Comparison results for inactivity times of k-out-of-n and general coherent systems with dependent components

Coherent systems, i.e., multicomponent systems where every component monotonically affects the working state or failure of the whole system, are among the main objects of study in reliability analysis. Consider a coherent system with possibly dependent components having lifetime T , and assume we know that it failed before a given time t > 0. Its inactivity time t −T can be evaluated under different conditional events. In fact, one might just know that the system has failed and then consider the inactivity time (t − T |T ≤ t), or one may also know which ones of the components have failed before time t, and then consider the corresponding system’s inactivity time under this condition. For all these cases, we obtain a representation of the reliability function of system inactivity time based on the recently defined notion of distortion functions. Making use of these representations, new stochastic comparison results for inactivity times of systems under the different conditional events are provided. These results can also be applied to order statistics which can be seen as particular cases of coherent systems (k-out-of-n systems, i.e., systems which work when at least k of their n components work)

Generalized Marshall-Olkin Distributions, and Related Bivariate Aging Properties

National Natural Science Foundation of China [10771090]A class of generalized bivariate Marshall-Olkin distributions, which includes as special cases the Marshall-Olkin bivariate exponential distribution and the Marshall-Olkin type distribution due to Muliere and Scarsini (1987) [19] are examined in this paper. Stochastic comparison results are derived, and bivariate aging properties, together with properties related to evolution of dependence along time, are investigated for this class of distributions. Extensions of results previously presented in the literature are provided as well. (C) 2011 Elsevier Inc. All rights reserved

Past Lifetime and Inactivity Time: from Entropy to Coherent Systems

Information Theory was originally proposed by Claude Shannon in 1948 in the landmark paper entitled "A Mathematical Theory of Communication". In this paper the concept of entropy was adopted for the first time in a field other than thermodynamics and statistical mechanics. Since then, the interest in entropy has grown more and more and the current literature now focuses mainly on the analysis of residual lifetime. However, in recent years the interest has 'changed direction'. New notions of entropy have been introduced and are used to describe the past lifetime and the inactivity time of a given system or of a component that is found not to be working at the current time. Moreover inferences about the history of a system may be of interest in real life situations. So, the past lifetime and the inactivity time can also be analysed in the context of the theory of coherent systems

Weighted mean inactivity time function with applications

The concept of mean inactivity time plays a crucial role in reliability, risk theory and life testing. In this regard, we introduce a weighted mean inactivity time function by considering a non-negative weight function. Based on this function, we provide expressions for the variance of transformed random variable and the weighted generalized cumulative entropy. The latter concept is an important measure of uncertainty which is shift-dependent and is of interest in certain applied contexts, such as reliability or mathematical neurobiology. Moreover, based on the comparison of mean inactivity times of a certain function of two lifetime random variables, we introduce and study a new stochastic order in terms of the weighted mean inactivity time function. Several characterizations and preservation properties of the new order under shock models, random maxima and renewal theory are discussed.Comment: 25 page