Ordering conditional lifetimes of coherent systems (original) (raw)

Conditional residual lifetimes of coherent systems

Statistics & Probability Letters, 2013

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at least j and at most k − 1 (j < k) components have failed by time t. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.

A note on the conditional residual lifetime of a coherent system under double monitoring

Communications in Statistics - Theory and Methods, 2018

In this note, we derive some mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at time t 1 the jth failures has occurred and at time t 2 the kth failures (j < k) have not occurred yet. Based on the mixture representations, we then discuss the stochastic comparisons of the conditional residual lifetimes of two coherent systems with i.i.d. components.

On the residual lifelengths of the remaining components in a coherent system

Metrika, 2013

In this note, we consider a coherent system with the property that, upon failure of the system, some of its components remain unfailed in the system. Under this condition, we study the residual lifetime of the live components of the system. Signature based mixture representation of the joint and marginal reliability functions of the live components are obtained. Various stochastic and aging properties of the residual lifetime of such components are investigated. Some characterization results on exponential distributions are also provided.

Conditional residual lifetimes of coherent systems under double monitoring

Communications in Statistics, 2016

In this paper, we derive mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at least j and at most k − 1 (j < k) components have failed by time t. Based on these mixture representations, we then discuss stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.

On the conditional residual life and inactivity time of coherent systems

Journal of Applied Probability, 2014

In this paper we derive mixture representations for the reliability functions of the conditional residual life and inactivity time of a coherent system with n independent and identically distributed components. Based on these mixture representations we carry out stochastic comparisons on the conditional residual life, and the inactivity time of two coherent systems with independent and identical components.

Comparisons and bounds for expected lifetimes of reliability systems

2010

Sharp bounds on expectations of lifetimes of coherent and mixed systems composed of elements with independent and either identically or non-identically distributed lifetimes are expressed in terms of expected lifetimes of components. Similar evaluations are concluded for the respective mean residual lifetimes. In the IID case, improved inequalities dependent on a concentration parameter connected to the Gini dispersion index are obtained. The results can be used to compare systems with component lifetimes ordered in the convex ordering. In the INID case, some refined bounds are derived in terms of the expected lifetimes of series systems of smaller sizes, and the expected lifetime of single unit for the equivalent systems with IID components. The latter can be further simplified in the case of weak Schur-concavity and Schur-convexity of the system generalized domination polynomial.

On conditional residual lifetime and conditional inactivity time of k-out-of-n systems

In designing structures of technical systems, the reliability engineers often deal with the reliability analysis of coherent systems. Coherent system has monotone structure function and all components of the system are relevant. This paper considers some particular models of coherent systems having identical components with independent lifetimes. The main purpose of the paper is to study conditional residual lifetime of coherent system, given that at a fixed time certain number of components have failed but still there are some functioning components. Different aging and stochastic properties of variables connected with the conditional residual lifetimes of the coherent systems are obtained. An expression for the parent distribution in terms of conditional mean residual lifetime is provided. The similar result is obtained for the conditional mean inactivity time of the failed components of coherent system. The conditional mean inactivity time of failed components presents an interest in many engineering applications where the reliability of system structure is important for designing and constructing of systems. Some illustrative examples with given particular distributions are also presented.

Tail hazard rate ordering properties of order statistics and coherent systems

Naval Research Logistics, 2007

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k-out-of-n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions.

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems

Journal of Applied Probability, 2013

Under the assumption of independent and identically distributed (i.i.d.) components, the problem of the stochastic comparison of a coherent system having used components and a used coherent system has been considered. Necessary and sufficient conditions on structure functions have been provided for the stochastic comparison of a coherent system having used/inactive i.i.d. components and a used/inactive coherent system. As a consequence, for r-out-of-n systems, it has been shown that systems having used i.i.d. components stochastically dominate used systems in the likelihood ratio ordering.

Coherent systems based on sequential order statistics

Naval Research Logistics (NRL), 2011

The sequential order statistics (SOS) are a good way to model the lifetimes of the components in a system when the failure of a component at time t affects the performance of the working components at this age t. In this article, we study properties of the lifetimes of the coherent systems obtained using SOS. Specifically, we obtain a mixture representation based on the signature of the system. This representation is used to obtain stochastic comparisons. To get these comparisons, we obtain some ordering properties for the SOS, which in this context represent the lifetimes of k-out-of-n systems. In particular, we show that they are not necessarily hazard rate ordered.