Signatures in Markov-Modulated Processes

□ In recent years, signatures are widely used for analysis of coherent systems consisting of unreliable components. If component lifetimes are independent and identically distributed, then system lifetime distribution function is a convex combination of distribution functions of order statistics for component lifetimes. Coefficients of this convex combination are called signatures. This article considers the case when a system operates in a so-called random environment, i.e., component failure rates are jointly modulated by a finite-state continuous-time Markov chain. In this model, component lifetimes remain exchangeable. An expression for distribution function of time to system failure is derived. Here, a crucial role is played by an elaborated procedure of deriving a distribution function of order statistics for system component lifetimes. A numerical example illustrates the suggested approach and analyzes the influence of random environment on the distribution function of system lifetime.     

On the effect of dependency in information properties of series and parallel systems

Information measures of reliability systems has been widely studied in the statistical and reliability literatures. These findings were obtained when lifetimes of components are independent and identically distributed. But, there is no context about the information properties of such systems when lifetimes of components are dependent. In this paper, we explore properties of the entropy and Kullback–Leibler discrimination information for series and parallel system’s lifetimes when lifetimes of components are dependent and identically distributed. Specifically, we give some results on the entropy of series systems when lifetimes of components are positive or negative quadrant dependence. Moreover, several results are obtained about the entropy ordering properties related to other well known stochastic orders. To illustrate the quality of the given results, some examples are also given.     

Stochastic comparisons of two-unit repairable systems

Abstract

We consider two models of two-unit repairable systems: cold standby system and warm standby system. We suppose that the lifetimes and repair times of the units are all independent exponentially distributed random variables. Using stochastic orders we compare the lifetimes of systems under different assumptions on the parameters of exponential distributions. We also consider a cold standby system where the lifetimes and repair times of its units are not necessarily exponentially distributed.     

On NBUE property of one component system supported by an inactive standby and a repair facility

The paper deals with the aging property of a one-component system supported by an identical, inactive standby and a perfect repair facility. We assume that all the lifetimes and repair times induced by the operating and under-repair components are mutually independent and repair times are arbitrary. It is shown that the lifetime of a system that begins with one (both) operative component (s) having NWUE (NBUE) lifetimes is NWUE (NBUE).     

Bayesian prediction of minimal repair times of a series system based on hybrid censored sample of components’ lifetimes under Rayleigh distribution

In this paper, we develop Bayesian predictive inferential procedures for prediction of repair times of a series system, applying a minimal repair strategy, using the information contained in an independent observed hybrid censored sample of the lifetimes of the components of the system, assuming the underlying distribution of the lifetimes to be Rayleigh distribution. An illustrative real data example and a simulation study are presented for the purpose of illustration and comparison of the proposed predictors.     

Maximum likelihood analysis of masked series system lifetime data

Component lifetime parameters of a series system are estimated from system lifetimes and masked cause of failure observations. The time and cause of system failures are assumed to follow a competing risks model. The masking probabilities of the minimum random subsets are not subjected to the symmetry assumption. Sufficient regularity conditions are provided, justifying the maximum likelihood analysis. Maximum likelihood estimates of both the lifetime parameters and masking probabilities are generically computed via an EM algorithm. An appropriate set of asymptotically pivotal quantities are also derived. Such maximum likelihood based estimates are further refined by bootstrap. The developed techniques are illustrated by numerical examples of independent Weibull component lifetimes with distinct scale and shape parameters.     

Likelihood inference for the component lifetime distribution based on progressively censored parallel systems data

Point and interval estimators for the scale parameter of the component lifetime distribution of a k-component parallel system are obtained when the component lifetimes are assumed to be independently and identically exponentially distributed. We prove that the maximum likelihood estimator of the scale parameter based on progressively Type-II censored system lifetimes is unique and can be obtained by a fixed-point iteration procedure. In particular, we illustrate that the Newton–Raphson method does not converge for any initial value. Furthermore, exact confidence intervals are constructed by a transformation using normalized spacings and other component lifetime distributions including Weibull distribution are discussed.     

On Redundancy Allocation to Series and Parallel Systems of Two Components

Abstract

For two components and one standby redundancy, we develop a characterization on the hazard rate order and the reversed hazard rate order of the redundant system lifetime in the context of mutually independent components lifetimes. Also, the likelihood ratio order is derived on the lifetime of the series system with two components lifetimes and two matched active redundancies lifetimes both following the proportional hazard model.     

A note on the mean residual life function of a coherent system with exchangeable or nonidentical components

This article investigates some properties of the mean residual life function of (n−k+1)-out-of-n systems, when the lifetimes of the system components are independent random variables but not necessarily identically distributed and when the joint distribution of the component lifetimes is exchangeable, extending the results of Asadi and Goliforushani (2008) [On the mean residual life function of coherent systems. IEEE Transactions on Reliability 57 (4) 574-580] for the case of independent and identically distributed components. The extension to a coherent system with exchangeable components is also given.     

Distributions of the lifetimes of system components operating under an unknown common environment

SUMMARY Families of joint distributions for describing the lifetimes of a system of components that operate under an unknown environment, when the environment follows a Weibull distribution, are derived. The reliability function for this system is calculated and several properties of the aforementioned joint distributions are investigated.     

