‎\documentclass[fleqn,11pt]{article}‎ ‎\usepackage{mathtools}‎ ‎\usepackage{ctable}‎ ‎\usepackage{amsmath}‎ ‎\usepackage{amsfonts}‎ ‎\usepackage{amssymb}‎ ‎\usepackage{latexsym}‎ ‎\usepackage{float}‎ ‎%\usepackage{fancyhdr}‎ ‎\usepackage{amssymb}‎ ‎\usepackage[justification=centering]{caption}‎ ‎%\usepackage{setspace}‎ ‎%\usepackage[dvips]{graphicx}‎ ‎\usepackage{lscape}‎ ‎\usepackage{pdflscape}‎ ‎\usepackage{float}‎ ‎\usepackage{mathrsfs,amsmath}‎ ‎\usepackage{longtable}‎ % ‎necessary for end of proof symbol according to Jim‎ ‎%\usepackage{rotate,epsfig}% necessary for figures‎ ‎\renewcommand{\baselinestretch}{1.3}‎ ‎\setlength{\textwidth}{16cm}‎ ‎\setlength{\textheight}{21cm}‎ ‎\setlength{\oddsidemargin}{-0.2cm}‎ ‎\setlength{\evensidemargin}{-0.2cm}‎ ‎\setlength{\mathindent}{0.2in} %equations intended by one inch‎ ‎\setlength{\jot}{.2in} %increases space b/w lines of eqnarray by‎ .‎2in‎ % ‎necessary for end of proof symbol according to Jim‎ ‎%\usepackage{rotate,epsfig}% necessary for figures‎ ‎\renewcommand{\baselinestretch}{1.3}‎ ‎\setlength{\textwidth}{16cm}‎ ‎\setlength{\textheight}{21cm}‎ ‎\setlength{\oddsidemargin}{-0.2cm}‎ ‎\setlength{\evensidemargin}{-0.2cm}‎ ‎\setlength{\mathindent}{0.2in} %equations intended by one inch‎ ‎\setlength{\jot}{.2in} %increases space b/w lines of eqnarray by‎ .‎2in‎ ‎\newtheorem{theorem}{\bf Theorem}‎ ‎\newtheorem{proposition}{\bf Proposition}‎ ‎\newtheorem{notation}{\bf Notation}‎ ‎\newtheorem{lemma}{\bf Lemma}‎ ‎\newtheorem{proof}{\bf Proof}‎ ‎\newtheorem{corollary}{\bf Corollary}‎ ‎\newtheorem{definition}{\bf Definition}‎ ‎\newtheorem{remark}{\bf Remark}‎ ‎\newtheorem{example}{\bf Example‎ } ‎\newtheorem{result}{\bf Result}‎ ‎\newtheorem{algorithm}{\bf Algorithm}‎ ‎\newtheorem{exmp}{Example}[section]‎ ‎\usepackage{amssymb}‎ ‎\usepackage{color}‎ ‎\newcommand{\ph}{\textcolor{blue}}‎ ‎%\newcommand{\ph}{\textcolor{red}}‎ ‎\newcommand{\gr}{\textcolor{green}}‎ ‎%***********************************************‎ ‎\begin{document}‎ ‎\title{ \textbf{ Analyzing the Cumulative Residual Extropy of System Inactivity Times and‎ Revealing System Complexity }} ‎\author{‎ ‎Zohreh Pakdaman$^1$\footnote{Corresponding author.\newline E-mail‎: ‎{\it zpakdaman@hormozgan.ac.ir } (Z‎. ‎Pakdaman), ‎{\it reza.alizadehn@gmail.com} (R‎. ‎Alizadeh Noughabi).} and‎ ‎Reza Alizadeh Noughabi$^1$‎ ‎\\‎ ‎$^1$\,{\small {\it Department of Statistics‎, ‎University of Hormozgan‎, ‎Hormozgan‎, ‎Iran}}\vspace{-0.2cm}\\‎ ‎%{\small {\it P‎. ‎O‎. ‎Box 3995‎, ‎Hormozgan 7916193145,\ Iran}}‎ } ‎%\date{}‎ ‎\maketitle‎ ‎\vspace{-0.4cm}‎ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ‎\begin{abstract}‎ In this paper, we employ an information-theoretic methodology known as cumulative residual extropy to assess and compare the ‎systems inactivity time‎. We develop mixture representations to model the cumulative residual extropy of system inactivity times and examine how this metric varies across various systems. Our analysis involves comparing these systems using stochastic ordering techniques and stochastically ordered conditional coefficient vectors. Furthermore, we establish bounds for the cumulative residual extropy of systems inactivity time. In addition, we introduce the Jensen-cumulative residual extropy divergence as a new measure for evaluating system complexity. To illustrate the applicability of our findings, we compute and compare the cumulative residual extropy and Jensen-cumulative residual extropy divergence for system inactivity times in an Exponential distribution framework. Finally, we determine the optimal system configuration by utilizing the signature criterion, derived from Jensen-cumulative residual extropy, within the context of the Exponential model. ‎\end{abstract}‎ ‎\vskip 1mm \noindent{\bf Keywords and Phrases}‎: ‎Cumulative residual extropy‎, ‎Jensen-cumulative residual extropy‎, ‎Mixed system‎, ‎Order statistics‎, ‎Stochastic order‎, ‎System signature‎, Systems inactivity time.\\‎ ‎\noindent{\bf AMS 2000 Subject Classification}‎: ‎Primary 62N05;‎ ‎Secondary 62F10‎. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ‎\section{Introduction}‎‎ ‎A coherent system is defined as a system where every component plays an essential role, and the relationship between the system's structure and its function is monotonic. The collection of all possible stochastic combinations of coherent systems of a fixed size is referred to as a mixed system. For an in-depth discussion, see Barlow and Proschan (1981). System signatures are valuable tools for the analysis and comparison of engineering systems. The conceptual foundation of system signatures is outlined in Samaniego (2007). Moreover, Samaniego (1985) demonstrated that the lifetime distribution of a coherent system, composed of ‎$‎n‎$‎‏ independent and identically distributed (IID) components, can be described by a function that depends exclusively on the system's architecture. For a system of order $n$, where the component lifetimes $Y_i$​ are IID random variables with a common distribution, the signature of the system is represented by an $n$-dimensional probability vector $\boldsymbol{s}=(s_1,\cdots,s_n)$. The ‎$‎i‎$‎-th element of this vector is defined as $s_{i} =P(T(n)=Y_{i:n})$, where ‎$i=1,\cdots,n$, and the condition $\sum_{i=1}^{n}s_i=1$ holds. Here, $T(n)$ represents the lifetime of the system, and $Y_{i:n}$​ denotes the lifetime of the $‎i‎$-th weakest component in the system. Let ‎$\bar{G}_{T(n)}(t)=1-{G}_{T(n)}(t)$‎ denote the survival function (SF) of the system