Joint Integrated Importance Measure for Multi-State Transition Systems

Joint reliability importance (JRI) evaluates the interaction of two components in contributing to the system reliability in a system. Traditional JRI measures mainly concern the change of the system reliability caused by the interactive change of the reliabilities of the two components and seldom consider the probability distributions, transition rates of the object component states, and system performance. This article extends the JRI concept of two components from multi-state systems to multi-state transition systems and mainly focuses on the joint integrated importance measure (JIIM) which considers the transition rates of component states. Firstly, the concept and physical meaning of JIIM in binary systems are described. Secondly, the JIIM for deterioration process (JIIMDP) and the JIIM for maintenance process (JIIMMP) in multi-state systems are studied respectively. The corresponding characteristics of JIIMDP and JIIMMP for series and parallel systems are also analyzed. Finally, an application to an offshore electrical power generation system is given to demonstrate the proposed JIIM.     

Prediction of the residual failure processes based on the process history

In the reliability area, the concept of the residual lifetime of a non repairable system is very important and its property has been intensively studied. In this article, we define the “residual failure process” for a repairable system and study its stochastic properties thoroughly. The detailed discussions are given when the corresponding failure process is a renewal process. An illustrative example is also discussed.     

Generalized Random Sign and Alert Delay models for imperfect maintenance

This paper considers the modelling of the process of Corrective and condition-based Preventive Maintenance, for complex repairable systems. In order to take into account the dependency between both types of maintenance and the possibility of imperfect maintenance, Generalized Competing Risks models have been introduced in “Doyen and Gaudoin (J Appl Probab 43:825–839, 2006)”. In this paper, we study two classes of these models, the Generalized Random Sign and Generalized Alert Delay models. A Generalized Competing Risks model can be built as a generalization of a particular Usual Competing Risks model, either by using a virtual age framework or not. The models properties are studied and their parameterizations are discussed. Finally, simulation results and an application to real data are presented.     

Age Replacement Policy with a Safety Constraint via the Bayesian Method

Consider a system that is subject to shocks that arrive according to a non homogeneous Poisson process. As the shocks occur, the system has m + 1 failure modes including the following: (i) a non repairable failure (catastrophic) mode that calls for a replacement and (ii) m repairable failure (non catastrophic) modes that are rectified by minimal repairs. In this article, we propose an age-replacement model with minimal repair based on using the natural conjugate prior of Bayesian method. In addition, a safety constraint is considered to control the risk of occurring catastrophic failures in a specified time interval. The minimum-cost replacement policy is studied in terms of its existence and safety constraint. A numerical example is also presented to illustrate the proposed model.     

A Study on Reliability for A Two-Item Cold Standby Markov Repairable System with Neglected Failures

This article studies reliability for a Markov repairable two-item cold standby system with neglected failures. In the system, if a failed time of the system is too short (less than a given critical value) to cause the system to fail, then the failed time may be omitted from the downtime record, i.e., the failure effect could be neglected. In ion-channel modeling, this situation is called the time interval omission problem. The availability indices and the mean downtime are presented as two measures of reliability for this repairable system. Some numerical examples are shown to illustrate the results obtained in this article.     

An optimal age-replacement policy for a simple repairable system with delayed repair

In this article, a simple repairable system (i.e., a repairable system consisting of one component and one repairman) with delayed repair is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, using the geometric process repair model, we consider a replacement policy T based on system age under which the system is replaced when the system age reaches T. Our problem is to determine an optimal replacement policy T*, such that the average cost rate (i.e., the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy T* can be determined by minimizing the average cost rate of the system. Finally, a numerical example is given to illustrate some theoretical results and the model's applicability.     

A comparison of non-homogeneous Markov regression models with application to Alzheimer's disease progression

Markov regression models are useful tools for estimating the impact of risk factors on rates of transition between multiple disease states. Alzheimer's disease (AD) is an example of a multi-state disease process in which great interest lies in identifying risk factors for transition. In this context, non-homogeneous models are required because transition rates change as subjects age. In this report we propose a non-homogeneous Markov regression model that allows for reversible and recurrent disease states, transitions among multiple states between observations, and unequally spaced observation times. We conducted simulation studies to demonstrate performance of estimators for covariate effects from this model and compare performance with alternative models when the underlying non-homogeneous process was correctly specified and under model misspecification. In simulation studies, we found that covariate effects were biased if non-homogeneity of the disease process was not accounted for. However, estimates from non-homogeneous models were robust to misspecification of the form of the non-homogeneity. We used our model to estimate risk factors for transition to mild cognitive impairment (MCI) and AD in a longitudinal study of subjects included in the National Alzheimer's Coordinating Center's Uniform Data Set. Using our model, we found that subjects with MCI affecting multiple cognitive domains were significantly less likely to revert to normal cognition.     

