Stochastic Ordering and Reliability Analysis of Inactivity Lifetime with a Cold Standby

SYNOPTIC ABSTRACT

The article discusses truncated inactivity lifetimes for systems with cold standby units. The most common bivariate lifetime distribution was used to clarify these systems. The essay concentrates on the Farlie-Gumbel-Morgenstern family for building a dependence structure between components and to study the properties of suggested systems. The article also compares systems with independent identically distributed components and dependent identically distributed components having the same survival function by using various stochastic orders. Moreover, some applications of the theoretical results of reliability theory are studied.

KEY WORDS AND PHRASES:

• Stochastic orders
• cold standby
• distribution copula
• inactivity lifetimes

1. Introduction

Suppose T is a positive random variable that denotes the lifetime of a unit or equipment, having density function f, absolutely continuous distribution function F, and survival function $\overline{F}$ with respect to the Lebesgue measure. In many reliability problems, it is interesting to consider variables of the following types. $$T_{\left(t\right)}=[t-T|T\le t],t\in \left\{x;x\in F_T\left(x\right)<1\right\},$$ denote the time elapsed after failure until time t, given that the unit has already failed at time t, $T_{(t)}$ having distribution function $F_{(t)}(s)=P[t-T\le s|T\le t]$, and which is known in literature as inactivity time or past lifetime. In recent times, the random variable $T_{(t)}$ has received considerable attention in the literature; see El-Bassiouny and Alwasel (2003), Lai and Xie (2006), Nair and Sudheesh (2010), Mahdy (2012, 2016), Rezaei, Gholizadeh, and Izadkhah (2015), and Kundu and Sarkar (2017).

Consider a probability density function f(t) for a lifetime random variable X with distribution function $F(t)$ and survival function $\overline{F}(t)=1-F$ $(t),$ $t\in {\mathbb{R}}^{+}.$ In addition, we assume that the mean life ${\mu }_X={\int }_0^{\infty }\overline{F}(u)du$ and variance ${\sigma }_X^2$ are finite. Likewise, let Y be the second lifetime random variable with density function $g(t)$, distribution function $G(t)$, and survival function $\overline{G}(t)=1-G(t)$; $t\in {\mathbb{R}}^{+}$. Furthermore, the mean life ${\mu }_Y={\int }_0^{\infty }\overline{G}(u)du$ and variance ${\sigma }_Y^2$ are both assumed to be finite. Let $l_X=\inf \left\{t\in {\mathbb{R}}^{+}:F(t)>0\right\},u_X=\sup \left\{t\in {\mathbb{R}}^{+}:F(t)<1\right\},{\Omega }_X=(l_X,u_X),l_Y=\inf \left\{t\in {\mathbb{R}}^{+}:G(t)>0\right\},u_Y=\sup \left\{t\in {\mathbb{R}}^{+}:G(t)<1\right\},$ and ${\Omega }_Y=(l_Y,u_Y)$. Let X have a reversed hazard rate function ${\tilde{r}}_F(t)=f(t)/F(t),$ $t>l_X$, and Y have a reversed hazard rate function ${\tilde{r}}_G(t)=g(t)/G(t),$ $t>l_Y.$ Then, the mean inactivity lifetime functions and variance past lifetime functions are defined by, $$\begin{array}{c}{m}_F\left(t\right)=\{\begin{array}{cc}\frac{1}{F\left(t\right)}{\int }_0^{t}F\left(u\right)du,& \text{if} t>l_X,\\ 0,& \text{if} t\le l_X,\end{array}\\ m_G\left(t\right)=\{\begin{array}{cc}\frac{1}{G\left(t\right)}{\int }_0^{t}G\left(v\right)dv,& \text{if} t>l_Y,\\ 0,& \text{if} t\le l_Y,\end{array}\\ {\sigma }_{F_{\left(x\right)}}^2=\{\begin{array}{cc}\frac{2{\int }_0^x{\int }_0^{y}F\left(u\right)\mathit{\text{dudy}}}{F\left(x\right)}-m_F^2\left(x\right),& \text{if} x>l_X,\\ 0,& \text{if} x\le l_X,\end{array}\end{array}$$ and $${\sigma }_{G_{\left(x\right)}}^2=\{\begin{array}{cc}\frac{2{\int }_0^x{\int }_0^{y}G\left(u\right)\mathit{\text{dudy}}}{G\left(x\right)}-m_G^2\left(x\right),& \text{if} x>l_Y,\\ 0,& \text{if} x\le l_Y,\end{array}$$ respectively.

The following definitions are essential for this study.

Definition 1.

The distribution function $F(.)$ of the random variable X is said to have the following characteristics.

1. A decreasing reversed hazard rate $(\mathit{\text{DRHR}})$, if ${\tilde{r}}_F(t)$ is a decreasing function in t, or if $F(t),$ is logarithmically concave in $t\in R^{+}.$

2. An increasing strong mean past lifetime class, if ${\int }_0^{t}xF(x)dx/F(t)$ is non-decreasing for all $t>l_X$.

The following stochastic orders are defined in Nanda, Singh, Misra, and Paul (2003), Shaked and Shanthikumar (2007), Mahdy (2009, 2012), and Kayid and Izadkhah (2014).

Definition 2.

Let X₁ and X₂ be two non-negative and absolutely continuous random variables, with distribution functions $F_1(.)$ and $F_2(.)$, density functions $f_1(.)$ and $f_2(.)$, reliability functions ${\overline{F}}_1(.)$ and ${\overline{F}}_2(.),$ reversed hazard rate functions ${\tilde{r}}_{F_1}(.)$ and ${\tilde{r}}_{F_2}(.)$, mean past lifetime functions $m_{F_1}(.)$ and $m_{F_2}(.)$, and variance past lifetime functions ${\sigma }_{F_{1(x)}}^2(.)$ and ${\sigma }_{F_{2(x)}}^2(.)$, respectively. Hence, X₁ is smaller than or equal to X₂ in the following cases as,

1. A usual stochastic order ($X_1{\le }_{ST}{X}_2),$ if ${\overline{F}}_1(x)\le {\overline{F}}_2(x)$ for all $x\ge 0$.

2. A hazard rate order ($X_1{\le }_{hr}{X}_2$) if ${\overline{F}}_1(u)/{\overline{F}}_2(u)$ is decreasing in $u\in {\mathbb{R}}^{+}$.

3. A reversed residual lifetime order ($X_1{\le }_{RHR}{X}_2$) if ${\tilde{r}}_{F_1}$ $(x_2)\le$ ${\tilde{r}}_{F_2}(x_2)$ for all $x_2>0$ or $\left\{F_1(x_1)/F_1(x_2)\right\}\ge \left\{F_2(x_1)/F_2(x_2)\right\}$ for all $x_1\le x_2$.

4. A mean past lifetime order ($X_1{\le }_{MP}{X}_2$) if $m_{F_1}(x_2)\ge m_{F_2}(x_2)$ for all $x_2>0$.

5. A variance past lifetime order ($X_1{\le }_{VP}{X}_2$) if ${\sigma }_{F_{1(x)}}^2(x_2)\ge {\sigma }_{F_{2(x)}}^2(x_2)$ for all $x_2>0$.

6. A strong mean past lifetime order ($X_1{\le }_{SMP}{X}_2$) if ${\int }_0^{x_2}x{F}_1(x)dx/F_1(x_2)\ge {\int }_0^{x_2}x{F}_2(x)dx/F_2(x_2)$ for all $x_2\in {\mathbb{R}}^{+}$.

