Reliability of coherent systems whose operating life is defined by the lifetime and power of the components
Ismihan Bayramoglu
TL;DR
This paper develops a reliability framework for coherent systems where component lifetimes $X_i$ interact with decaying power contributions $W_i(t)=W_i\varphi(t)$, with $\varphi$ nonincreasing and $X$ and $W$ potentially dependent via a joint law. The core idea is to quantify operational reliability through the joint survival of the $r$th order statistic and the concomitants of the surviving components, yielding the central function $\mathbf{Q}(t,s)$ and the normalizing term $D(s)$ to obtain $\mathbf{P}_s(t)=\mathbf{Q}(t,s)/D(s)$. The paper derives exact integral expressions for $\mathbf{Q}(t,s)$, analyzes the $n-r+1$-out-of-$n$ system with a power constraint $s$, and defines the operational mean residual life $\Psi_{r:n,s}(t)$ with an explicit pdf for $T_{r:n;s}$. Through special cases (independence, exponential, Pareto, and copula-based dependence) and graphical examples, the work demonstrates how the joint lifetimes and concomitants govern reliability under a power threshold, providing a practical, adjustable framework for systems where both degradation and power contribution matter.
Abstract
We consider systems whose lifetime is measured by the time of physical degradation of components, as well as the degree of power each component contributes to the system. The lifetimes of the components of the system are random variables. The power that each component contributes to the system is the product of a random variable and a time-decreasing stable function. The operational reliability of these systems is investigated and shown that it is determined by the joint lifetime functions of the order statistics and their concomitants. In addition to general formulas, examples are given using some known life distributions, and graphs of the operation life functions are shown.