Bi-directional system analysis under copula-coverage approach

Reliability assessment is a major step, toward the development of fault-tolerant computing system. In different fields of engineering and physical/ailed sciences, researchers or engineers have several reliability approaches for the better performance of the system. By assuming different types of time trends, failure modes and repair effects, the legion stochastic model has been developed for repairable system. This study shows a novel concept for three state fault tolerant repairable systems with two types of repair. This research predicts the effect of the coverage factor using Gumbel-Hougaard family of copula approach on reliability characteristics of the designed system.     

Mixture Multi-state Markov Regression Model

Although heterogeneity across individuals may be reduced when a two-state process is extended into a multi-state process, the discrepancy between the observed and the predicted for some states may still exist owing to two possibilities, unobserved mixture distribution in the initial state and the effect of measured covariates on subsequent multi-state disease progression. In the present study, we developed a mixture Markov exponential regression model to take account of the above-mentioned heterogeneity across individuals (subject-to-subject variability) with a systematic model selection based on the likelihood ratio test. The model was successfully demonstrated by an empirical example on surveillance of patients with small hepatocellular carcinoma treated by non-surgical methods. The estimated results suggested that the model with the incorporation of unobserved mixture distribution behaves better than the one without. Complete and partial effects regarding risk factors on different subsequent multi-state transitions were identified using a homogeneous Markov model. The combination of both initial mixture distribution and homogeneous Markov exponential regression model makes a significant contribution to reducing heterogeneity across individuals and over time for disease progression.     

Selective Maintenance in System Reliability with Random Costs of Repairing and Replacing the Components

In this paper we consider the problem of determining the optimum number of repairable and replaceable components to maximize a system's reliability when both, the cost of repairing the components and the cost of replacement of components by new ones, are random. We formulate it as a problem of non-linear stochastic programming. The solution is obtained through Chance Constrained programming. We also consider the problem of finding the optimal maintenance cost for a given reliability requirement of the system. The solution is then obtained by using Modified E-model. A numerical example is solved for both the formulations.     

An integrated model for economic design of chi-square control chart and maintenance planning

Considered process in this article is a two-stage dependent process. Each item in this process has two quality characteristics as x and y while x and y are related to the stage 1 and 2, respectively. Each stage has two operational states as the in-control state and out-of-control state and transition time from the in-control state to the out-of-control state follows a general continues distribution function. The process is monitored using a chi-square control chart. An integrated model that coordinates the decisions related to the economic design of the used control chart and maintenance planning is presented. For the evaluation of the integrated model performance, a stand-alone maintenance model is also presented, and the performance of these two models is compared with each other.     

The alert-delay competing risks model for maintenance analysis

The paper considers the modelling of the dependency between corrective maintenance and condition-based preventive maintenance of complex repairable systems. A new model of dependent competing risks is proposed, called the alert-delay (AD) model. This model has different properties from that of the delay-time, repair-alert and proportional warning constant inspection models and happens to fit some data sets which could not be fitted by the previous models. The main features of the AD model are derived: probabilistic properties and statistical analysis. Simulation results and an application to real data are presented.     

On collapsing categories in two-way contingency tables

In this paper, we examine the relationships between log odds rate and various reliability measures such as hazard rate and reversed hazard rate in the context of repairable systems. We also prove characterization theorems for some families of distributions viz. Burr, Pearson and log exponential models. We discuss the properties and applications of log odds rate in weighted models. Further we extend the concept to the bivariate set up and study its properties.     

A Simulation Study Comparing Modeling Approaches in an Illness-Death Multi-State Model

In many medical studies, there are covariates that change their values over time and their analysis is most often modeled using the Cox regression model. However, many of these time-dependent covariates can be expressed as an intermediate event, which can be modeled using a multi-state model. Using the relationship of time-dependent (discrete) covariates and multi-state models, we compare (via simulation studies) the Cox model with time-dependent covariates with the most frequently used multi-state regression models. This article also details the procedures for generating survival data arising from all approaches, including the Cox model with time-dependent covariates.     