7. A likelihood ratio order ($X_1{\le }_{LR}{X}_2)$ if $f_1(u)f_2(v)\ge f_1(v)f_2(u)$ for all $u\le v$.

Suppose $\mathbb{Y}=\psi (Y_1,\dots ,Y_n)$ is the lifetime of a coherent system, and $Y_1,\dots ,Y_n$ are dependent lifetimes with a common survival function $\overline{G}(y)=\mathrm{Pr}\left[Y_i>y\right]$. Then, the system survival function can be presented as, $${\overline{G}}_{\mathbb{Y}}\left(y\right)=\phi \left(\overline{G}\left(y\right)\right),$$ where $\phi$ depends only on ψ and on the survival copula of $(Y_1,\dots ,Y_n).$ Lehmann (1966) considered the definition of the bivariate dependence of two random variables $\mathbb{A}$ and $\mathbb{B}$ as, (1) $$\mathrm{Pr}\left(\mathbb{A}>a,\mathbb{B}>b\right)\ge \mathrm{Pr}\left(\mathbb{A}>a\right)\mathrm{Pr}\left(\mathbb{B}>b\right).$$(1)

A solution methodology is provided and demonstrated to determine optimal design configurations for nonrepairable coherent systems with three types of redundancies: cold standby, warm standby, and hot standby. The cold standby redundancy method involves having one system as a backup for another identical primary system where in the response time is of minimal concern. In the warm standby redundancy method, the secondary system runs in the background of the primary system, and it is used when time is somewhat critical. Finally, hot redundancy is used when the process must not go down for even a brief moment under any circumstance and where the primary and secondary systems run simultaneously.

A large number of studies have discussed standby systems and their extensions and applications, including Yang and Dhillon (1995), Wang, Lai, and Li (2003), Meng, Yuan, and Yin (2006), Eryilmaz (2011, 2012, 2013), Tannous, Xing, Rui, Xie, and Ng (2011), Wu and Wu (2011), Xing, Tannous, and Dugan (2012), Zhai, Peng, Xing, and Yang (2013), Levitin, Xing, and Dai (2014a, Levitin, Xing, & Dai, 2014b, Levitin, Xing, Johnson, & Dai, 2015), Liu, Cui, Wen, and Guo (2016), Kaur and Gupta (2017), and Mendes, Ribeiro, and Coit (2017).

Throughout this study, S_U is used to indicate the single unit system, $C_U$ is used to indicate the cold standby unit, A_U is used to indicate active equipment, SD_U is used to indicate standby equipment, $COD$ is used to indicate the conditional distribution function, and RL is used to indicate the residual lifetime of the system.

Let S_U be equipped with C_U, and suppose $\mathbb{A}$ denotes the lifetime of A_U and $\mathbb{B}$ denotes the $S{D}_U.$ Obviously, the lifetime of the whole system corresponds to the random variable $(RV)$ $U=\mathbb{A}+\mathbb{B}$. Let the system have survived to the age of s₁ and already failed by time s₂; then, the inactivity lifetime of a system can be defined as, $${\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}=\left\{s_2-\mathbb{A}-\mathbb{B}|s_1\le \mathbb{A}+\mathbb{B}\le s_2\right\},$$ for $s_2\in {\mathbb{R}}^{+}$ and $s_1\in {\mathbb{R}}^{+}.$ Here, ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}$ implies only the survival of the system at time s₁ and failure by time s₂, but no information is included about which unit survives at time s₁. Note, that the system has benefited from the alternative element (standby unit). Therefore, it could exceed the age of s₁, but it fails before time s₂ even after we use the standby unit.

To the author’s knowledge, the inactivity lifetime for a system with SD_U has been scarcely investigated from a theoretical point of view. Therefore, we define it by analyzing and studying the properties of the following conditional random variables. (2) $${\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)=\left\{s_2-\mathbb{A}-\mathbb{B}|\mathbb{A}\le s_1,s_1\le \mathbb{B}\le s_2\right\},$$(2) and (3) $${\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)=\left\{s_2-\mathbb{A}-\mathbb{B}|s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right\},$$(3) where ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ represents the inactivity lifetime of the system, considering that A_U has a breakdown before time s₁; on the other side, the full system works with the effect of the standby unit $(\mathbb{B})$ and fails before time $s_2.$ Similarly, ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ represents the inactivity lifetime of the system under the condition that the system works at time s₁ with the active component and fails before time s₂. Clearly, ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ and ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ are more helpful than ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}$.

For study and analysis of systems ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ and ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ with identically distributed (ID) components, we consider two cases wherein the relationship between the two components $\mathbb{A}$ and $\mathbb{B}$ is either dependent or independent. The most common bivariate lifetime distribution (bivariate generalized exponential distribution; BGE) was used to clarify these systems. Moreover, the Farlie-Gumbel-Morgenstern (FGM) family is extensively used in building a dependence structure between the marginal distributions; we use it to build the distribution functions of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ and ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ when $\mathbb{A}$ and $\mathbb{B}$ are dependent. Furthermore, we derive $\mathbb{E}\left[{\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)\right]$ and $\mathbb{E}\left[{\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)\right]$ by using the FGM Copula. We compare the systems with independent identically distributed components $(\mathbb{I}{\mathbb{D}}_s)$ and dependent identically distributed components $(\mathbb{D}{\mathbb{D}}_s)$ having the same survival function by using various stochastic orders. In addition, we provide some applications of theoretical results using reliability theory.

The remainder of the article is organized as follows. Section 2 describes the main results, and Section 3 discusses and includes some reliability applications of the new lifetime’s classes.

2. The Main Results

Suppose $\mathbb{A}\in {\mathbb{R}}^{+}$ and $\mathbb{B}\in {\mathbb{R}}^{+}$ are two dependent lifetime random variables with joint cumulative distribution function $F_{\mathbb{A},\mathbb{B}}(a,b)=\mathrm{Pr}\{\mathbb{A}<a,\mathbb{B}<b\}$ and marginal distributions $F_{\mathbb{A}}(a)=\mathrm{Pr}\{\mathbb{A}<a\}$ and $F_{\mathbb{B}}(b)=\mathrm{Pr}\{\mathbb{B}<b\}$, for $a,b>0$.

Now, we can derive the distribution function of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ by the following theorem.

Theorem 1.

The conditional distribution function of $\mathbb{B}+\mathbb{A}$ given, $$\left\{\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right\}$$ can be represented as follows.

1. For $s_1\le u<2s_1$, (4) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{u-s_2}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)\\ +{\int }_{u-\theta s_1}^{s_2}\mathrm{Pr}\left(\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)].\end{array}$$(4)

2. For $u\ge 2s_1,\theta \in (0,1)$, and $2s_1<s_2$, (5) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{u-\theta s_1}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)\\ +{\int }_{u-\theta s_1}^{s_2}\mathrm{Pr}\left(\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)].\end{array}$$(5)

Proof.

By considering the conditioning on $\mathbb{A}$ and letting $u-b<s_2$ or $u-b\ge s_2$, we have the following results. (6) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u,\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ ={\int }_{s_1\le b\le s_2}\mathrm{Pr}\left(\mathbb{A}<u-b,\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ ={\int }_{s_1\le b\le s_2,u-b>\theta s_1}\mathrm{Pr}\left(\mathbb{A}<u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ +{\int }_{s_1\le b\le s_2,u-b<\theta s_1}\mathrm{Pr}\left(\mathbb{A}<s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right).\end{array}$$(6)

For $s_1\le u<2s_1$ and $u-b>\theta s_1>s_2$, we have $u-b>s_2$, and by Equation (6)(6) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u,\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ ={\int }_{s_1\le b\le s_2}\mathrm{Pr}\left(\mathbb{A}<u-b,\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ ={\int }_{s_1\le b\le s_2,u-b>\theta s_1}\mathrm{Pr}\left(\mathbb{A}<u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ +{\int }_{s_1\le b\le s_2,u-b<\theta s_1}\mathrm{Pr}\left(\mathbb{A}<s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right).\end{array}$$(6) we have, (7) $${\int }_{s_1\le b\le s_2,u-b>\theta s_1}\mathrm{Pr}\left(\mathbb{A}<u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)={\int }_{s_1}^{u-s_2}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right),$$(7) and for $s_1\le b\le s_2$ and $u-b<\theta s_1$, we get, (8) $${\int }_{s_1\le b\le s_2,u-b<\theta s_1}\mathrm{Pr}\left(\mathbb{A}<s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)={\int }_{u-\theta s_1}^{s_2}\mathrm{Pr}\left(\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)].$$(8)