with lifetime $T(n)$, where ‎${G}_{T(n)}(t)‎$ is the cumulative distribution function (CDF) of the random variable $T(n)$. According to Samaniego (2007), this SF can be expressed as: ‎\begin{equation}\label{1}‎ ‎\bar{G}_{T(n)}(t)=P({T(n)}>t)=\sum_{i=1}^{n}s_{i}P(Y_{i:n}>t)‎. ‎\end{equation}‎ This formula reveals how the survival probability of the system, ‎$‎‎\bar{G}_{T(n)}(t)‎‎$, is determined by the system signature and the survival probabilities of the individual components. ‎In recent years‎, ‎researchers have shown increasing interest in studying the reliability properties of coherent systems using a signature vector‎. ‎Kochar et al‎. ‎(1999) proposed some applications of the signature vector to compare coherent systems with IID components‎. ‎Additionally‎, ‎Navarro et al‎. ‎(2005‎, ‎2007) utilized the concept of the signature vector to compare coherent systems when the components are not necessarily independent‎. ‎%Zhang (2010) obtained some stochastic results for the conditional coherent system‎.‎ ‎In reliability theory, the historical performance of a system often provides valuable insights into its operational characteristics. For example, consider a coherent system with a lifetime $T(n)$. If the system fails before a specified time $t>0$, the operator may evaluate the duration that has elapsed since the system's failure. This analysis involves the conditional random variable $t-T(n)|T(n)\leq t$, commonly referred to as the system inactivity time (SIT). Specifically, when the system is observed at time $t$, and it is found to have failed (i.e., $T(n)\leq t$), the SIT captures the interval $t-T(n)$, conditioned on the event $T(n)\leq t$.‎‎‎ This concept is closely related to "autopsy data," which refers to insights gathered by examining the conditions of a system's components post-failure. For further information, readers can consult sources such a‏s Bhattacharya and Samaniego (2010)‎, ‎‎ ‎Gåsemyr and Natvig (1998),‎ Gåsemyr and Natvig (2001)‎, and Meilijson (1981). ‎ Zhang and Jiaotong (2010) established the representation similar to Equation \eqref{1} for the‎ % ‎the reliability function of a working system‎, ‎i.e‎. ‎for the distribution of the system lifetime $T(n)$ given that it is known that $T(n)>t$ or‎, ‎equivalently‎, ‎for the‎ SF of SIT $t-T(n)|T(n)‎\leq‎ t$ as follows‎ ‎\begin{equation}\label{202}‎ ‎P(t-T(n)|T(n)‎\leq ‎t‎)=\sum\limits_{i=1}^{n}q_{i}(t)P(t-Y_{i:n}>x|Y_{i:n}‎\leq ‎t)‎, ‎\end{equation}‎ ‎where $q_{i}(t)=\frac{s_{i}P(Y_{i:n}‎\leq ‎ ‎t)}{P(T(n)‎\leq‎ t)}$ for $i=1,\ldots,n$ is conditional coefficient vector given $T(n)‎\leq ‎t$‎. Zhang and Jiaotong (2010) studied the stochastic orders and various properties of Equation \eqref{202}‎. ‎One purpose of this paper is to investigate the information properties of SIT from the viewpoint of cumulative residual extropy‎. ‎ ‎In information theory, entropy quantifies the uncertainty linked to a random variable. This concept, introduced by Shannon in 1948, is mathematically defined for a continuous random variable $X$ with a probability density function (PDF) $f(x)$ as: ‎$‎ ‎{H}(X) =-E(\log f(X))‎ ‎$‎ where $``\log"$ denotes the natural logarithm. In 2015, Lad and colleagues introduced an alternative measure of uncertainty, termed "extropy." Designed as a counterpart to entropy, extropy has been referenced in academic discussions as representing attributes such as intelligence, functional order, vitality, energy, life, experience, capacity, and a drive for improvement and growth within living or organizational systems. Lad et al. (2015) defined the extropy of a random variable $X$ with PDF $f(x)$ as follows:‎ ‎\begin{equation*}\label{2}‎ ‎J(X)=-\frac{1}{2}\int_{-\infty}^{+\infty}f^2(x)dx=-\frac{1}{2}\int_{0}^{1}f(F^{-1}(u))du‎. ‎\end{equation*}‎‎ Numerous studies have explored extropy and its applications in analyzing the informational properties of reliability systems. Qiu and Jia (2018a) interpreted the extropy of a system's residual lifetime as a measure of remaining uncertainty and examined its properties within order statistics. In a separate work, Qiu (2017) investigated characterization results and symmetric properties of extropy concerning order statistics and record values. Additionally, Qiu and Jia (2018b) proposed two estimators for calculating the extropy of absolutely continuous random variables. For more in-depth research on the informational aspects of system extropy, readers can consult works such as Qiu (2017), Jose and Sathar (2019), Jahanshahi et al. (2019), and Chakraborty and Pradhan (2023). ‎ Cumulative residual entropy (CRE) was initially introduced by Rao et al. in 2004, based on the SF of $X$. Later, in 2020, Jahanshahi et al. proposed cumulative residual extropy (CRJ), which is defined as:‎ ‎\begin{equation}\label{kesi}‎ ‎ \xi J\left( X \right) =‎ - ‎\frac{1}{2}\int_0^\infty {{{\bar F}^2}\left( x \right)dx}‎. ‎\end{equation}‎ ‎ This study explores the CRJ of the SIT using Equation \eqref{202}. The paper is structured as follows: Section 2 introduces an expression for the CRJ of SIT based on the conditional coefficients vector, along with derived stochastic comparisons and bounds. Section 3 develops a new divergence measure to study the complexity of mixed systems. In Section 4, the CRJ of mixed systems is compared under an exponential system with three IID components, considering various configurations. Section 5 focuses on identifying the optimal signature vector to simultaneously minimize the Jensen-cumulative residual extropy divergence and costs. Finally, Section 6 provides concluding remarks summarizing the study's key findings.