Nonparametric estimation in the illness-death model using prevalent data

We study nonparametric estimation of the illness-death model using left-truncated and right-censored data. The general aim is to estimate the multivariate distribution of a progressive multi-state process. Maximum likelihood estimation under censoring suffers from problems of uniqueness and consistency, so instead we review and extend methods that are based on inverse probability weighting. For univariate left-truncated and right-censored data, nonparametric maximum likelihood estimation can be considerably improved when exploiting knowledge on the truncation distribution. We aim to examine the gain in using such knowledge for inverse probability weighting estimators in the illness-death framework. Additionally, we compare the weights that use truncation variables with the weights that integrate them out, showing, by simulation, that the latter performs more stably and efficiently. We apply the methods to intensive care units data collected in a cross-sectional design, and discuss how the estimators can be easily modified to more general multi-state models.     

Joint coverage probability in a simulation study on continuous-time Markov chain parameter estimation

Parameter dependency within data sets in simulation studies is common, especially in models such as continuous-time Markov chains (CTMCs). Additionally, the literature lacks a comprehensive examination of estimation performance for the likelihood-based general multi-state CTMC. Among studies attempting to assess the estimation, none have accounted for dependency among parameter estimates. The purpose of this research is twofold: (1) to develop a multivariate approach for assessing accuracy and precision for simulation studies (2) to add to the literature a comprehensive examination of the estimation of a general 3-state CTMC model. Simulation studies are conducted to analyze longitudinal data with a trinomial outcome using a CTMC with and without covariates. Measures of performance including bias, component-wise coverage probabilities, and joint coverage probabilities are calculated. An application is presented using Alzheimer's disease caregiver stress levels. Comparisons of joint and component-wise parameter estimates yield conflicting inferential results in simulations from models with and without covariates. In conclusion, caution should be taken when conducting simulation studies aiming to assess performance and choice of inference should properly reflect the purpose of the simulation.     

Network reliability for multipath TCP networks with a retransmission mechanism under the time constraint

It is essential to reduce data latency and guarantee quality of service for modern computer networks. The emerging networking protocol, Multipath Transmission Control Protocol, can reduce data latency by transmitting data through multiple minimal paths (MPs) and ensure data integrity by the packets retransmission mechanism. The bandwidth of each edge can be considered as multi-state in computer networks because different situations, such as failures, partial failures and maintenance, exist. We evaluate network reliability for a multi-state retransmission flow network through which the data can be successfully transmitted by means of multiple MPs under the time constraint. By generating all minimal bandwidth patterns, the proposed algorithm can satisfy these requirements to calculate network reliability. An example and a practical case of the Pan-European Research and Education Network are applied to demonstrate the proposed algorithm.     

Estimation in a General Repair Model Based on Left-truncated Data

We consider a repairable system with general repairs introduced by Last & Szekli (1998a ). Apart from simple special cases this model leads to a strong dependency among the observed failure times. Our aim is to estimate the underlying failure time distribution and its cumulative hazard given that the failure process has been observed up to the n th failure. We use non-parametric estimators of Kaplan–Meier and Nelson–Aalen type. We prove strong uniform consistency of the estimators as n tends to infinity. Further results on weak convergence are derived. Neither stationarity nor mixing conditions are required.     

Bayesian estimation and prediction of the intensity of the power law process

In analyzing failure data pertaining to a repairable system, perhaps the most widely used parametric model is a nonhomogeneous Poisson process with Weibull intensity, more commonly referred to as the Power Law Process (PLP) model. Investigations relating to inference of parameters of the PLP under a frequentist framework abound in the literature. The focus of this article is to supplement those findings from a Bayesian perspective, which has thus far been explored to a limited extent in this context. Main emphasis is on the inference of the intensity function of the PLP. Both estimation and future prediction are considered under traditional as well as more complex censoring schemes. Modern computational tools such as Markov Chain Monte Carlo are exploited efficiently to facilitate the numerical evaluation process. Results from the Bayesian inference are contrasted with the corresponding findings from a frequentist analysis, both from a qualitative and a quantitative viewpoint. The developed methodology is implemented in analyzing interval-censored failure data of equipments in a fleet of marine vessels.     

Modelling of dependence between critical failure and preventive maintenance: The repair alert model

We consider the competing risks problem for a repairable unit which at each sojourn may be subject to either a critical failure, or a preventive maintenance (PM) action, where the latter will prevent the failure. It is reasonable to expect a dependence between the failure mechanism and the PM regime. The paper presents a new model, called the repair alert model, for handling such cases. This model is a special case of random signs censoring, which was introduced by Roger Cooke [1993. The total time on test statistic and age-dependent censoring. Statist. Probab. Lett., 18, 307–312]. The pleasant feature of random signs censoring is that the marginal distribution of the failure time is identifiable. The repair alert model introduces the so-called repair alert function, which characterizes the “alertness” of the maintenance crew, and which is shown to be uniquely identifiable from field data. Statistical estimation is considered both nonparametrically and parametrically.