In addition, (9) $$\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)=F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1).$$(9)

It follows from Equations (7–9) that the required result (i) is provided. When, $$\mathcal{F}=\left\{s_1\le b\le s_2,u-b>\theta s_1\right\},$$

$u\ge 2s_1,\theta \in (0,1),$ and $2s_1<s_2$, we have, (10) $$\mathcal{F}=\left\{u-\theta s_1\le b\le s_1\right\}.$$(10)

Similarly, (11) $$\mathcal{M}=\left\{s_2\le b\le u-\theta s_1\right\}$$(11) is also valid for the following relation. $$\mathcal{M}=\left\{s_1\le b\le s_2,u-b<\theta s_1\right\}.$$

Substituting Equations (9–11) into Equation (6)(6) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u,\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ ={\int }_{s_1\le b\le s_2}\mathrm{Pr}\left(\mathbb{A}<u-b,\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ ={\int }_{s_1\le b\le s_2,u-b>\theta s_1}\mathrm{Pr}\left(\mathbb{A}<u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ +{\int }_{s_1\le b\le s_2,u-b<\theta s_1}\mathrm{Pr}\left(\mathbb{A}<s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right).\end{array}$$(6) , we obtain (ii).

Based on relation Equation (1)(1) $$\mathrm{Pr}\left(\mathbb{A}>a,\mathbb{B}>b\right)\ge \mathrm{Pr}\left(\mathbb{A}>a\right)\mathrm{Pr}\left(\mathbb{B}>b\right).$$(1) , we can say that there is positive dependence between $\mathbb{A}$ and $\mathbb{B}$ if, $$\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\ge \mathrm{Pr}\left(\mathbb{A}<s_1\right)\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right).$$

However, $$\mathrm{Pr}\left(\mathbb{A}<s_1\right)\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)=F_{\mathbb{A}}\left(s_1\right)\left(F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)\right),$$ and then, (12) $$\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\ge F_{\mathbb{A}}\left(s_1\right)\left(F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)\right).$$(12)

In addition, based on the alternative positive dependence introduced by Lai and Xie (2006), we can say that $\mathbb{A}$ is left-tail decreasing in $\mathbb{B}$ in $\left[s_1,s_2\right]$ if, $$\mathrm{Pr}\left(\mathbb{A}<s_1|s_1\le \mathbb{B}\le s_2\right) \text{is} \text{decreasing} \text{in} s_2.$$

Under this definition, we can rewrite relation Equation (12)(12) $$\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\ge F_{\mathbb{A}}\left(s_1\right)\left(F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)\right).$$(12) as, $$\mathrm{Pr}\left(\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\ge F_{\mathbb{A}}\left(s_1\right).$$

By using Theorem (1), we can determine the average value of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ as, (13) $$\begin{array}{c}{\varphi }_T\left(s_2\right)=\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)\right]\\ ={\int }_0^{\infty }\mathrm{Pr}\left(s_2-\mathbb{A}-\mathbb{B}>u|\mathbb{A}\le s_1,s_1\le \mathbb{B}\le s_2\right)du\\ ={\int }_0^{\infty }\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}<s_2-u|\mathbb{A}\le s_1,s_1\le \mathbb{B}\le s_2\right)du.\end{array}$$(13)

Corollary 1.

Let $\mathbb{A}$ denote the lifetime of an equipment with distribution function $F_{\mathbb{A}}(.),a\in R^1$, and survival function ${\overline{F}}_{\mathbb{A}}(.)=1-F_{\mathbb{A}}(.)$. Similarly, let $\mathbb{B}$ be the standby random lifetime with distribution function $F_{\mathbb{B}}(.)$ and survival function ${\overline{F}}_{\mathbb{B}}(.)=1-F_{\mathbb{B}}(.).$ When $\mathbb{A}$ and $\mathbb{B}$ are independent, we have the COD of $\mathbb{B}+\mathbb{A}$ given $\left\{\mathbb{A}\le s_1,s_1\le \mathbb{B}\le s_2\right\}$, which can be obtained as, $$\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)=1.$$

When $s_1\le u<2s_1,u\ge 2s_1$, and $2s_1<s_2,$ we have, (14) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{2s_1}{F}_{\mathbb{A}}\right(u-b\left)d{F}_{\mathbb{B}}\right(b)+F_{\mathbb{A}}(s_1\left)\left[F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(2s_1\right)\right]\right],\end{array}$$(14) where, $$F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)=F_{\mathbb{A}}\left(s_1\right)\left[F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)\right].$$

The cold standby redundancy method involves having one system as a backup for another identical primary system wherein the response time is of minimal concern. When $\mathbb{A}$ and $\mathbb{B}$ are independent and we consider the cold system, it is evident that the inactivity time of the system is the inactivity time of $\mathbb{B}$. Since the system will fail before time $s_2$ and there is residual lifetime after s₁, the inactivity time of the system is the inactivity time of $\mathbb{B}$ at time $\left[s_1,s_2\right]$; consequently, the inactivity time of system ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ is obtained as follows. (15) $$\begin{array}{c}{}_{\mathbb{A}}{\varphi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)\right]=E\left[s_2-\mathbb{B}|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_{s_1}^{s_2}\frac{F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(15)

The residual of system after s₁ can be represented as, (16) $$\begin{array}{c}{}_{\mathbb{A}}{\psi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[s_2-\mathbb{A}|\mathbb{A}<s_1\right]+E\left[\mathbb{B}-s_1|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_0^{s_1}\frac{F_{\mathbb{A}}\left(u\right)du}{F_{\mathbb{A}}\left(s_1\right)}+{\int }_{s_1}^{s_2}\frac{\left(s_2-s_1\right)F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(16)

The BGE is discussed in Kundu and Gupta (2009) and has been utilized in many applications, such as analysis of lifetime data. In the following application, we clarify the above results by using BGE.

If $\mathbf{U}=(U_1,U_2)$ indicates bivariate random variables and $\mathbf{U}\in BGE(\alpha ,\beta ,\theta )$, then the joint probability density function of $(U_1,U_2)$ for $u_1\in {\mathbb{R}}^{+}$ and u₂ $\in {\mathbb{R}}^{+}$ is, $$g_{{\mathbf{U}}_1,{\mathbf{U}}_2}\left(u_1,u_2\right)=\{\begin{array}{cc}{g}_1\left(u_1,u_2\right)& \mathrm{i}\mathrm{f}0<u_1<u_2<\infty ,\\ g_2\left(u_1,u_2\right)& \mathrm{i}\mathrm{f}0<u_2<u_1<\infty ,\\ g_0\left(u\right)& \mathrm{i}\mathrm{f}0<u_1=u_2<\infty ,\end{array}$$ where, $$\begin{array}{c}{g}_1\left(u_1,u_2\right)=\beta \left(\alpha +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha +\theta -1}{\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right),\\ g_2\left(u_1,u_2\right)=\alpha \left(\beta +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha -1}{\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta +\theta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right),\\ g_0\left(u\right)=\theta {\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u\right)\right)}^{\alpha +\beta +\theta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u\right).\end{array}$$

Theorem 2.