Dynamic reliability models with conditional proportional hazards

A dynamic approach to the stochastic modelling of reliability systems is further explored. This modelling approach is particularly appropriate for load-sharing, software reliability, and multivariate failure-time models, where component failure characteristics are affected by their degree of use, amount of load, or extent of stresses experienced. This approach incorporates the intuitive notion that when a set of components in a coherent system fail at a certain time, there is a jump from one structure function to another which governs the residual lifetimes of the remaining functioning components, and since the component lifetimes are intrinsically affected by the structure function which they constitute, then at such a failure time there should also be a jump in the stochastic structure of the lifetimes of the remaining components. For such dynamically-modelled systems, the stochastic characteristics of their jump times are studied. These properties of the jump times allow us to obtain the properties of the lifetime of the system. In particular, for a Markov dynamic model, specific expressions for the exact distribution functions of the jump times are obtained for a general coherent system, a parallel system, and a series-parallel system. We derive a new family of distribution functions which describes the distributions of the jump times for a dynamically-modelled system.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.     

Stochastic properties and parameter estimation for a general load-sharing system

We study here a general load-sharing parallel system in which the lifetimes of the components of the system are arbitrary continuous random variables. The system functions if at least one component in the system functions and the surviving unit shares the whole load. Some sufficient conditions are obtained for the usual stochastic order between two different load-sharing systems. We then consider the optimal allocation problem of one load standby in a series system with two independent components. Finally, the maximum likelihood estimation of the parameters for some specific systems is discussed.     

On hazard rate ordering of parallel systems with two independent components

This note builds a sufficient condition for the hazard rate ordering between lifetimes of parallel systems with two independent components having proportional hazard rates. Some comparisons on lifetimes of such systems with general components are also obtained.     

Stochastic comparisons of parallel and series systems of dependent components equipped with starting devices

This paper deals with series and parallel systems of dependent components equipped with starters. We study the hazard rate order, the dispersive order and the usual stochastic order of system lifetimes in the context of component lifetimes having proportional hazard rates. The main results either generalize or extend corresponding conclusions of Joo and Mi (2010) and Da, Ding, and Li (2010).     

Necessary and sufficient conditions for stochastic orders between (n − r + 1)-out-of-n systems in proportional hazard (reversed hazard) rates model

Consider two (n ? r + 1)-out-of-n systems, one with independent and non-identically distributed components and another with independent and identically distributed components. When the lifetimes of components follow the proportional hazard rates model, we establish a necessary and sufficient condition for the usual stochastic order to hold between the lifetimes of these two systems. For the special case of r = 2, some generalized forms of this result to the hazard rate, dispersive and likelihood ratio orders are also obtained. Moreover, for the case when the lifetimes of components follow the proportional reversed hazard rates model, we derive some similar results for comparing the lifetimes of two systems . Applications of the established results to different situations are finally illustrated.     

Estimation of Hazard, Density and Survivor Functions for Randomly Censored Data

Maximum likelihood estimation and goodness-of-fit techniques are used within a competing risks framework to obtain maximum likelihood estimates of hazard, density, and survivor functions for randomly right-censored variables. Goodness-of- fit techniques are used to fit distributions to the crude lifetimes, which are used to obtain an estimate of the hazard function, which, in turn, is used to construct the survivor and density functions of the net lifetime of the variable of interest. If only one of the crude lifetimes can be adequately characterized by a parametric model, then semi-parametric estimates may be obtained using a maximum likelihood estimate of one crude lifetime and the empirical distribution function of the other. Simulation studies show that the survivor function estimates from crude lifetimes compare favourably with those given by the product-limit estimator when crude lifetimes are chosen correctly. Other advantages are discussed.     

Stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components

Abstract

In this paper, we discuss stochastic comparisons of series and parallel systems with independent heterogeneous lower-truncated Weibull components. When a system with possibly different shape and scale parameters and its matrix of parameters changes to another matrix in a certain mathematical sense, we study the hazard rate order of lifetimes of series systems and the usual stochastic order of lifetimes of parallel systems.     

Hazard Rate Properties of Systems with Two Components

In this article, we study the hazard rate ordering of lifetimes of two-component systems (series and parallel) by considering some bivariate distributions for the joint distribution of component lifetimes. Such that, we aim to investigate the lifetimes of systems with stochastically dependent and with stochastically independent components, the lifetimes of the components, and a stochastic ordering relation between these lifetimes. In addition to these, the mononotonicity of the hazard rates of the parallel and the series systems for the bivariate Farlie–Gumbel–Morgenstern (FGM) family is studied.     

Bayesian Analysis of Masked Data in Step-stress Accelerated Life Testing

This article considers a k level step-stress accelerated life testing (ALT) on series system products, where independent Weibull-distributed lifetimes are assumed for the components. Due to cost considerations or environmental restrictions, causes of system failures are masked and type-I censored observations might occur in the collected data. Bayesian approach combined with auxiliary variables is developed for estimating the parameters of the model. Further, the reliability and hazard rate functions of the system and components are estimated at a specified time at use stress level. The proposed method is illustrated through a numerical example based on two priors and various masking probabilities.