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

Abstract

By considering k-out-of-n systems with independent and nonidentically distributed components, we discuss stochastic monotone properties of the residual life and the inactivity time. We then present some stochastic comparisons of two systems based on the residual life and inactivity time.

Some characterization results for parallel systems with two heterogeneous exponential components

“…exponential random variables. Since then, many researchers have worked on this topic, including Proschan and Sethuraman [8], Kochar and Rojo [9], Dykstra et al [10], Khaledi and Kochar [11,12], Bon and Pǎltǎnea [13], Kochar and Xu [14], Pǎltǎnea [15], Zhao and Balakrishnan [16], Zhao et al [17,18], and Joo and Mi [19].…”

New results on comparisons of parallel systems with heterogeneous gamma components

“…(independent and non-identically distributed) exponential random variables. Subsequently, many researchers have worked on this topic, including Proschan and Sethuraman (1976), Boland et al (1994), Kochar and Rojo (1996), Dykstra et al (1997), Khaledi and Kochar (2000), Sun and Zhang (2005), Bon and Pǎltǎnea (2006), Kochar and Xu (2007), Pǎltǎnea (2008), Balakrishnan (2009, 2010a,b), Zhao et al (2008, Joo and Mi (2010), Mao and Hu (2010), and Khaledi et al (2011).…”

On conditional residual lifetime and conditional inactivity time of k-out-of-n systems

“…A problem of interest is to get information about the history of the system, e.g., when the components of the system have failed. Motivated by this, Asadi (2006) defined and investigated the concept of mean inactivity time (MIT) of failed components of a parallel system, at the system level, as E(t − X i:n | X n:n ≤ t), 1 ≤ i ≤ n. Tavangar and Asadi (2010) have extended the Asadi's (2006) results to (n − k + 1)-out-of-n structures and defined the MIT of components of such a system as E(t − X i:n | X k:n ≤ t), 1 ≤ i ≤ k ≤ n. By considering (n − k + 1)-out-of-n systems with independent and non-identically distributed components, Zhao et al (2008) study the inactivity time (t − X i:n | X k:n ≤ t < X k+1:n ) given that the system had failed but the (k + 1)th component 1 ≤ i < k ≤ n, is working at time t ≥ 0. In this paper, we also consider the mean time that has elapsed from the failure of a component with lifetime X i:n , i = 1, 2, ..., j, given that the component with lifetime X j:n has failed at or before time t, but the system is working at t; that is, the random variable…”