If $\mathbf{U}=(U_1,U_2)$ indicates bivariate random variables and $\mathbf{U}\in BGE(\alpha ,\beta ,\theta )$, we obtain the mean the inactivity lifetime of the system given that the active unit has failed before time s₁, which is denoted by $U_{1(s_1,s_2)}^{U_2}(1)$ as follows. $$U_{1\left(s_1,s_2\right)}^{U_2}\left(1\right)={\varphi }^{-1}\left(s_1,s_2\right)\times \eta \left(u_1,u_2\right),$$ when $$\eta \left(u_1,u_2\right)=\{\begin{array}{cc}\beta \left(\alpha +\theta \right)\left[s_2-{\varkappa }_1\left(s_1,s_2,\alpha ,\beta ,\theta \right)\right]& if0<u_1<u_2<\infty ,\\ \left(1/2\right)s_2^2(s_2-2s_1)\beta (\alpha +\theta )-\phi \left(s_1,\alpha ,\beta ,\theta \right)& if0<u_2<u_1<\infty ,\\ \frac{\theta s_2^2}{2}-s_2{s}_1\theta -{\varkappa }_2\left(s_1,\alpha ,\beta ,\theta \right)& if0<u_1=u_2<\infty ,\end{array}$$ where $$\begin{array}{c}\varphi \left(s_1,s_2\right)=G_k\left(s_1,s_2\right)-G_k\left(s_1,s_1\right),\\ =\frac{\alpha +\beta +\theta }{\alpha +\beta }{\left(1-e^{-d}\right)}^{\theta }{\left(1-e^{-s_1}\right)}^{\alpha }\left[{\left(1-e^{-s_2}\right)}^{\beta }-{\left(1-e^{-s_1}\right)}^{\beta }\right],\\ {\varkappa }_1\left(s_1,s_2,\alpha ,\beta ,\theta \right)=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{k\Gamma \left(\beta -k\right)k!}\left[\frac{1}{k}{e}^{-k{s}_2}-\frac{1}{k}+s_2{e}^{-k{s}_1}\right],\\ {\varkappa }_2\left(s_1,\alpha ,\beta ,\theta \right)=\frac{s_2\theta \left(s_2-2s_1\theta \right){\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)}^{\alpha +\beta +\theta }}{2\left(\alpha +\beta +\theta \right)},\\ \varpi \left(s_1,\alpha ,\beta ,\theta \right)=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma \left(\beta +\theta +1\right)}{\Gamma \left(\beta +\theta -k+1\right)k!}{e}^{-\theta s_1\left(k-\beta -\theta \right)}\left[e^{s_2\left(\theta +\beta \right)}-e^{k{s}_2}\right],\\ \psi \left(s_1,\alpha ,\beta ,\theta \right)={\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)}^{\alpha }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-s_2\left(\beta +\theta \right)\right),\\ \phi \left(s_1,\alpha ,\beta ,\theta \right)=\psi \left(s_1,\alpha ,\beta ,\theta \right)\left[s_2{\left(\text{exp}\hspace{0.17em}\left(s_2\right)-1\right)}^{\beta +\theta }-\frac{{\left(-1\right)}^{\beta +\theta }}{\beta +\theta -k}\varpi \left(s_1,\alpha ,\beta ,\theta \right)\right],\\ \text{and}\\ d=min\left\{s_1,s_2\right\}.\end{array}$$

Proof.

Since $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}\mathrm{Pr}\left(U_1\le s_2-u-u_2|U_2=u_2\right)d{G}_{U_2}\left(u_2\right)du\\ =\{\begin{array}{cc}{s}_2\beta \left(\alpha +\theta \right)& \mathrm{i}\mathrm{f}0<u_1<u_2<\infty ,\\ \left(1/2\right)s_2^2(s_2-2s_1)\beta (\alpha +\theta )& \mathrm{i}\mathrm{f}0<u_2<u_1<\infty ,\\ \frac{\theta s_2^2}{2}-s_2{s}_1\theta & \mathrm{i}\mathrm{f}0<u_1=u_2<\infty ,\end{array}\end{array}$$ if $\left|x\right|<1$ and $j\notin {\mathbb{Z}}_{+}$, it follows the series given below (Nadarajah & Kotz, 2004, pp. 324, Equation (1.7)). (17) $${\left(1-x\right)}^j=\sum _{y=0}^{\infty }\frac{{\left(-1\right)}^y\Gamma \left(j+1\right)}{\Gamma \left(j-y+1\right)y!}{x}^y,$$(17) and (18) $${\left(1-e^{-y}\right)}^j=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma \left(j+1\right)}{\Gamma \left(j-k+1\right)k!}{e}^{-yk}.$$(18)

Then, by using Equation (18)(18) $${\left(1-e^{-y}\right)}^j=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma \left(j+1\right)}{\Gamma \left(j-k+1\right)k!}{e}^{-yk}.$$(18) , we can obtain (19) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_1\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}\beta \left(\alpha +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha +\theta -1}\\ \times {\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{\Gamma \left(\beta -k\right)k!}{e}^{-yk}\mathit{\text{dydu}}\\ =\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{\Gamma \left(\beta -k\right)k!}{\int }_0^{\infty }\frac{e^{-\left(s_2-u\right)k}-e^{-s_{1}k}}{-k}du.\end{array}$$(19)

Therefore, Equation (19)(19) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_1\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}\beta \left(\alpha +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha +\theta -1}\\ \times {\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{\Gamma \left(\beta -k\right)k!}{e}^{-yk}\mathit{\text{dydu}}\\ =\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{\Gamma \left(\beta -k\right)k!}{\int }_0^{\infty }\frac{e^{-\left(s_2-u\right)k}-e^{-s_{1}k}}{-k}du.\end{array}$$(19) implies that, (20) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_1\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ =\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{k\Gamma \left(\beta -k\right)k!}\left[\frac{1}{k}{e}^{-k{s}_2}-\frac{1}{k}+s_2{e}^{-k{s}_1}\right].\end{array}$$(20)

In addition, $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_2\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}\alpha \left(\beta +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha -1}\\ \times {\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta +\theta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }\alpha \left(\beta +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha -1}{\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta +\theta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right)du.\end{array}$$

We write the above expression as, (21) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_2\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ ={\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)}^{\alpha }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-s_2\left(\beta +\theta \right)\right)\\ \times \left[{\int }_0^{s_2}{\left(\text{exp}\hspace{0.17em}\left(s_2\right)-1\right)}^{\beta +\theta }du-{\int }_0^{s_2}{e}^{\left(\beta +\theta \right)\left(u+s_1\theta \right)}{\left(e^{s_2-u-s_1\theta }-1\right)}^{\beta +\theta }du\right]\\ =\psi \left(s_1,\alpha ,\beta ,\theta \right)\left[s_2{\left(\text{exp}\hspace{0.17em}\left(s_2\right)-1\right)}^{\beta +\theta }-\frac{{\left(-1\right)}^{\beta +\theta }}{\beta +\theta -k}\varpi \left(s_1,\alpha ,\beta ,\theta \right)\right],\end{array}$$(21) where $$\varpi \left(s_1,\alpha ,\beta ,\theta \right)=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma \left(\beta +\theta +1\right)}{\Gamma \left(\beta +\theta -k+1\right)k!}{e}^{-\theta s_1\left(k-\beta -\theta \right)}\left[e^{s_2\left(\theta +\beta \right)}-e^{k{s}_2}\right],$$ and $$\psi \left(s_1,\alpha ,\beta ,\theta \right)={\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)}^{\alpha }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-s_2\left(\beta +\theta \right)\right).$$