‎ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ‎\section{CRJ for the SIT}‎‎ ‎Let $Y_1, \ldots, Y_n$ represent the lifetimes of IID components, with the CDF $G$ for a coherent system of order $n$ characterized by the signature vector $\boldsymbol{s}$. Let $T(n)$ be the lifetime of this system, with the PDF $g_{T(n)}(.)$ and the cdf $G_{T(n)}(.)$. Additionally, assume the system starts at time $t = 0$ and is operational at time $t > 0$. The goal is to compute the extropy of the used system, or equivalently, the extropy of the residual lifetime $t - T(n) \mid T(n) \leq t$.‎ For simplicity in notation, we define $t - T(n) \mid T(n) \leq t$, $t - Y_{i:n} \mid Y_{i:n} \leq t$, and $t - Y_i \mid Y_i \leq t$ for $i = 1, \ldots, n$ as $IT(n,t)$, $IY_{i:n,t}$, and $IY_{i,t}$, respectively, throughout the rest of the paper. Additionally, assume that the PDF and CDF of $IT(n,t)$ (or $IY_{i:n,t}$) are $g_{IT(n,t)}(.)$ and $G_{IT(n,t)}(.)$ (or $g_{i:n,t}(.)$ and $G_{i:n,t}(.)$), respectively. ‎From \eqref{202} and Zhang and Jiaotong (2010)‎, ‎we have‎ ‎\begin{equation}\label{3}‎ ‎\bar{G}_{IT(n,t)}(x)= \sum\limits_{i=1}^{n}q_{i}(t)\bar{G}_{i:n,t}(x)‎, ‎\end{equation}‎ ‎and‎ ‎\begin{equation*}\label{15}‎ ‎g_{IT(n,t)}(x)= \sum\limits_{i=1}^{n}q_{i}(t)g_{i:n,t}(x)‎, ‎\end{equation*}‎ where $q_i(t) = \frac{s_i P(Y_{i:n} \leq t)}{G_{T(n)}(t)}$ for $i = 1, \ldots, n$, and $\bar{G}_{IT(n,t)}(x) = 1 - G_{IT(n,t)}(x)$ and $\bar{G}_{T(n)}(t) = 1 - G_{T(n)}(t)$ are the SFs corresponding to the systems with lifetimes $IT(n,t)$ and $‎‎‎T(n)$, respectively. In this section, we derive an expression for the CRJ of the SIT, given the signature vector $\boldsymbol{s}$. Using equations \eqref{kesi} and \eqref{3}, the CRJ for $IT(n,t)$ is expressed as ‎\begin{eqnarray}\label{4}‎ ‎\xi J(IT(n,t))&=&‎ -‎\frac{1}{2}\int_{0}^{+\infty}\bar{G}^2_{IT(n,t)}(x)dx\nonumber\\‎ ‎&=&‎ -‎\frac{1}{2}\int_{0}^{t}\Bigg(\sum\limits_{i=1}^{n}q_{i}(t)‎ ‎\frac{\sum\limits_{j=i}^{n}{n \choose j}G(x)^{j}\bar{G}(x)^{n-j} }{{G}_{i:n}(t)}‎ ‎\Bigg)^2dx\nonumber\\‎ ‎&=&-\frac{1}{2}\int_{0}^{G(t)}\frac{\Bigg(\sum\limits_{i=1}^{n}q_{i}(t)‎ ‎\frac{\sum\limits_{j=i}^{n}{n \choose j}v^{j}(1-v)^{n-j} }{{G}_{i:n}(t)}‎ ‎\Bigg)^2}{g(G^{-1}(v))}dv\nonumber\\‎ ‎&=&‎ ‎-\frac{1}{2}\int_{0}^{G(t)}\frac{{{G}}^2_{V_t}(v)}{g(G^{-1}(v))}dv,‎ ‎\end{eqnarray}‎ ‎where‎ ‎\begin{equation*}\label{5}‎ ‎{{G}}_{V_t}(v)=\sum\limits_{i=1}^{n}q_{i}(t){{G}}_{V_{t,i}}(v)‎, ‎\end{equation*}‎ ‎and‎ ‎\begin{equation*}\label{9}‎ ‎{{G}}_{V_{t,i}}(v)=\sum\limits_{j=i}^{n}{n \choose j}\frac{v^{j}(1-v)^{n-j} }{{G}_{i:n}(t)},~‎~ 0 0, $ and the CDF is $ G_Y(y) = \frac{\gamma(\alpha, \lambda y)}{\Gamma(\alpha)}, $ where $\Gamma(‎\alpha‎)=\int_{0}^{‎+\infty‎}t^{\alpha -1}e^{-t} dt$ represents the complete gamma function, and ‎$‎\gamma(\alpha, \lambda y)=\int_{0}^{ \lambda y}t^{‎\alpha -1‎}e^{-t} dt$ is the incomplete gamma function. This system can be viewed as a uniform mixture of series and parallel systems, each consisting of ‎$‎n‎$‎‏ components, with each configuration selected with a probability of $\frac{1}{2}$​. The following is established ‎\begin{eqnarray*}‎ ‎{{G}}_{V}(v)&=&\sum\limits_{i=1}^{n}s_{i}{G}_{V_{i}}(v)\nonumber\\‎ ‎&=&\frac{1}{2}\left[1+v^n-(1-v)^n\right]‎, ‎\end{eqnarray*}‎ ‎\begin{eqnarray*}‎ {G}_{T(n)}(t)&=&\sum_{i=1}^{n}s_{i}P(Y_{i:n}‎\leq ‎t)\nonumber\\‎ ‎&=&\frac{1}{2}\left[1-\left(1-\frac{\gamma(\alpha,\lambda t)}{\Gamma(\alpha)}\right)^n+\left(\frac{\gamma(\alpha,\lambda t)}{\Gamma(\alpha)}\right)^n\right]‎, ‎\end{eqnarray*}‎ ‎and $g(G^{-1}(v))=\frac{\lambda}{\Gamma(\alpha)}\left(\gamma^{-1}\left(\alpha,\Gamma(\alpha)v\right)\right)^{\alpha-1}e^{-\gamma^{-1}\left(\alpha,\Gamma(\alpha)v\right)}$‎, ‎where $\gamma^{-1}\left(\alpha,\Gamma(\alpha)v\right)$‎ ‎is the inverse of the incomplete gamma function $\gamma(\alpha,\Gamma(\alpha)v)$‎. As a result, by applying Equation \eqref{110}, we derive ‎\begin{equation}\label{exa1}‎ ‎\xi J(IT(n,t))=-\frac{\Gamma(\alpha)}{2\lambda\left[1-\left(1-\frac{\gamma(\alpha,\lambda t)}{\Gamma(\alpha)}\right)^n+\left(\frac{\gamma(\alpha,\lambda t)}{\Gamma(\alpha)}\right)^n\right]^2}\int_{0}^{\frac{\gamma(\alpha,\lambda t)}{\Gamma(\alpha)}}\frac{\left[1+v^n-(1-v)^n\right]^2}{\left(\gamma^{-1}\left(\alpha,\Gamma(\alpha)v\right)\right)^{\alpha-1}e^{-\gamma^{-1}\left(\alpha,\Gamma(\alpha)v\right)}}dv‎. ‎\end{equation}‎ ‎\begin{figure}[H]‎ ‎\centering‎ ‎\vspace*{-0.6cm}‎ ‎\includegraphics[scale=0.3]{example1.eps}‎ ‎\vspace*{-0.5cm}\caption{ \footnotesize{Plot of $\xi‎ ـ‎J(T(n,t))$ in Example \ref{examplereviewer1}‎. ‎}}\label{figexa1}‎ ‎\end{figure}‎‎‎ ‎In Figure \ref{figexa1}, we illustrate the CRJ for Example \ref{examplereviewer1}, using the parameters $\lambda = \frac{3}{5}$, $\alpha = 2$, and different values of $n$. The observed trend shows that as the number of components increases, the CRJ of the SIT decreases. This outcome is consistent with the expectation that systems with lower CRJ values are deemed more reliable. Additionally, based on the monotonicity property of coherent systems, it follows that augmenting the number of components leads to an improvement in system reliability. ‎\end{example}‎ ‎%where‎ ‎%\begin{equation}\label{11}‎ ‎%g_{V}(v)=\sum\limits_{i=1}^{n}s_{i}‎ ‎%\frac{v^{i-1}(1-v)^{n-i} }{B(i,n-i+1)},~‎~ ‎G(t)0$‎. ‎%{{(v)}} ${\bf p}\leqslant _{hr}{\bf q}$ if and only if ${\sum\limits_{i=j}^{n}p_{i}}\Big/{\sum\limits_{i=j}^{n}q_{i}}$ is decreasing in $j$‎, ‎for $j=1,\cdots,n$.