On unimodality of the lifetime distribution of coherent systems with failure-dependent component lifetimes

Abstract

We study the conditions for unimodality of the lifetime distribution of a coherent system when the ordered component lifetimes in the system are described by generalized order statistics. Results for systems with independent and identically distributed lifetimes of components are included in this setting. The findings are illustrated with some examples for different types of systems. In particular, coherent systems with strictly bimodal density functions are presented in the case of independent standard uniform distributed lifetimes of components. Furthermore, we use the results to derive a sharp upper bound on the expected system lifetime in terms of the mean and the standard deviation of the underlying distribution.

Comparisons of the Expectations of System and Component Lifetimes in the Failure Dependent Proportional Hazard Model

“…The assumption is not very restrictive, because coherent systems with non-unimodal signatures are very rare. An example of a system of size 5 with bimodal signature was presented in Jasiński et al (2009), and the construction was extended to higher dimensions by Bieniek and Burkschat (2018). In the proof, we apply a characterization of unimodality based on sign change behavior (cf., e.g., Marshall and Olkin 2007, proof of Proposition B.2., p. 99).…”