According to Equations (20)(20) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_1\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ =\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{k\Gamma \left(\beta -k\right)k!}\left[\frac{1}{k}{e}^{-k{s}_2}-\frac{1}{k}+s_2{e}^{-k{s}_1}\right].\end{array}$$(20) and (21)(21) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_2\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ ={\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)}^{\alpha }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-s_2\left(\beta +\theta \right)\right)\\ \times \left[{\int }_0^{s_2}{\left(\text{exp}\hspace{0.17em}\left(s_2\right)-1\right)}^{\beta +\theta }du-{\int }_0^{s_2}{e}^{\left(\beta +\theta \right)\left(u+s_1\theta \right)}{\left(e^{s_2-u-s_1\theta }-1\right)}^{\beta +\theta }du\right]\\ =\psi \left(s_1,\alpha ,\beta ,\theta \right)\left[s_2{\left(\text{exp}\hspace{0.17em}\left(s_2\right)-1\right)}^{\beta +\theta }-\frac{{\left(-1\right)}^{\beta +\theta }}{\beta +\theta -k}\varpi \left(s_1,\alpha ,\beta ,\theta \right)\right],\end{array}$$(21) , after some simplifications, we have, $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_2-u-\theta s_1}^{s_2}\mathrm{Pr}\left(U_1\le s_1|U_2=u_2\right)d{G}_{U_2}\left(u_2\right)du\\ =\{\begin{array}{cc}\beta \left(\alpha +\theta \right){\varkappa }_1\left(s_1,s_2,\alpha ,\beta ,\theta \right)& \mathrm{i}\mathrm{f}0<u_1<u_2<\infty ,\\ \phi \left(s_1,\alpha ,\beta ,\theta \right)& \mathrm{i}\mathrm{f}0<u_2<u_1<\infty ,\\ {\varkappa }_2\left(s_1,\alpha ,\beta ,\theta \right)& \mathrm{i}\mathrm{f}0<u_1=u_2<\infty ,\end{array}\end{array}$$ where $$\begin{array}{c}\varphi \left(s_1,s_2\right)=G_k\left(s_1,s_2\right)-G_k\left(s_1,s_1\right)\\ =\frac{\alpha +\beta +\theta }{\alpha +\beta }{\left(1-e^{-d}\right)}^{\theta }{\left(1-e^{-s_1}\right)}^{\alpha }\left[{\left(1-e^{-s_2}\right)}^{\beta }-{\left(1-e^{-s_1}\right)}^{\beta }\right],\\ {\varkappa }_1\left(s_1,s_2,\alpha ,\beta ,\theta \right)=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{k\Gamma \left(\beta -k\right)k!}\left[\frac{1}{k}{e}^{-k{s}_2}-\frac{1}{k}+s_2{e}^{-k{s}_1}\right],\\ \text{and}\\ {\varkappa }_2\left(s_1,\alpha ,\beta ,\theta \right)=\frac{s_2\theta \left(s_2-2s_1\theta \right){\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)}^{\alpha +\beta +\theta }}{2\left(\alpha +\beta +\theta \right)}.\end{array}$$

By using Theorem 2.3 in Kundu and Gupta (2009), we have, $$G_{U_1,U_2}(u_1,u_2)=\frac{\alpha +\beta }{\alpha +\beta +\theta }{G}_k\left(u_1,u_2\right)+\frac{\theta }{\alpha +\beta +\theta }{G}_j\left(u_1,u_2\right),$$ where for $d=min\left\{u_1,u_2\right\},$ $$G_j\left(u_1,u_2\right)={\left(1-e^{-d}\right)}^{\alpha +\beta +\theta },$$ and $$G_k\left(u_1,u_2\right)=\frac{\alpha +\beta +\theta }{\alpha +\beta }{\left(1-e^{-u_1}\right)}^{\alpha }{\left(1-e^{-u_2}\right)}^{\beta }{\left(1-e^{-d}\right)}^{\theta }-\frac{\theta }{\alpha +\beta }{\left(1-e^{-d}\right)}^{\alpha +\beta +\theta },$$ where $G_j(.,.)$ and $G_k(.,.)$ are the singular and absolute continuous parts, respectively. With $G_{U_1,U_2}(u_1,u_2)$ and by using Equations (13)(13) $$\begin{array}{c}{\varphi }_T\left(s_2\right)=\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)\right]\\ ={\int }_0^{\infty }\mathrm{Pr}\left(s_2-\mathbb{A}-\mathbb{B}>u|\mathbb{A}\le s_1,s_1\le \mathbb{B}\le s_2\right)du\\ ={\int }_0^{\infty }\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}<s_2-u|\mathbb{A}\le s_1,s_1\le \mathbb{B}\le s_2\right)du.\end{array}$$(13) , (17)(17) $${\left(1-x\right)}^j=\sum _{y=0}^{\infty }\frac{{\left(-1\right)}^y\Gamma \left(j+1\right)}{\Gamma \left(j-y+1\right)y!}{x}^y,$$(17) , and (19)(19) $$\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}{g}_1\left(u_1,u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-y}\beta \left(\alpha +\theta \right){\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1\right)\right)}^{\alpha +\theta -1}\\ \times {\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_2\right)\right)}^{\beta -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-u_1-u_2\right)d{u}_{1}d{u}_{2}du\\ ={\int }_0^{\infty }{\int }_{s_1}^{s_2-u}\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{\Gamma \left(\beta -k\right)k!}{e}^{-yk}\mathit{\text{dydu}}\\ =\beta \left(\alpha +\theta \right)\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k\Gamma (\beta )}{\Gamma \left(\beta -k\right)k!}{\int }_0^{\infty }\frac{e^{-\left(s_2-u\right)k}-e^{-s_{1}k}}{-k}du.\end{array}$$(19) , we can complete the proof.

Proposition 1.

If $\mathbf{X}=(X,Y)$ indicates bivariate Weibull random variables with the joint probability density function of, $$f_{X,Y}\left(x,y\right)=\frac{{\beta }_1{\beta }_2}{{\eta }_1{\eta }_2}{\left(\frac{x}{{\eta }_1}\right)}^{{\beta }_1-1}{\left(\frac{y}{{\eta }_2}\right)}^{{\beta }_2-1}{\alpha }_{1x}{\alpha }_{2y},$$ if $0<x,0<y$ and $f_{\mathbf{X},\mathbf{Y}}(x,y)=0$ otherwise. By Equation (15)(15) $$\begin{array}{c}{}_{\mathbb{A}}{\varphi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)\right]=E\left[s_2-\mathbb{B}|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_{s_1}^{s_2}\frac{F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(15) , we can compute the average value of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ by the transformation technique as follows. $${}_X{\varphi }_Y\left(s_1|s_2\right)=\frac{\left(s_2-s_1\right)-\frac{{\eta }_2}{{\beta }_2}\left({\psi }_2\left(s_1\right)-\phi \left(s_2\right)\right)}{\kappa \left(s_1,s_2\right)}.$$

By using Equation (16)(16) $$\begin{array}{c}{}_{\mathbb{A}}{\psi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[s_2-\mathbb{A}|\mathbb{A}<s_1\right]+E\left[\mathbb{B}-s_1|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_0^{s_1}\frac{F_{\mathbb{A}}\left(u\right)du}{F_{\mathbb{A}}\left(s_1\right)}+{\int }_{s_1}^{s_2}\frac{\left(s_2-s_1\right)F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(16) , we get the residual of system after s₁ as, $${}_X{\psi }_Y\left(s_1|s_2\right)=\frac{s_1-\frac{{\eta }_1}{{\beta }_1}\left(\Gamma \left(\frac{1}{{\beta }_1}\right)-{\psi }_1\left(s_1\right)\right)}{1-{\alpha }_{1s_1}}+\frac{\frac{{\eta }_2}{{\beta }_2}\left({\psi }_2\left(s_1\right)-\phi \left(s_2\right)\right)-\left(s_2-s_1\right){\alpha }_{2s_2}}{\kappa \left(s_1,s_2\right)},$$ where $$\begin{array}{c}{\psi }_i\left(s_1\right)=\Gamma \left(\frac{1}{{\beta }_i},{\left(\frac{s_1}{{\eta }_i}\right)}^{{\beta }_i}\right), i=1,2,\\ \phi \left(s_2\right)=\Gamma \left(\frac{1}{{\beta }_2},{\left(\frac{s_2}{{\eta }_2}\right)}^{{\beta }_2}\right),\\ \kappa \left(s_1,s_2\right)=e^{-{\left(\frac{s_2}{{\eta }_2}\right)}^{{\beta }_2}}-e^{-{\left(\frac{s_1}{{\eta }_2}\right)}^{{\beta }_2}},\\ {\alpha }_{ij}=e^{-{\left(\frac{j}{{\eta }_i}\right)}^{{\beta }_i}},i=1,2\text{;} j=x,y,s_1,s_2,\end{array}$$ and $\Gamma (.,.)$ is the upper incomplete gamma.