\\‎ ‎%{{(vi)}} ${\bf p}\leqslant _{lr}{\bf q}$ if and only if ${p_{i}}/q_{i}$ is decreasing in $i$‎, ‎for $i=1,\cdots,n$ when $p_{i},q_{i}>0$‎. ‎\end{definition}‎ ‎%Based on this similarity‎, ‎some results in Qiu and Jia (2018)‎ ‎%satisfy for the extropy of mixed used system‎. ‎ ‎In the subsequent analysis, let us define the model function of the system as ${\cal{M}}_{IT(n,t),Y,G‎, ‎\boldsymbol{ q}(t)}=\{T(n,t),Y,G(.)‎, ‎\boldsymbol{q}(t) \} $, where the system is associated with the lifetimes of ‎$‎n‎$ ‎IID‎ components, ‎$Y_{1},\ldots,Y_{n}$​, each with a common CDF $G$. The SIT with lifetime ‎$‎‎T(n)$‎ is denoted by $IT(n,t)$, and the vector ‎$\boldsymbol{ q}(t)=(q_1(t)‎, q_2(t),\ldots,q_n(t))$ represents the set of coefficients defined in Equation \eqref{3}. Furthermore, we assume that the random variable $Y$ corresponds to one of the components' lifetimes, $Y_{1},\ldots,Y_{n}$​. ‎ In the following result, by using Equation ‎\eqref{1110}‎, we focus on investigating the behavior of the CRJ measure as a function of $t$, and aim to derive the conditions under which the uncertainty of the SIT decreases over time. ‎\begin{result}‎‎ ‎Let ${\cal{M}}_{IT(n,t),Y,G‎, ‎\boldsymbol{q}(t)}$​ represent the model function associated with the SIT. If the expression $\frac{{{G}}^2_{V}(v)}{g(G^{-1}(v))}$ is decreasing for $v\geq 0$, then it follows that the quantity $ \xi J(IT(n,t))$ is increasing with respect to $t\geq 0$. ‎\end{result}‎ ‎{\bf{Proof.}}‎ ‎%Using Equation \eqref{10}‎, ‎the proof is similar to the proof of Theorem 2.5 of Qiu and Jia (2018a)‎. ‎Assume that $0 \leq t_1\leq t_2$‎. ‎For $G(t_1)\leq v\leq G(t_2)$ and $G(t_2)\leq v\leq 1$‎, ‎we have $\frac{g_{{t_1}}(v)}{g_{{t_2}}(v)}=\frac{{G}(t_2)}{{G}(t_1)}$‎, ‎$i=1,\ldots,n$ and $\frac{g_{{t_1}}(v)}{g_{{t_2}}(v)}=0$‎, ‎$i=1,\ldots,n$‎, ‎respectively‎. Thus, we can infer that $U_{{t_1}}\leqslant_{lr}U_{{t_2}}$‎. Consequently, we obtain ‎$U_{{t_1}}\leqslant_{st}U_{{t_2}}$‎. ‎Since $\frac{{{G}}^2_{V}(v)}{g(G^{-1}(v))}$ is deccreasing‎, ‎we have $E\left[\frac{{{G}}^2_{V}(U_{t_1})}{g(G^{-1}(U_{t_1}))}\right]\leq E\left[\frac{{{G}}^2_{V}(U_{t_2})}{g(G^{-1}(U_{t_2}))}\right]$‎. ‎Since $0<\frac{1}{{G}(t_2)}\leq\frac{1}{{G}(t_1)}$ and $0<\frac{1}{{G}_{T(n)}(t_2)}\leq\frac{1}{{G}_{T(n)}(t_1)}$‎, we arrive at the desired conclusion by observing that‎ ‎\begin{eqnarray*}‎ ‎\xi J(IT(n,t_1))&=&-\frac{1}{2{G}(t_1){G}^2_{T(n)}(t_1)}E\Big[\frac{{{G}}^2_{V}(U_{t_1})}{g(G^{-1}(U_{t_1}))}\Big]\\&‎\leq‎ &‎ -‎\frac{1}{2{G}(t_2){G}^2_{T(n)}(t_2)}E\Big[\frac{{{G}}^2_{V}(U_{t_2})}{g(G^{-1}(U_{t_2}))}\Big]= \xi J(IT(n,t_2))‎. ‎\end{eqnarray*}‎ ‎$\hfill \Box$‎ ‎ The condition in Result 1, stating that $\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$​ is decreasing in $v$, is sufficient but not necessary. This is demonstrated through the following example. ‎\begin{example}\label{examm22}‎‎ ‎Given the assumptions in Example \ref{examplereviewer1} and the corresponding parameter values, we present the plot of $H(v)=\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$​ versus $v$ in Figure \ref{ex2}. From the plot, it is evident that $H(v)$ is an increasing function for $v\in (0,1)$. However, as shown in Figure \ref{figexa1}, $ \xi J(IT(n,t))$ demonstrates an increasing trend with respect to $t$. This observation leads us to conclude that the condition in Result 1, which posits that $\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$​ is decreasing with respect to $v$, is sufficient but not essential.‎ ‎\end{example}‎ ‎\begin{figure}[H]‎ ‎\centering‎ ‎\vspace*{-0.6cm}‎ ‎\includegraphics[scale=0.3]{ex22.eps}‎ ‎\vspace*{-0.5cm}\caption{ \footnotesize{ Plot of $H(v)=\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$ in Example \ref{examm22}‎. ‎}}\label{ex2}‎ ‎\end{figure}‎ Below, we present an example demonstrating that the condition ``$\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$‎ ‎is decreasing in $v$‎" in Result 1 can be fulfilled.‎ ‎\begin{example}\label{exm33}‎‎ ‎Given the assumptions in Example \ref{examplereviewer1} with parameters $\alpha=0.01$‎, ‎$\lambda=0.1$‎, ‎and $n=10$, we present the plot of $H(v)=\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$​ against $v$ in Figure \ref{exex22}. The plot shows that $H(v)$ increases within the interval $v\in (0,1)$. On the other hand, as shown in Figure \ref{crjexex22}, ξ$ \xi J(IT(n,t))$ exhibits a decreasing trend as $t$ increases. Thus, we conclude that the condition in Result 1 holds true. ‎\end{example}‎ ‎\begin{figure}[H]‎ ‎\centering‎ ‎\vspace*{-0.6cm}‎ ‎\includegraphics[scale=0.3]{Gg.eps}‎ ‎\vspace*{-0.5cm}\caption{ \footnotesize{ Plot of $H(v)=\frac{{G}^2_{V}(v)}{g(G^{-1}(v))}$ in Example \ref{exm33}‎. ‎}}\label{exex22}‎ ‎\end{figure}‎ ‎\begin{figure}[H]‎ ‎\centering‎ ‎\vspace*{-0.6cm}‎ ‎\includegraphics[scale=0.3]{alfa001n10lan01-CRJ.eps}‎ ‎\vspace*{-0.5cm}\caption{ \footnotesize{ Plot of $ \xi J(T(n,t))$ in Example \ref{exm33}‎. ‎}}\label{crjexex22}‎ ‎\end{figure}‎ The dispersive order is a fundamental and widely explored stochastic order, primarily used to assess the "variability" or "dispersion" of random variables. This order is crucial for comparing the spread or diversity within probability distributions (Jeon et al., 2006; Kochar, 2012; Shaked and Shanthikumar, 2007). It is intricately linked to the notion of log-concavity, and numerous studies have delved into the log-concavity characteristics of ordered random variables. For a thorough review of these studies, one may refer to Kochar and Korwar (1996), Dykstra et