The Joint Signature of Coherent Systems with Shared Components

Abstract

System signatures are useful tools in the study and comparison of coherent systems. In this paper, we define and study a similar concept, called the joint signature, for two coherent systems which share some components. Under an independent and identically distributed assumption on component lifetimes, a pseudo-mixture representation based on this joint signature is obtained for the joint distribution of the lifetimes of both systems. Sufficient conditions are given based on the respective joint signatures of two pairs of systems, each with shared components, to ensure various forms of bivariate stochastic orderings between the joint lifetimes of the two pairs of systems.

On the stochastic and dependence properties of the three-state systems

“…For signature based properties of coherent systems, we refer the reader, among others, to Asadi and Goliforushani (2008), Eryilmaz and Zuo (2010), Eryilmaz (2011b), Kelkin Nama et al (2013, Khaledi and Shaked (2007), Navarro et al (2008aNavarro et al ( , 2010Navarro et al ( , 2008b, Samaniego (1985Samaniego ( , 2007 and Zhang (2010a,b).…”

Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

“…lifetimes with common continuous distribution F, Navarro et al [26] established an explicit representation of the joint distribution of the pair (T 1 , T 2 ) of system lifetimes. The representation in [26] displays the joint distribution G(t 1 , t 2 ) = P(T 1 ≤ t 1 , T 2 ≤ t 2 ) of system lifetimes separately, for t 1 ≤ t 2 or for t 1 > t 2 , as a function of distribution-free matrices of coefficients S or S * and the marginal distributions of the order statistics of component lifetimes.…”

“…Worthy of special mention is the fact that the representations in [26] do not involve the joint behavior of these order statistics. As we will see, this is a key difference between this representation and the main result in the present work.…”

On a mixture representation of the conditional inactivity time of a coherent system

“…, X n . For more details on this topic, one may refer to Boland (2001), Samaniego (2007), Triantafyllou and Koutras (2008), Samaniego et al (2009), Navarro et al (2010, Mahmoudi and Asadi (2011), etc. The mixture representation of the reliability function of the coherent system has been found to be useful to obtain stochastic comparison results and many researchers have discussed this topic; see, for example, Navarro et al (2005), Li and Zhang (2008), Navarro et al (2008), Zhang (2010b), Eryılmaz (2011a,b) and Burkschat and Navarro (2013).…”