In the following result, we study the distribution function of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ defined by Equation (3)(3) $${\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)=\left\{s_2-\mathbb{A}-\mathbb{B}|s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right\},$$(3) .

Theorem 3.

Suppose $\mathbb{A}$ and $\mathbb{B}$ are dependent lifetime random variables with the joint cumulative distribution function $F_{\mathbb{A},\mathbb{B}}(a,b)$ and marginal distributions $F_{\mathbb{A}}(a)$ and $F_{\mathbb{B}}(b)$, for $a,b>0$. Hence, the conditional distribution function of $\mathbb{B}+\mathbb{A}$ given $\left\{s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right\}$ is, $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right)\\ ={\phi }_{\mathbb{A},\mathbb{B}}^{-1}\left(s_1,s_2\right)\left[{\int }_{s_1}^{u-\theta s_1}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)\\ +{\int }_{u-\theta s_1}^{s_2}\mathrm{Pr}\left(s_1\le \mathbb{A}\le s_2|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)],\end{array}$$ where $${\phi }_{\mathbb{A},\mathbb{B}}\left(s_1,s_2\right)=F_{\mathbb{A},\mathbb{B}}(s_2,s_2)-F_{\mathbb{A},\mathbb{B}}(s_2,s_1)-F_{\mathbb{A},\mathbb{B}}(s_1,s_2)+F_{\mathbb{A},\mathbb{B}}(s_1,s_1).$$

Proof.

Consider $u-b<\theta s_1$ or $u-b\ge \theta s_1$, where $\theta \in (0,1)$. Then, we have, $$\begin{array}{l}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u,s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right)\\ ={\int }_{s_1\le b\le s_2}\mathrm{Pr}\left(\mathbb{A}\le u-b,s_1\le \mathbb{A}\le s_2|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ ={\int }_{s_1\le b\le s_2,u-b>\theta s_1}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)\\ +{\int }_{s_1\le b\le s_2,u-b<\theta s_1}\mathrm{Pr}\left(s_1\le \mathbb{A}\le s_2|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right),\end{array}$$ and $$\begin{array}{c}\mathrm{Pr}\left(s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right)=F_{\mathbb{A},\mathbb{B}}(s_2,s_2)-F_{\mathbb{A},\mathbb{B}}(s_2,s_1)\\ -F_{\mathbb{A},\mathbb{B}}(s_1,s_2)+F_{\mathbb{A},\mathbb{B}}(s_1,s_1).\end{array}$$

Thus, the required result is provided.

By using Theorem (3), the average value of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ can be presented as the following formula. $$\begin{array}{l}\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)\right]={\int }_0^{\infty }\mathrm{Pr}\left(s_2-u\ge \mathbb{A}+\mathbb{B},s_1\le \mathbb{A}\le s_2,s_1\le \mathbb{B}\le s_2\right)du\\ ={\phi }_{\mathbb{A},\mathbb{B}}^{-1}\left(s_1,s_2\right)\left[{\int }_0^{\infty }{\int }_{s_1}^{s_2-u-\theta s_1}\mathrm{Pr}\left(\mathbb{A}\le s_2-u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)du\\ +{\int }_0^{\infty }{\int }_{s_2-u-\theta s_1}^{s_2}\mathrm{Pr}\left(s_1\le \mathbb{A}\le s_2|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)du].\end{array}$$

Corollary 2.

Let $\mathbb{A}$ denote the lifetime of an equipment with distribution function $F_{\mathbb{A}}(.),$ $a\in R^1$ and survival function ${\overline{F}}_{\mathbb{A}}(.)=1-F_{\mathbb{A}}(.)$. Similarly, let $\mathbb{B}$ be a standby random lifetime with distribution function $F_{\mathbb{B}}(.)$ and survival function ${\overline{F}}_{\mathbb{B}}(.)=1-F_{\mathbb{B}}(.).$ When $\mathbb{A}$ and $\mathbb{B}$ are independent, we have the average value of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ as follows. (22) $$\begin{array}{c}\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)\right]={\overline{\phi }}_{\mathbb{A},\mathbb{B}}^{-1}\left(s_1,s_2\right){\int }_0^{\infty }{\int }_{s_1}^{s_2-u-\theta s_1}{F}_{\mathbb{A}}(s_2-u-b)d{F}_{\mathbb{B}}\left(b\right)du\\ +\frac{{\int }_0^{\infty }{F}_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_2-u-\theta s_1\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)},\end{array}$$(22) where $${\overline{\phi }}_{\mathbb{A},\mathbb{B}}\left(s_1,s_2\right)=\left(F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)\right)\left(F_{\mathbb{A}}\left(s_2\right)-F_{\mathbb{A}}\left(s_1\right)\right).$$

The Farlie-Gumbel-Morgenstern (FGM) family is extensively used in building the dependence structure between marginal distributions. It is defined as, $${\mathbb{D}}_{\beta }^{FGM}\left(a_1,a_2\right)=a_1{a}_2+\beta a_1{a}_2\left(1-a_1\right)\left(1-a_2\right),a_1{a}_2\in \left[0,1\right],$$ where $\beta \in \left[-1,1\right].$ This section establishes explicit expressions for systems by using the FGM Copula. Now, the density function of the FGM Copula can be obtained as, $$\begin{array}{l}D\left(F_{\mathbb{A}},F_{\mathbb{B}}\right)={\partial }^2\mathbb{D}\left(F_{\mathbb{A}},F_{\mathbb{B}}\right)/\partial F_{\mathbb{A}}\partial F_{\mathbb{B}}\\ =1+\beta \left(1-2F_{\mathbb{A}}\left(a\right)\right)-2\beta F_{\mathbb{B}}\left(b\right)\left(1-2F_{\mathbb{A}}\left(a\right)\right).\end{array}$$

Now, we derive $\mathbb{E}\left[{\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)\right]$ by using the FGM Copula as follows. $$\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)\right]={\kappa }_1^{-1}\left(s_1,s_2\right)\left[\begin{array}{c}{\int }_0^{\infty }{\int }_{s_1}^{s_2-u}{\int }_0^{s_2-u-b}D\left(F_{\mathbb{A}},F_{\mathbb{B}}\right)f_{\mathbb{B}}\left(b\right)f_{\mathbb{A}}\left(a\right)\mathit{\text{dadbdu}}\\ +{\int }_0^{\infty }{\int }_{s_2-u-\theta s_1}^{s_2}{\int }_0^{s_1}D\left(F_{\mathbb{A}},F_{\mathbb{B}}\right)f_{\mathbb{B}}\left(b\right)f_{\mathbb{A}}\left(a\right)\mathit{\text{dadbdu}}\end{array}\right],$$ where ${\kappa }_1(s_1,s_2)=\mathbb{D}(F_{\mathbb{A}}(s_1),F_{\mathbb{B}}(s_2))F_{\mathbb{A}}(s_1)F_{\mathbb{B}}(s_2)-\mathbb{D}(F_{\mathbb{A}}(s_1),F_{\mathbb{B}}(s_1))F_{\mathbb{A}}(s_1)F_{\mathbb{B}}(s_1).$