al. (1997), and Khaledi and Kochar (2000). In the subsequent result, we illustrate that by increasing the variability of the baseline CDF of components in a system, we induce an inequality in the CRJ of the SIT. ‎\begin{result}\label{resultt2}‎‎ ‎Let ${\cal{M}}_{IT^{Y}(n,t),Y,G,\boldsymbol{ q^Y}(t)}$ and ${\cal{M}}_{IT^{X}(n,t),X,F‎, ‎{\boldsymbol{q}}^{X}(t)}$ represent two distinct model functions describing two SysIT, both with identical signature vectors $\pmb s=(s_1,...,s_n)$. Let $S_X$ and $S_Y$ denote the domains of the random variables $X$ and $Y$, respectively. We define $l_X = \inf\{x‎ : ‎x \in S_X\}$ and‎ ‎$u_X = \sup\{x‎ : ‎x \in S_X\}$ for the support of $X$, and similarly for $Y$, where $l_X$ and $u_X$ represent the lower and upper bounds of the support of $X$, respectively. If $X\leqslant_{disp}Y$ and $u_X=u_Y<+\infty$, then it holds that $ \xi J(IT^{X}(n,t))\geq \xi J(IT^{Y}(n,t))$.‎ ‎\end{result}‎ ‎{\bf{Proof.}}‎‎ Given that $X\leqslant_{disp}Y$ according to Theorem 3.B.13 (b) in Shaked and Shanthikumar (2007), it implies that $X‎\geqslant‎_{st}Y$. As a result, we can infer that‎ ‎\begin{equation*}\label{7}‎ ‎\int_{0}^{G(t)}\frac{{{G}}^2_{V}(v)}{g(G^{-1}(v))}dv\geq\int_{0}^{F(t)}\frac{{{G}}^2_{V}(v)}{f(F^{-1}(v))}dv‎. ‎\end{equation*}‎ As a consequence, we arrive at the conclusion that‎ ‎\begin{equation*}\label{7777}‎ ‎\frac{1}{{G}^2_{T^{Y}(n)}(t)}\int_{0}^{G(t)}\frac{{{G}}^2_{V}(v)}{g(G^{-1}(v))}dv\geq‎ ‎\frac{{G}^2_{T^{X}(n)}(t)}{{G}^2_{T^{Y}(n)}(t)}.\frac{1}{{G}^2_{T^{X}(n)}(t)}\int_{0}^{F(t)}\frac{{{G}}^2_{V}(v)}{f(F^{-1}(v))}dv‎. ‎\end{equation*}‎ ‎%{\bar{G}^2_{T^{\prime}(n)}(t)}{\bar{G}^2_{T(n)}(t)}J(IT^{\prime}(n,t))\leq J(IT(n,t))‎ ‎%\end{equation}‎‎ ‎Using Equation \eqref{110}, the proof is concluded. It is important to highlight that, according to Theorem 3.B.26 in Shaked and Shanthikumar (2007), if $X\leqslant_{disp}Y$, then $X_{i:n}\leqslant_{disp}Y_{i:n}$​. From Theorem 3.B.13 (a) of Shaked and Shanthikumar (2007), we then deduce that $X_{i:n}‎\geqslant‎_{st}Y_{i:n}$​. Consequently, it follows that $\frac{{G}^2_{T^{X}(n)}(t)}{{G}^2_{T^{Y}(n)}(t)}‎\geq‎ 1$. ‎%From \eqref{7777}‎, ‎we get‎ ‎%\begin{eqnarray}\label{7777}‎ ‎%-\frac{1}{2\bar{G}^2_{T(n)}(t)}\int_{G(t)}^{1}\frac{G^2_{V}(v)}{g(G^{-1}(v))}dv&\leq‎ ‎%&-\frac{\bar{G}^2_{T^{\prime}(n)}(t)}{\bar{G}^2_{T(n)}(t)}.\frac{1}{2\bar{G}^2_{T^{\prime}(n)}(t)}\int_{F(t)}^{1}\frac{G^2_{V}(v)}{f(F^{-1}(v))}dv\\&\leq &‎ -‎\frac{1}{2\bar{G}^2_{T^{\prime}(n)}(t)}\int_{F(t)}^{1}\frac{G^2_{V}(v)}{f(F^{-1}(v))}dv‎ ‎%\end{eqnarray*}‎ ‎$\hfill \Box$‎ ‎%\begin{result}\label{R222}‎ ‎%Assume that‎ %${\cal{M}}_{T(n,t),Y,G, ‎\boldsymbol{p}(t)}$ and ${\cal{M}}_{T^{\prime}(n,t),X,F‎, ‎{\boldsymbol{ p}}^{\prime}(t) }$ be two model functions of two mixed used systems‎. ‎%Let $m_X(t)=\inf_{F(t)\leq v\leq1}\frac{{{\bar{G}}}^2_{V}(v)}{(1-v)^2}$‎, ‎$M_Y(t)=\sup_{G(t)\leq v\leq1}\frac{{{\bar{G}}}^2_{V}(v)}{(1-v)^2}$‎, ‎$\frac{m_{X}(t)\bar{G}^2_{T(n)}(t)}{\bar{G}^2_{T^{\prime}(n)}(t)}\geq M_{Y}(t)$‎, ‎and $Y\leqslant_{st}X$‎. ‎If $\xi J(X(t))\leq\xi J(Y(t))$‎, ‎then $\xi J(T(n,t))\geq \xi J(T^{\prime}(n,t))$‎, ‎where‎ % ‎$X(t)=X-t|X>t$ and $Y(t)=Y-t|Y>t$‎. ‎%\end{result}‎ ‎%{\bf{Proof.}}‎ ‎%Since $\xi J(X(t))\leq \xi J(Y(t))$‎, ‎we have‎ ‎%\begin{equation}\label{12}‎ %\int_{F(t)}^{1}\frac{(1-v)^2}{f(F^{-1}(v))}dv\geq\frac{\bar{F}^2(t)}{\bar{G}^2(t)}\int_{G(t)}^{1}\frac{(1-v)^2}{g(G^{-1}(v))}dv. ‎%\end{equation}‎ ‎%It follows from \eqref{4} that‎ ‎%\begin{eqnarray*}‎ ‎%&&\xi J(T(n,t))-\xi J(T^{\prime}(n,t))\nonumber\\‎ ‎%&= &‎ ‎%-\frac{1}{2\bar{G}^2_{T(n)}(t)}\int_{G(t)}^{1}\frac{{\bar{G}}^2_{V}(v)}{(1-v)^2}\frac{(1-v)^2}{g(G^{-1}(v))}dv‎ ‎%+\frac{1}{2\bar{G}^2_{T^{\prime}(n)}(t)}\int_{F(t)}^{1}\frac{{{\bar{G}}}^2_{V}(v)}{(1-v)^2}\frac{(1-v)^2}{f(F^{-1}(v))}dv‎ ‎%\nonumber\\‎ ‎%&\geq &‎ ‎%-\frac{M_{Y}(t)}{2\bar{G}^2_{T(n)}(t)}\int_{G(t)}^{1}\frac{(1-v)^2}{g(G^{-1}(v))}dv‎ +‎\frac{m_{X}(t)}{2\bar{G}^2_{T^{\prime}(n)}(t)}‎ ‎%\int_{F(t)}^{1}\frac{(1-v)^2}{f(F^{-1}(v))}dv‎ ‎%\nonumber\\&\geq & \Big(-M_{Y}(t)+\frac{m_{X}(t)\bar{F}^2(t)\bar{G}^2_{T^{\prime}(n)}(t)}{\bar{G}^2(t)\bar{G}^2_{T(n)}(t)}\Big)\frac{1}{\bar{G}^2_{T(n)}(t)}\int_{G(t)}^{1}\frac{(1-v)^2}{g(G^{-1}(v))}dv‎ ‎%\nonumber\\&\geq &‎ %\Big(\frac{m_{X}(t)\bar{G}^2_{T(n)}(t)}{\bar{G}^2_{T^{\prime}(n)}(t)}-M_{Y}(t)\Big)\frac{1}{\bar{G}^2_{T(n)}(t)}\int_{G(t)}^{1}\frac{(1-v)^2}{g(G^{-1}(v))}dv. ‎%\end{eqnarray*}‎ ‎%Based on the assumption that $\frac{m_{X}(t)\bar{G}^2_{T(n)}(t)}{\bar{G}^2_{T^{\prime}(n)}(t)}\geq M_{Y}(t)$ and $Y\leqslant_{st}X$‎, ‎the proof is now complete‎. ‎%$\hfill \Box$‎ ‎In the subsequent theorems, we analyze and compare the CRJ of two SysIT, utilizing the coefficient vector $\pmb q (t)=(q_1(t),\cdots,q_n(t))$ and the properties of the order statistics of their respective components. ‎\begin{theorem}\label{R11}‎‎ ‎Consider ${\cal{M}}_{IT^{Y}(n,t),Y,F‎, ‎{\boldsymbol{ q}}^{Y}(t)}$ and ${\cal{M}}_{IT^{X}(n,t),X,F‎, ‎{\boldsymbol{ q}}^X(t) }$ as the model functions for the two SysIT of two distinct systems. ‎\begin{itemize}‎ ‎\item[(i)] If ‎${\boldsymbol{ q}}^Y(t)\leqslant_{st}{\boldsymbol{q}}^{X}(t) $ then $\xi J(IT^{Y}(n,t))‎\leq‎ \xi J(IT^{X}(n,t))$‎. ‎\item[(ii)] If ‎${\boldsymbol{ q}}^{Y}(t)\leqslant_{hr}{\boldsymbol{q}}^{X}(t) $ then $\xi J(IT^{Y}(n,t))‎\leq‎ \xi J(IT^{X}(n,t))$‎. ‎\item[(iii)]‎ ‎If ‎${\boldsymbol{ q}}^{Y}(t)\leqslant_{lr}{\boldsymbol{q}}^{X}(t) $ then $\xi J(IT^{Y}(n,t))‎\leq‎ \xi J(IT^{X}(n,t))$‎. ‎\end{itemize}‎ ‎\end{theorem}‎ ‎{\bf{Proof.