On the mean past lifetime of the components of a parallel system

Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

“…or exchangeable dependent components, especially k-outof-n systems. For instance, see [1], [8], [9], [10], [12], [13], [15], [21], [22], [24], [26], [27], and [28].…”

Results on the past lifetime of (n − k + 1)-out-of-n structures with nonidentical components

“…In recent years several authors have studied the reliability and aging properties of (n − k + 1)-out-of-n systems. Among others, we refer to Asadi (2006), Asadi and Bayramoglu (2006), Bairamov and Arnold (2007), Khaledi and Shaked (2007), Li and Zhang (2008), Li and Zhao (2008), Navarro and Hernandez (2008), , Gurler and Bairamov (2009), Samaniego et al (2009), Navarro and Balakrishnan (2010), Zhang (2010) and Navarro and Shaked (2010). In a recent paper, Kochar and Xu (2010) studied the stochastic properties of the residual lifetime of (n − k + 1)-out-of-n systems under the condition that the components of the system have independent and nonidentical lifetimes.…”

“…Most of the results in the literature are dealing with the case in which the system has independent and identical components, i.e the X i 's are independent and have the same distribution function F. Under these conditions Asadi (2006) considered the past lifetime of the components of a parallel system and defined the concept of mean past lifetime (MPL) of the components, denoted by M k n (t), as follows M k n (t) = E(t − X k:n |X n:n ≤ t), k = 1, 2, . .…”

“…This expectation shows the mean time since the failure of the component with lifetime X k:n given that the system has already failed at time t or sometime before t. Several properties of M k n (t) are explored by Asadi (2006). Tavangar and Asadi (2010) extended the result of Asadi (2006) to (n − k + 1)-out-of-n system and defined the MPL of the components of the system as follows M l,k n (t) = E (t − X l:n | X k:n ≤ t) , l = 1, 2, .…”

Conditional Ordering of k-out-of-n Systems with Independent But Nonidentical Components

“…For more details about autopsy data, we refer the reader to Meilijson (1981) and Natvig (1998), (2001). Asadi (2006) studied the mean inactivity time E(t − X n:n | X n:n ≤ t) of a parallel system given that the system failed at or before time t > 0. More generally, under the assumption that the remaining (n − k) components continue to work and are still subject to failure after the failure of the system, Khaledi and Shaked (2006) …”