In addition, $$\begin{array}{c}\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)\right]={\kappa }_2^{-1}\left(s_1,s_2\right)\left[{\int }_0^{\infty }{\int }_{s_1}^{s_2-u-\theta s_1}{\int }_0^{s_2-u-b}f\left(\mathbb{A}|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)du\\ +{\int }_0^{\infty }{\int }_{s_2-u-\theta s_1}^{s_2}{\int }_{s_1}^{s_2}D\left(F_{\mathbb{A}},F_{\mathbb{B}}\right)f_{\mathbb{A}}\left(a\right)f_{\mathbb{B}}\left(b\right)\mathit{\text{dadbdu}}].\end{array}$$

This can be rewritten as, $$\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)\right]={\kappa }_2^{-1}\left(s_1,s_2\right)[{\mu }_1\left(s_1,s_2\right)+{\mu }_2\left(s_1,s_2\right)],$$ where $$\begin{array}{c}{\mu }_1\left(s_1,s_2\right)={\int }_0^{\infty }{\int }_{s_1}^{s_2-u-\theta s_1}{\int }_0^{s_2-u-b}\left[\begin{array}{c}1+\beta \left(1-2F_{\mathbb{A}}\left(a\right)\right)\\ -2\beta F_{\mathbb{B}}\left(b\right)\left(1-2F_{\mathbb{A}}\left(a\right)\right)\end{array}\right]f_{\mathbb{A}}\left(a\right)f_{\mathbb{B}}\left(b\right)\mathit{\text{dadbdu}},\\ {\mu }_2\left(s_1,s_2\right)={\int }_0^{\infty }{\int }_{s_2-u-\theta s_1}^{s_2}{\int }_{s_1}^{s_2}\left[\begin{array}{c}1+\beta \left(1-2F_{\mathbb{A}}\left(a\right)\right)\\ -2\beta F_{\mathbb{B}}\left(b\right)\left(1-2F_{\mathbb{A}}\left(a\right)\right)\end{array}\right]f_{\mathbb{A}}\left(a\right)f_{\mathbb{B}}\left(b\right)\mathit{\text{dadbdu}},\\ {\kappa }_2\left(s_1,s_2\right)=F_{\mathbb{A}}^2\left(s_2\right)F_{\mathbb{B}}^2\left(s_2\right)\left\{1+\beta \left[1-F_{\mathbb{A}}\left(s_2\right)\right]\left[1-F_{\mathbb{B}}\left(s_2\right)\right]\right\}\\ -F_{\mathbb{A}}^2\left(s_2\right)F_{\mathbb{B}}^2\left(s_1\right)\left\{1+\beta \left[1-F_{\mathbb{A}}\left(s_2\right)\right]\left[1-F_{\mathbb{B}}\left(s_1\right)\right]\right\}\\ -F_{\mathbb{A}}^2\left(s_1\right)F_{\mathbb{B}}^2\left(s_2\right)\left\{1+\beta \left[1-F_{\mathbb{A}}\left(s_1\right)\right]\left[1-F_{\mathbb{B}}\left(s_2\right)\right]\right\}\\ +F_{\mathbb{A}}^2\left(s_1\right)F_{\mathbb{B}}^2\left(s_1\right)\left\{1+\beta \left[1-F_{\mathbb{A}}\left(s_1\right)\right]\left[1-F_{\mathbb{B}}\left(s_1\right)\right]\right\}.\end{array}$$

Next, we provide the main results related to stochastic orders. First, we compare a system with $\mathbb{I}{\mathbb{D}}_s$ and $\mathbb{D}{\mathbb{D}}_s$ having the same survival function.

Theorem 4.

Let ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}$ =$\phi (X_1,Y_1)$ be a random variable of system ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ based on $\mathbb{D}{\mathbb{D}}_s$ lifetimes $(X_1,Y_1)$ with the same distribution function $\mathbf{F}$. Suppose ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ =$\phi (X_2,Y_2)$ is a random variable of system ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$ based on $\mathbb{I}{\mathbb{D}}_s$ lifetimes $(X_2,Y_2)$ with the same distribution function $\mathbf{F}.$ Let ${\varphi }_{\mathbb{D}\mathbb{D}}$ and ${\varphi }_{\mathbb{I}\mathbb{D}}$ be their respective domination functions. Thus, the following properties are obtained.

1. ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{ST}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ for all F if and only if ${\varphi }_{\mathbb{D}\mathbb{D}}(x)/$ ${\varphi }_{\mathbb{I}\mathbb{D}}(x)$ is a decreasing function in $(0,1)$, and for all $\beta \in {\mathbb{R}}^{+}.$

2. ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{hr}({\ge }_{hr})$ ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ for all F if and only if $\varphi { }_{\mathbb{D}\mathbb{D}}(x)/$ ${\varphi }_{\mathbb{I}\mathbb{D}}(x)$ is a decreasing (increasing) function in x, and for all $\beta \in {\mathbb{R}}^{+}.$

3. ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{RHR}({\ge }_{RHR})$ ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ for all F if and only if $(1-{\varphi }_{\mathbb{D}\mathbb{D}}(x))/$ $(1-{\varphi }_{\mathbb{I}\mathbb{D}}(x))$ is a decreasing (increasing) function in x, and for all $\beta \in {\mathbb{R}}^{+}.$

4. ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{MP}({\ge }_{MP})$ ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ for all F if and only if ${\int }_0^t(1-{\varphi }_{\mathbb{D}\mathbb{D}}(x))dx/$ ${\int }_0^t(1-{\varphi }_{\mathbb{I}\mathbb{D}}(x))dx$ is decreasing (increasing) for all t, and for all $\beta \in {\mathbb{R}}^{+}.$

Proof.

According to Equation (4)(4) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{u-s_2}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)\\ +{\int }_{u-\theta s_1}^{s_2}\mathrm{Pr}\left(\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)].\end{array}$$(4) and Equation (14)(14) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{2s_1}{F}_{\mathbb{A}}\right(u-b\left)d{F}_{\mathbb{B}}\right(b)+F_{\mathbb{A}}(s_1\left)\left[F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(2s_1\right)\right]\right],\end{array}$$(14) and as mentioned earlier in Definitions (1 and 2), we can get the complete proof.

Proposition 2.

Let ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}$ =$\phi ({\mathbb{A}}_1,{\mathbb{B}}_1)$ be a random variable of system Equation (1)(1) $$\mathrm{Pr}\left(\mathbb{A}>a,\mathbb{B}>b\right)\ge \mathrm{Pr}\left(\mathbb{A}>a\right)\mathrm{Pr}\left(\mathbb{B}>b\right).$$(1) based on $\mathbb{D}{\mathbb{D}}_s$ lifetimes $({\mathbb{A}}_1,{\mathbb{B}}_1)$ with the same distribution function F. Suppose ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ $=\phi ({\mathbb{A}}_2,{\mathbb{B}}_2)$ is a random variable of system (${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(2)$) based on $\mathbb{I}{\mathbb{D}}_s$ lifetimes $({\mathbb{A}}_2,{\mathbb{B}}_2)$ with the same distribution function $\mathbf{F}.$ Let ${\varphi }_{\mathbb{D}\mathbb{D}}$ and ${\varphi }_{\mathbb{I}\mathbb{D}}$ be their respective domination functions. Thus, the following properties are obtained.

• ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{LR}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ for all $\mathbf{F}$ if and only if ${\varphi }_{\mathbb{D}\mathbb{D}}({\varphi }_{{}_{\mathbb{I}\mathbb{D}}}^{-}1x)$ is convex in $(0,1).$

• ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{LR}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}\iff {\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{hr}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}\iff {\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{ST}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}.$

Proof.