}}‎ The results can be directly derived from Theorems 2, 3, and 4 in Zhang and Jiaotong (2010). ‎$\hfill \Box$‎ ‎%\begin{theorem}\label{R11}‎ %‎Let ${\cal{M}}_{T(n,t),Y,G‎, ‎\boldsymbol{ p}(t)}$ be a model functions of the mixed used system‎. %‎\begin{itemize}‎ %‎\item[(i)] If $Y_{i:n}\leqslant_{st}(Y_{i:n}-t|Y_{i:n}>t)$‎, ‎then $\xi J(T(n))\geq \xi J(T(n,t))$‎. % ‎\item[(ii)] If $(Y_{i:n}-t_1|Y_{i:n}>t_1)\leqslant_{st}(Y_{i:n}-t_2)|Y_{i:n}>t_2)$‎, ‎then $\xi J(T(n,t_1))\geq \xi J(T(n,t_2))$‎. %‎\end{itemize}‎ %\end{theorem}‎ %‎{\bf{Proof.}}‎ %‎The results follow immediately from Theorem 2.4‎. ‎(a) and (b) of Navarro et al‎. ‎(2008)‎. %‎$\hfill \Box$‎ ‎\begin{result}\label{resultt333}‎ ‎Let ${\cal{M}}_{IT(n,t),Y,G‎, ‎\boldsymbol{q}(t)}$ be the model function of the SIT, ‎‎then $\xi J(IY_{1:n,t})\geq \xi J(IT(n,t))$‎. ‎\end{result}‎ ‎{\bf{Proof.}}‎ %‎Bagai and Kochar (1986 ) stated that $X\leqslant_{disp}Y$ if $X\leqslant_{hr}Y$ and either $X$ or $Y$ is DFR‎. %‎From this fact and Result \ref{resultt2}‎, ‎it is easy to prove Result \ref{resultt333}‎. It is straightforward to observe that $IY_{1:n,t}‎\geqslant‎_{hr}IT(n,t) $ by Theorem 3 of Zhang and Jiaotong ‎(2010)‎.‎ % ‎Since $Y_{1:n,t}$ is‎ %‎DFR provided that $Y$ is DFR‎,‎‎ As a result, we obtain ‎$\xi J(IY_{1:n,t})\geq \xi J(IT(n,t))$‎. ‎$\hfill \Box$‎ ‎%\begin{result}‎ % ‎Let‎ % ‎${\cal{M}}_{T(n,t),Y,G‎, ‎{\bf s}‎, ‎\bar{G}_{T(n)}}=\{T(n,t),Y,G(.)‎, ‎{\bf s}‎, ‎\bar{G}_{T(n)}(.) \} $ be a model function of a mixed used system‎. ‎%Then $J(T(n,t))=J^D(T(n,t))$ for all ${\bf s}$‎, ‎$n$ and $t$ if and only if distribution $F$ is symmetric‎. ‎%\end{result}‎ Determining the precise information of a system can be difficult because of the large number of components and its intricate structure. Consequently, it becomes crucial to establish bounds for the CRJ of the SIT, allowing for an approximation of its behavior in complex scenarios. The following theorems provide such bounds on the CRJ of the SIT. ‎\begin{theorem}\label{BB11}‎ ‎Let $\xi J(IT(n,t))$ be the CRJ of the SIT with the model function‎ ‎${\cal{M}}_{IT(n,t),Y,G‎, ‎\boldsymbol{ q}(t)}$‎. ‎Then‎ ‎$$\xi J(IY_t)\frac{{G}^2(t)}{{G}^2_{T(n)}(t)}M_Y(t)< \xi J(IT(n,t))<\xi J(IY_t)\frac{{G}^2(t)}{{G}^2_{T(n)}(t)}m_Y(t),$$‎ ‎where $m_Y(t)= \inf_{00$ represents a constant rate parameter.‎ % Using Equation (2) from Zhang and Jiaotong (2010), the expression for the inactivity time of a system of order $n$ with i.i.d. exponential components is derived as follows %‎\begin{equation}\label{20}‎ %‎\bar{G}_{IT(n,t)}(t)= \sum\limits_{i=1}^{n}q^*_{i}(t)\bar{G}_{i:n,t}(x)‎, %‎\end{equation}‎ %‎where $q^*_1(t)‎, q^*_2(t),\ldots, q^*_n(t)$ are coefficients such that $\sum\limits_{i=1}^{n}p^*_{i}(t)=1$ and $p^*_{i}(t)=\frac{a_i\bar{G}_{1:i}(t)}{\bar{G}_{T(n)}(t)}$ for $i=1,\cdots,n$‎, ‎where $\pmb a=(a_1,\cdots,a_n)$ is domination vector; see Navarro et al‎. ‎(2007)‎. %‎This result follows directly from the lack of memory property of exponential distribution‎.‎ By utilizing Equation \eqref{4}, the CRJ divergence for the inactivity time of a system of order $n$ is expressed as ‎\begin{equation}\label{2000}‎ ‎\xi J(IT(n,t))=-\frac{1}{2}\int_{0}^{1}\frac{\Big(\sum\limits_{i=1}^{n}q^*_{i}(t)\bar{G}_{V_i}(v) \Big)^2}{\beta(1-v)}dv‎. ‎\end{equation}‎ ‎%Due to the lack of memory property of the exponential distribution‎, ‎this result is obvious‎.‎ Let the coefficient vector in Equation \eqref{2000} be denoted by $\boldsymbol{ q}^*(t)=(q^*_1(t)‎, q^*_2(t),\ldots,q^*_n(t))$‎.‎ We calculate the coefficient vectors ‎$\boldsymbol{ q}^*(t)‎‎$‎ for the inactivity time of a system of order 3 with 1-3 IID exponential components. The results are presented in Table \ref{tab1}. Specifically, this table is a reformulation of Table 1 in Zhang and Jiaotong (2010), which outlines the CRJ divergence for the inactivity time of a system with 1-3 exponential IID components. ‎\begin{table}[H]‎‎‎‎‎ ‎\small{‎ ‎\caption{ The vectors of coefficients $\boldsymbol{q}(t)$ in \eqref{3.1} for coherent systems with 1-3 IID ‎exponential components.}‎\label{tab2} ‎\begin{tabular}{l| l| l| l} \hline‎ ‎System & $T(3)=\phi(Y_1‎, ‎Y_2,Y_3)$ & $\boldsymbol{ q}^‏*(t)$ & ‎CRJ‎ \\ \hline‎ ‎\rule{0pt}{5ex}‎ ‎1 & $T_1(3)=Y_{1}$ & $\left(\frac{1+e^{-\lambda t}+e^{-2\lambda t}}{3},\frac{1+e^{-\lambda t}-2e^{-2\lambda t}}{3},\frac{1-2e^{-\lambda t}+e^{-2\lambda t}}{3}\right)$ & $J(IT_1(3,t))$\\‎ ‎\rule{0pt}{5ex}‎ ‎2& $T_2(3)=\min\lbrace Y_{1},Y_{2}\rbrace$ & $\left(\frac{2(1+e^{-\lambda t}+e^{-2\lambda t})}{3(1+e^{\lambda t})},\frac{1+e^{-\lambda t}-2e^{-2\lambda t}}{3(1+e^{\lambda t})},0\right)$ & $J(IT_2(3,t))$ \\‎ ‎\rule{0pt}{5ex}‎ ‎3& $T_3(3)=\max\lbrace Y_{1},Y_{2}\rbrace$&$\left(0,\frac{1+2e^{-\lambda t}}{3},\frac{2(1-e^{-\lambda t})}{3}\right)$ & $J(IT_3(3,t))$\\‎ ‎\rule{0pt}{5ex}‎ ‎4& $T_4(3)=\min\lbrace Y_{1},Y_{2},Y_{3‎ ‎}\rbrace$ & $(1,0,0)$ & $J(IT_4(3,t))$\\‎ ‎\rule{0pt}{5ex}‎ ‎5& $T_5(3)=\min\lbrace Y_{1},\max\lbrace Y_{2},Y_{3}\rbrace\rbrace$&$\left(\frac{1+e^{-\lambda t}+e^{-2\lambda t}}{3(1+e^{-\lambda t}-e^{-2\lambda t})},\frac{2(1+e^{-\lambda t}-2e^{-2\lambda t})}{1+e^{-\lambda t}-e^{-2\lambda t}},0\right)$ & $J(IT_5(3,t))$\\‎ ‎\rule{0pt}{5ex}‎ ‎6&$T_6(3)=Y_{2:3}$(2-out-of-3)&$\left(0,1,0\right)$ & $J(IT_6(3,t))$\\‎ ‎\rule{0pt}{5ex}‎ ‎7&$T_7(3)=\max\lbrace Y_{1},\min\lbrace Y_{2},Y_{3}\rbrace\rbrace$&$\left(0,\frac{2(1+2e^{-\lambda t})}{3(1+e^{-\lambda t})},\frac{1-e^{-\lambda t}}{3(1+e^{-\lambda t})}\right)$ & $J(IT_7(3,t))$\\‎ ‎\rule{0pt}{5ex}‎ ‎8&$T_8(3)=\max\lbrace Y_{1},Y_{2},Y_{3}\rbrace$&$\left(0,0,1\right)$ & $J(IT_8(3,t))$\\ \hline‎ ‎\end{tabular}‎ }‎ ‎\end{table}‎ ‎\\‎ ‎%Let we use the transformation $U_i=G(Y_i)$ for the components lifetime of the aforementioned system for‎ %$i=1,\ldots,n$. ‎%The expression $‎ ‎%\frac{v^{i-1}(1-v)^{n-i} }{B(i,n-i+1)}$ that appeared in equation \eqref{4} is the pdf of $i$th order %statistic $W_{i:n}=G(Y_{i;n})$ of the components with lifetime $U_i$‎, ‎$i=1,\ldots,n$ which are uniformly %distributed on $(G(t),1)$ with pdf $g_{u}(y)=\frac{1}{\bar{G(t)}}$‎, ‎$G(t)0.048$‎, ‎we have‎ ‎{\scriptsize‎ ‎\begin{equation}\label{eq3}‎ ‎\xi J(T_4(3,t))\geq \xi J(T_2(3,t))\geq \xi J(T_5(3,t))\geq \xi J(T_6(3,t))\geq \xi J(T_7(3,t))\geq \xi J(T_1(3,t))\geq \xi J(T_3(3,t))\geq \xi J(T_8(3,t))‎. ‎\end{equation}‎ } ‎Also‎, ‎inequalities \eqref{eq1}‎, ‎\eqref{eq2}‎, ‎and \eqref{eq3} confirm Theorem 1‎. ‎For example $\boldsymbol{ p}^*_2(t)\leqslant_{st}\boldsymbol{ p}^*_5(t)$ and $\boldsymbol{ p}^*_3(t)\leqslant_{st}\boldsymbol{ p}^*_8(t)$‎, ‎therefore from inequality \eqref{eq1}‎, ‎we have‎ ‎$ \xi J(T_2(3,t))\geq \xi J(T_5(3,t))$ and $\xi J(T_3(3,t))\geq \xi J(T_8(3,t))$‎, ‎respectively‎. ‎Note that $\boldsymbol{ p}^*_2(t)$‎, ‎$\boldsymbol{ p}^*_5(t)$‎, ‎$\boldsymbol{ p}^*_3(t)$‎, ‎and $\boldsymbol{ p}^*_8(t)$ are vectors of coefficients for systems 2‎, ‎5‎, ‎3‎, ‎and 8 in Table \ref{tab1}‎, ‎respectively‎. ‎\begin{figure}[H]‎ ‎\centering‎ ‎\includegraphics[scale=0.95]{correct.eps}‎ ‎\vspace*{-0.5cm}\caption{ \footnotesize{The CRJ of the inactivity time of systems of order 3 with IID exponential components in Table \ref{tab1}‎. }} ‎\label{fig2222}‎ ‎\end{figure}‎ ‎The JCRJ divergence for the group of systems listed in Table 1 is depicted in Figure 3‎. ‎This group of systems comprises three IID components‎, ‎each with an exponential lifetime having a mean of 0.1‎. ‎In Figure 3‎, ‎it can be observed that system 4 is a 3-out-of-3 system with a JCRJ divergence of zero‎. ‎System 4 is the least complex system among the systems listed in Table 1‎. ‎Furthermore‎, ‎it is evident from Figure 3 that as the complexity of the systems increases‎, ‎the JCRJ divergence also increases‎. ‎The JCRJ divergence of systems 5 and 6 are the same for $t>0.048$‎. ‎As $t$ approaches infinity‎, ‎the JCRJ divergence of systems 5 and 6 tends to the JCRJ divergence of system 2‎. ‎Additionally‎, ‎as $t$ approaches infinity‎, ‎the JCRJ divergence of systems 7‎, ‎3‎, ‎and 8 tends to the JCRJ divergence of system 1‎. ‎One advantage of the JCRJ divergence measure is that we can compare the complexity of systems consisting of different numbers of IID components‎. ‎\begin{figure}[H]‎ ‎\centering‎ ‎\includegraphics[scale=0.95]{JCRJ.eps}‎ ‎\vspace*{-0.5cm}\caption{ \footnotesize{JCRJ of exponential mixed used systems of order 3 in Table \ref{tab1}‎. }} ‎\label{fig111}‎ ‎\end{figure}‎ ‎%\section{Nonparametric Estimators‎ } ‎%In this section‎, ‎we introduce a nonparametric approach to estimating the CRJ of the mixed used system defined in \eqref{4}‎. ‎Let us assume we have a sequence of $m$ i.i.d‎. ‎random variables $Y_r$ for $r=1,...,m$‎ ‎%with pdf $g(y)$‎, ‎cdf $G(y)$‎, ‎and survival function $\bar{G}(y)$‎. ‎Silverman (2018) proposed a kernel density estimator for the pdf $g(y)$ as follows‎ ‎%\begin{equation}‎ %g_m(y)=\frac{1}{mh_m}\sum\limits_{r=1}^{m}K\left(\frac{y-Y_r}{h_m}\right),~~x\in\mathbb{R}, ‎%\end{equation}‎ ‎%where $h_m$ is the bandwidth or smoothing parameter and $K(.)$ is the kernel function‎. ‎The bandwidth sequence‎ ‎%$h_m$ is chosen such that $mh_m\rightarrow \infty$ as $m\rightarrow \infty$‎. ‎This ensures that the kernel density estimator $g_m(y)$ converges to the true PDF $g(y)$ as the sample size increases‎. ‎The kernel function $K(.)$ is a‎ ‎%symmetric PDF with finite variance‎. ‎Some commonly used kernel functions include the normal (Gaussian)‎, ‎Epanechnikov‎, ‎and tricube kernels‎. ‎Hereafter‎, ‎we will use the normal kernel function‎. ‎Once we have the kernel density estimator $g_m(y)$‎, ‎we can define nonparametric kernel-based estimators of CRJ of the mixed used system as follows‎ ‎%\begin{eqnarray}\label{4}‎ ‎%\xi J_m(T(n,t))=‎ %-\frac{1}{2}\int_{G_m(t)}^{1}\frac{{\bar{G}}^2_{V_t,m}(v)}{g_m(G_m^{-1}(v))}dv, ‎%\end{eqnarray}‎ ‎%where‎ ‎%\begin{equation}\label{5}‎ %{\bar{G}}_{V_t,m}(v)=\sum\limits_{i=1}^{n}p_{i,m}(t){\bar{G}}_{V_{t,i,m}}(v), ‎%\end{equation}‎ ‎%and‎ ‎%\begin{equation}\label{9}‎ ‎%{\bar{G}}_{V_{t,i,m}}(v)=\sum\limits_{j=0}^{i-1}{n \choose j}\frac{v^{i}(1-v)^{n-i} }{\bar{G}_{i:n,m}(t)},~‎~ ‎G_m(t)0$ and vectors ${\bf a}$ and ${\bf c}$ can be chosen arbitrary within‎ ‎the context of two natural constraints‎: ‎$0t)}{P(T(n)> t)} R(T(n,t):Y_{i:n,t})$‎. ‎By substituting \eqref{JCRJJJ} into the criterion function (20)‎, ‎we can achieve our goal‎. ‎In Equation \eqref{111}‎, ‎we assume that $a_{i}=\frac{P(Y_{i:n}>t)}{P(T(n)> t)} R(T(n,t):Y_{i:n,t})$ and $c_{i}=C_{I}+n(A-B)+iB$ for $i=1,\ldots,n$‎, ‎where $C_{I}$ is an initial fixed cost of manufacturing the systems of interest‎, ‎$A$ is cost of‎ ‎an individual component‎, ‎$B$ is salvage value of a used but working component‎ ‎removed after system failure‎, ‎and $t$ is a fixed time‎. ‎For additional information about the values of $c_i$‎, ‎refer to Page 95 of chapter 7 in Samaniego (2007)‎. ‎Let the components of mixed used system have exponential distribution with SF $\bar{G}(y)=e^{-\beta y}$‎, ‎where $\beta$ is positive constant‎. ‎Let $C_{I}=0$ and $B=1$‎. ‎Consider the class of all mixed used systems based on $n$ components with IID lifetimes‎. ‎The goal is to identify a system that simultaneously minimizes the JCRJ (maximizes $(-1)\times$JCRJ) divergence as well decreases the costs‎. ‎Now the question come up is‎, ‎``What is the optimal signature vector subject to the criterion function \eqref{111}?"