Systems with Failure-Dependent Lifetimes of Components

Abstract

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k-out-of-n system, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

Coherent systems based on sequential order statistics

“…(2.1) can be used to compute the system reliability function. Burkschat [7] extends SOS to exchangeable multivariate distributions obtaining a similar representation for the coherent systems based on these generalized SOS.…”

Dynamic Signatures of Coherent Systems Based on Sequential Order Statistics

“…Related recent results in settings of generalized order statistics can be found in Xie and Hu (2008), Balakrishnan et al (2009), Burkschat andLenz (2009), andHashemi et al (2010). This follows from the distribution theoretical relation (X * i+1:n , .…”

“…Recently, the idea underlying sequential order statistics has been extended to general coherent systems (see Burkschat (2009) and Navarro and Burkschat (2011)). In order to describe component lifetimes in a system, where failures might change the lifetimes of remaining components, a particular dependence model for the components is assumed.…”

Reliability, signature, and relative quality functions of systems under time‐homogeneous load‐sharing models

“…In particular, following the works of Arjas and Norros,5,7 in the paper of Norros, 8 the more general version of the time-homogeneous case was considered where the joint distribution of the lifetimes X 1 , … , X n is not absolutely continuous. Such models have been looked at as multivariate extensions of the exponential distribution (see, eg, other works 8,13,27 ).…”

“…functions j (t|I; t 1 , … , t |I| ) do not even depend on the calendar time t. Such models can be seen as natural generalizations of the condition of independent and exponentially distributed lifetimes (see, in particular, the work of Norros 8 ). A number of different papers in the literature have been devoted to the reliability analysis of systems under related probability models for the lifetimes of components (see, in particular, other works 11,[13][14][15][16] ).…”