benchmarks.

Health state evaluation of an item: A general framework and graphical representation

This paper presents a general theoretical framework to evaluate the health state of an item based on condition monitoring information. The item's health state is defined in terms of its relative health level and overall health level. The former is evaluated based on the relative magnitude of the composite covariate and the latter is evaluated using a fractile life of the residual life distribution at the decision instant. In addition, a method is developed to graphically represent the degradation model, failure threshold model, and the observation history of the composite covariate. As a result, the health state of the monitored item can be intuitively presented and the evaluated result can be subsequently used in a condition-based maintenance optimization decision model, which is amenable to computer modeling. A numerical example is included to illustrate the proposed approach and its appropriateness.

Sensory-Updated Residual Life Distributions for Components With Exponential Degradation Patterns

Research on interpreting data communicated by smart sensors and distributed sensor networks, and utilizing these data streams in making critical decisions stands to provide significant advancements across a wide range of application domains such as maintenance management. In this paper, a stochastic degradation modeling framework is developed for computing and continuously updating residual life distributions of partially degraded components. The proposed degradation methodology combines population-specific degradation characteristics with component-specific sensory data acquired through condition monitoring in order to compute and continuously update remaining life distributions of partially degraded components. Two sensory updating procedures are developed and validated using real-world vibration-based degradation information acquired from rolling element thrust bearings. The results are compared with two benchmark policies and illustrate the benefits of the sensory updated degradation models proposed in this paper. Note for Practitioners-The proposed degradation-based prognostic methodology provides a comprehensive assessment of the current and future degradation states of partially degraded components by combining population-specific degradation or reliability information with real-time sensory health monitoring data. It is specifically beneficial for cases where degradation occurs in a cumulative manner and the degradation signal can be approximated by an exponential functional form. To implement this methodology, it is necessary: 1) to identify the physical phenomena associated with the evolution of the degradation process (spalling and wear herein); 2) choose the appropriate condition monitoring technology to monitor this phenomena (accelerometers); 3) identify a characteristic pattern in the sensory information to help develop a degradation signal (exponential growth); and 4) identify a failure threshold associated with the degradation signal. The first step in implementing this prognostic methodology is to obtain prior information related to stochastic parameters f the exponential model. This may require fitting some sample degradation signals with an exponential functional form and noting the values of the exponential parameters, or using subjective prior distributions. The second step is to acquire sensory information and begin updating the prior distribution. The updating frequency will dictate which expressions are used to compute the posterior distributions. Once the posterior means, variances, and correlation are computed, the truncated CDF of the residual life can be evaluated using (10) and (11). Note that the truncation is necessary to preclude negative values of the remaining life. Practitioners can implement this methodology using a simple spreadsheet. Since the residual life distributions are skewed, it is reasonable to utilize the median as a measure of the central tendency and, hence, an alternative estimate for the expected value of the remaining life

SOME RESULTS ABOUT MIT ORDER AND IMIT CLASS OF LIFE DISTRIBUTIONS

We investigate some new properties of mean inactivity time (MIT) order and increasing MIT (IMIT) class of life distributions. The preservation property of MIT order under increasing and concave transformations, reversed preservation properties of MIT order, and IMIT class of life distributions under the taking of maximum are developed. Based on the residual life at a random time and the excess lifetime in a renewal process, stochastic comparisons of both IMIT and decreasing mean residual life distributions are conducted as well.

Some general notions of stochastic orderings for weighted reliability and uncertainty measures with applications

In this note, stochastic comparisons of reliability measures and related functions are presented. Inequalities for uncertainty of a residual life distribution and certain modified cross-entropy or discrimination information measures under weighted models are established. Comparisons of the expected uncertainty about the remaining lifetime of a component for weighted conditional distributions and unweighted conditional distributions are presented.

Multivariate dynamic information

This paper develops measures of information for multivariate distributions when their supports are truncated progressively. The focus is on the joint, marginal, and conditional entropies, and the mutual information for residual life distributions where the support is truncated at the current ages of the components of a system. The current ages of the components induce a joint dynamic into the residual life information measures. Our study of dynamic information measures includes several important bivariate and multivariate lifetime models. We derive entropy expressions for a few models, including Marshall–Olkin bivariate exponential. However, in general, study of the dynamics of residual information measures requires computational techniques or analytical results. A bivariate gamma example illustrates study of dynamic information via numerical integration. The analytical results facilitate studying other distributions. The results are on monotonicity of the residual entropy of a system and on transformations that preserve the monotonicity and the order of entropies between two systems. The results also include a new entropy characterization of the joint distribution of independent exponential random variables.

Residual-life distributions from component degradation signals: A Bayesian approach

Real-time condition monitoring is becoming an important tool in maintenance decision-making. Condition monitoring is the process of collecting real-time sensor information from a functioning device in order to reason about the health of the device. To make effective use of condition information, it is useful to characterize a device degradation signal, a quantity computed from condition information that captures the current state of the device and provides information on how that condition is likely to evolve in the future. If properly modeled, the degradation signal can be used to compute a residual-life distribution for the device being monitored, which can then be used in decision models. In this work, we develop Bayesian updating methods that use real-time condition monitoring information to update the stochastic parameters of exponential degradation models. We use these degradation models to develop a closed-form residual-life distribution for the monitored device. Finally, we apply these degradation and residual-life models to degradation signals obtained through the accelerated testing of bearings.

Maximum dynamic entropy models

A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.

Maximum dynamic entropy models

A formal approach to produce a model for the data-generating distribution based on partial knowledge is the well-known maximum entropy method. In this approach, partial knowledge about the data-generating distribution is formulated in terms of some information constraints and the model is obtained by maximizing the Shannon entropy under these constraints. Frequently, in reliability analysis the problem of interest is the lifetime beyond an age t. In such cases, the distribution of interest for computing uncertainty and information is the residual distribution. The information functions involving a residual life distribution depend on t, and hence are dynamic. The maximum dynamic entropy (MDE) model is the distribution with the density that maximizes the dynamic entropy for all t. We provide a result that relates the orderings of dynamic entropy and the hazard function for distributions with monotone densities. Applications include dynamic entropy ordering within some parametric families of distributions, orderings of distributions of lifetimes of systems and their components connected in series and parallel, record values, and formulation of constraints for the MDE model in terms of the evolution paths of the hazard function and mean residual lifetime function. In particular, we identify classes of distributions in which some well-known distributions, including the mixture of two exponential distributions and the mixture of two Pareto distributions, are the MDE models.

Determination of burn‐in parameters and residual life for highly reliable products

AbstractToday, many products are designed and manufactured to function for a long period of time before they fail. Determining product reliability is a great challenge to manufacturers of highly reliable products with only a relatively short period of time available for internal life testing. In particular, it may be difficult to determine optimal burn‐in parameters and characterize the residual life distribution. A promising alternative is to use data on a quality characteristic (QC) whose degradation over time can be related to product failure. Typically, product failure corresponds to the first passage time of the degradation path beyond a critical value. If degradation paths can be modeled properly, one can predict failure time and determine the life distribution without actually observing failures. In this paper, we first use a Wiener process to describe the continuous degradation path of the quality characteristic of the product. A Wiener process allows nonconstant variance and nonzero correlation among data collected at different time points. We propose a decision rule for classifying a unit as normal or weak, and give an economic model for determining the optimal termination time and other parameters of a burn‐in test. Next, we propose a method for assessing the product's lifetime distribution of the passed units. The proposed methodologies are all based only on the product's initial observed degradation data. Finally, an example of an electronic product, namely contact image scanner (CIS), is used to illustrate the proposed procedure. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003