‎. ‎This problem can be formulated as‎ ‎$$‎ ‎\text{maximized}:‎~ ‎m_{d}({\bf s},{\bf a},{\bf c})=\frac{-\frac{1}{2}\sum\limits_{i=1}^{n}p^*_i(t)\int_{0}^{1}‎ ‎\Big(\sum\limits_{i=1}^{n}\sum\limits_{j=0}^{i-1}p^*_i(t){n \choose j}u^{i}(1-u)^{n-i}‎ -‎\sum\limits_{j=0}^{i-1}{n \choose j}u^{i}(1-u)^{n-i}\Big)^2~dx}{(\sum\limits_{i=1}^{n}(i+n(A-1))s_{i})^{d}}‎, ‎$$‎ ‎$$‎ ‎\text{subject‎~ ‎to}‎: ~‎\left\{‎ ‎\begin{array}{*{20}{l ccc}}‎ ‎\sum\limits_{i=1}^{n} s_{i}=1‎, ‎\\‎ ‎0\leq s_{i}\leq 1.\\‎ ‎\end{array}‎ ‎\right‎. ‎$$‎ ‎Numerical computations are needed to find the solutions‎. ‎We have done the computations to obtain the optimal signature of mixed used system with 5 independent exponentially distributed components for some selected values of $A$‎, ‎$d$‎, ‎and $\beta$‎. ‎The numerical results are presented in Table 3 which shows that‎, ‎there is no continuity in nature of the optimal signature‎. ‎From Table 2‎, ‎we observe that the optimal signature system is a series system for $d\leq 0.13$ and with increasing the mean lifetime of components‎, ‎the optimal signature system is a series system‎ ‎for $d\leq 0.5$‎. ‎Also‎, ‎we observe that for $d\geq 12$‎, ‎the optimal system is a mixture system with uniform signature ${\bf s}=(\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5},\frac{1}{5})$‎. ‎\begin{table}[H]‎ ‎\centering‎ ‎\tiny{‎ ‎\caption{Optimal signatures of mixed used system of order 5 with independent exponentially distributed components at time $t=0.1$.}‎ ‎\begin{tabular}{l l | c c c c c c c c c c c c} \hline‎ ‎& & \multicolumn{5}{c}{ $\beta=2$}&\multicolumn{5}{c}{$\beta=0.1$}\\‎ ‎\cmidrule(lr){3-7} \cmidrule(lr){8-12}‎ ‎$A$ & $d$ & $s_{1}$ & $s_{2}$& $s_{3}$& $s_{4}$& $s_{5}$& $s_{1}$ & $s_{2}$& $s_{3}$& $s_{4}$& $s_{5}$\\‎ ‎\specialrule{.1em}{.05em}{.05em}‎ ‎1.5 & $\leq 0.1$ & 1 & 0 & 0 & 0& 0 &1&0&0&0&0\\‎ ‎1.5 & 0.13 & 1 & 0 & 0 & 0& 0 & 1&0&0&0&0\\‎ ‎1.5 & 0.2 & 1 & 0 & 0 & 0& 0& 1&0&0&0&0 \\‎ ‎1.5 & 0.5 & 1 & 0 & 0 & 0& 0& 1&0&0&0&0\\‎ ‎1.5 & 0.9 & 0.9998 & 0 & 0 & 0& 0& 0.9231&0.0769&0&0&0\\‎ ‎1.5 & 1 & 0.9970 & 0.0002 & 0.0001 & 0& 0& 0.5820&0.0761&0.3355&0.0051&0.0013\\‎ ‎1.5 & 1.5 & 0.3130 & 0.2772 & 0.2270 & 0.1530 & 0.0298&0.6142&0.2602&0.0784&0.0319&0.0153\\‎ ‎1.5 & 2 & 0.3053 & 0.2770 & 0.2329 & 0.1612 & 0.0236&0.5938&0.2896&0.0708&0.0306&0.0153 \\‎ ‎1.5 & 5.5 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2&0.2&0.2&0.2&0.2&0.2 \\‎ ‎1.5 & 10 & 0.2& 0.2 & 0.2 & 0.2 & 0.2&0.2&0.2&0.2&0.2&0.2\\‎ ‎1.5 & $\geq 12$& 0.2 & 0.2 & 0.2 & 0.2 & 0.2&0.2&0.2&0.2&0.2&0.2\\‎ ‎\hline‎ ‎2 & $\leq 0.1$ & 1&0&0&0&0&1&0&0&0&0 \\‎ ‎2 & 0.13 &1&0&0&0&0&1&0&0&0&0 \\‎ ‎2 & 0.2 & 0.9999 & 0.0001 & 0 & 0& 0 &1&0&0&0&0 \\‎ ‎2 & 0.5 & 0.9998 & 0.0002 & 0 & 0& 0& 1&0&0&0&0 \\‎ ‎2 & 0.9 & 0.9997 & 0.0003 & 0 & 0& 0&1&0&0&0&0 \\‎ ‎2 & 1 & 0.9999 & 0.0001 & 0 & 0& 0&1&0&0&0&0 \\‎ ‎2 & 1.5 & 0 & 0 & 0 & 0.0001& 0.9999&09224&0.0763&0.0009&0.0002&0.0001 \\‎ ‎2 & 2 & 0 & 0 & 0.0002 & 0.0001& 0.9997 &0.5622&0.1246&0.3006&0.0100&0.0026\\‎ ‎2 & 5.5 & 0.0088 & 0.0112 & 0.0168 & 0.0399& 0.9233 &0.5936&0.2897&0.0708&0.0306&0.0153\\‎ ‎2 & 10 & 0.1970 & 0.1984 & 0.1999 & 0.2015 & 0.2032&0.2&0.2&0.2&0.2&0.2\\‎ ‎2 & $\geq 12$& 0.2 & 0.2 & 0.2 & 0.2 & 0.2& 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\‎ ‎\hline‎ ‎3 & $\leq 0.1$ & 1 & 0 & 0 & 0& 0& 1&0&0&0&0 \\‎ ‎3& 0.13 & 1 & 0 & 0 & 0& 0 & 1&0&0&0&0\\‎ ‎3& 0.2 & 1 & 0 & 0 & 0& 0& 1&0&0&0&0\\‎ ‎3& 0.5 & 0.8701 & 0.1299 & 0 & 0& 0 & 1&0&0&0&0\\‎ ‎3& 0.9 & 0.8701 & 0.1299 & 0 & 0& 0& 1&0&0&0&0 \\‎ ‎3& 1 & 0.8704 & 0.1295 & 0.0001 & 0& 0& 0.9230&0.0765&0.0002&0.0001&0.0001\\‎ ‎3& 1.5 & 0.8712 & 0.1283 & 0.0002 & 0.0002&0.0001&0.5448&0.2199&0.2026&0.0254&0.0073 \\‎ ‎3& 2 & 0.5936 & 0.2897 & 0.0708 & 0.3060& 0.1520 &0.5937&0.2896&0.0708&0.0306&0.0153\\‎ ‎3& 5.5 & 0.5772 & 0.3050 & 0.6770 & 0.3160 & 0.0185&0.5838&0.2896&0.0708&0.0306&0.0153\\‎ ‎3& 10 & 0.2 & 0.2 & 0.2 & 0.2& 0.2 &0.2 & 0.2 & 0.2 & 0.2& 0.2\\‎ ‎3 & $\geq 12$& 0.2 & 0.2 & 0.2 & 0.2 & 0.2&0.2 & 0.2 & 0.2 & 0.2 & 0.2\\‎ ‎\specialrule{.1em}{.05em}{.05em}‎ ‎\end{tabular}‎ } ‎\end{table}‎ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ‎\section{ Conclusion and future remarks}‎ ‎The purpose of this paper is to develop representations for the cumulative residual extropy (CRJ) of systems with an order of n‎, ‎under specific conditions related to the system state at time $t$‎. ‎We focused on one form of conditioning‎, ‎which is that the system is operational at time $t$ (i.e‎. ‎$T(n)>t$)‎. ‎When $T(n)>t$‎, ‎the system is considered a used system with a lifetime of $T(n)-t|T(n)>t$‎. ‎The representations of the CRJ for these conditional systems are valuable for analyzing the information properties of conditional mixed systems‎. ‎By using these representations‎, ‎we established some stochastic comparisons and boundaries for the CRJ of conditional systems‎. ‎We computed the CRJ of coherent systems with 3 IID components with exponentially distributed lifetimes‎, ‎and discovered that the series system has the highest CRJ‎, ‎while the parallel system has the lowest CRJ‎. ‎Additionally‎, ‎we introduced a divergence measure based on the CRJ of a mixed used system that evaluates the complexity of the system‎, ‎i.e.‎, ‎how much more complex the system is compared to a $k$-out-of-$n$ system with baseline CDF 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