By using the main results related to the likelihood ratio order introduced in Müller and Stoyan (2002), Hu and Zhuang (2005), and Khaledi and Shaked (2007), and according to Equation (4)(4) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{u-s_2}\mathrm{Pr}\left(\mathbb{A}\le u-b|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\right(b)\\ +{\int }_{u-\theta s_1}^{s_2}\mathrm{Pr}\left(\mathbb{A}\le s_1|\mathbb{B}=b\right)d{F}_{\mathbb{B}}\left(b\right)].\end{array}$$(4) and Equation (14)(14) $$\begin{array}{c}\mathrm{Pr}\left(\mathbb{A}+\mathbb{B}\le u|\mathbb{A}<s_1,s_1\le \mathbb{B}\le s_2\right)\\ =\frac{1}{F_{\mathbb{A},\mathbb{B}}(s_1,s_2)-F_{\mathbb{A},\mathbb{B}}(s_1,s_1)}\left[{\int }_{s_1}^{2s_1}{F}_{\mathbb{A}}\right(u-b\left)d{F}_{\mathbb{B}}\right(b)+F_{\mathbb{A}}(s_1\left)\left[F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(2s_1\right)\right]\right],\end{array}$$(14) as mentioned in Definitions (1 and 2), we can determine whether ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}$ and ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ are two systems with common survival functions $\mathbf{F}(.)$ as well as hazard rate functions $r_{{\mathbb{U}}_{\mathbb{D}\mathbb{D}}}(.)$ and $q_{{\mathbb{U}}_{\mathbb{I}\mathbb{D}}}(.),$ respectively. Therefore, we can obtain the complete proof.

Theorem 5.

Under the assumptions of Proposition 2, when ${\varphi }_{\mathbb{D}\mathbb{D}}$ and ${\varphi }_{\mathbb{I}\mathbb{D}}$ are differentiable, ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{LR}({\ge }_{LR})$ ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ if and only if ${\varphi }_{\mathbb{D}\mathbb{D}}^{\prime }(t)$/${\varphi }_{\mathbb{I}\mathbb{D}}^{\prime }(t)$ is increasing for all $t\in {\mathbb{R}}^{+}$.

Proof.

The results can be provided by using the same steps in Theorem 2.5 in Navarro et al. (2013).

Proposition 3.

(i) If ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{hr}({\ge }_{hr}){\mathbb{U}}_{\mathbb{I}\mathbb{D}}\Rightarrow \varphi ({\mathbb{U}}_{\mathbb{D}\mathbb{D}}){\ge }_{RHR}({\le }_{RHR})\varphi ({\mathbb{U}}_{\mathbb{I}\mathbb{D}}),$ where $\varphi$ is a continuous strictly decreasing function on ($l_X,u_X).$

(ii) If ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{RHR}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}\Rightarrow \varphi ({\mathbb{U}}_{\mathbb{D}\mathbb{D}}){\ge }_{hr}\varphi ({\mathbb{U}}_{\mathbb{I}\mathbb{D}})$ for any such function $\varphi .$

(iii) If ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}{\le }_{RHR}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}$ and if $\mathbb{A}\in \mathit{\text{DRHR}}$, which is independent of ${\mathbb{U}}_{\mathbb{D}\mathbb{D}}$ and ${\mathbb{U}}_{\mathbb{I}\mathbb{D}}$, we obtain $${\mathbb{U}}_{\mathbb{D}\mathbb{D}}+\mathbb{A}{\le }_{RHR}{\mathbb{U}}_{\mathbb{I}\mathbb{D}}+\mathbb{A}.$$

Proof.

From Theorem (2.2) in Ahmad, Kayid, and Pellerey (2005), Corollary (3.1) and Theorems (3.2 and 3.3) in Kayid and Ahmad (2004), and Nanda and Shaked (2001), we deduce the results.

3. Applications

Based on the approach presented in previous sections, the purpose of this study is to introduce some applications of theoretical results to dependence in reliability. We consider the independence relationship between the main system and the standby system. There is a dependence relationship between the components of the same system that are used in the same environment or that share the same load while supposing that components of the standby system are independent.

We assume that $T_{1\mathbb{A}},T_{2\mathbb{A}},\dots ,T_{n\mathbb{A}}$ $$and $T_{1\mathbb{B}},T_{2\mathbb{B}},\dots ,T_{n\mathbb{B}}$ are lifetimes of the $n$ components of two systems $\mathbb{A}$ and $\mathbb{B}$, respectively, and we suppose that $T_{i\mathbb{A}}\ge 0$ and $T_{i\mathbb{B}}\ge 0$, for $i=1,2,\dots ,n.$ The joint distribution functions of $T_{1\mathbb{A}},T_{2\mathbb{A}},\dots ,T_{n\mathbb{A}}$ and $T_{1\mathbb{B}},T_{2\mathbb{B}},\dots ,T_{n\mathbb{B}}$ are written as, $$F_{\mathbb{A}}(s_1,s_2,\dots ,s_n)=\mathrm{Pr}\left(T_{1\mathbb{A}}\le s_1,T_{2\mathbb{A}}\le s_2,\dots ,T_{n\mathbb{A}}\le s_n\right),$$ and $$F_{\mathbb{B}}(s_1,s_2,\dots ,s_n)=\mathrm{Pr}\left(T_{1\mathbb{B}}\le s_1,T_{2\mathbb{B}}\le s_2,\dots ,T_{n\mathbb{B}}\le s_n\right),$$ respectively. By using Sklar’s theorem (see, e.g., Nelsen, 2006), we get $F_{\mathbb{A}}(s_1,s_2,\dots ,s_n)$ and $F_{\mathbb{B}}(s_1,s_2,\dots ,s_n)$, which can be represented, respectively, as, $$F_{\mathbb{A}}(s_1,s_2,\dots ,s_n)=\kappa \left(F_{1\mathbb{A}}\left(s_1\right),F_{2\mathbb{A}}\left(s_2\right),\dots ,F_{n\mathbb{A}}\left(s_n\right)\right),$$ and $$F_{\mathbb{B}}(s_1,s_2,\dots ,s_n)=\kappa \left(F_{1\mathbb{B}}\left(s_1\right),F_{2\mathbb{B}}\left(s_2\right),\dots ,F_{n\mathbb{B}}\left(s_n\right)\right),$$ where κ is the connecting copula. We use the following lemma, which was introduced by Nelsen (2006).

Lemma 1.

If κ is a copula, then ${\kappa }_L\le \kappa \le {\kappa }_U,$ where $${\kappa }_L\left(F_{1i}\left(s_1\right),F_{2i}\left(s_2\right),\dots ,F_{ni}\left(s_n\right)\right)=\max \left(1-n+\sum _{j=1}^n{F}_{ji},0\right),$$ and $${\kappa }_U\left(F_{1i}\left(s_1\right),F_{2i}\left(s_2\right),\dots ,F_{ni}\left(s_n\right)\right)=\min \left(F_{1i},F_{2i},\dots ,F_{ni}\right),$$ for all $i=\mathbb{A},\mathbb{B}$. Functions κ_L and κ_U are called Frèchet–Hoeffding bounds or minimal and maximal copulas, respectively. Now, the average of ${\kappa }_L(F_{1i}(s_1),F_{2i}(s_2),\dots ,F_{ni}(s_n))$ is, $${\mathbb{M}}_{{\kappa }_L}\left(F_{1i}\left(s_1\right),F_{2i}\left(s_2\right),\dots ,F_{ni}\left(s_n\right)\right)=\sum _{j=1}^n{F}_{ji}/n,$$ and the mean function of κ_U can be written as $${\mathbb{M}}_{{\kappa }_U}\left(F_{1i}\left(s_1\right),F_{2i}\left(s_2\right),\dots ,F_{ni}\left(s_n\right)\right)=\min \left(F_{1i},F_{2i},\dots ,F_{ni}\right),$$ for all $i=\mathbb{A}$ and $\mathbb{B}$.

Application 1. Suppose a system with lifetimes $\mathbb{A}=\min (T_{1\mathbb{A}},T_{2\mathbb{A}},\dots ,T_{m\mathbb{A}})$ and $\mathbb{B}=\min (T_{1\mathbb{B}},T_{2\mathbb{B}},\dots ,T_{n\mathbb{B}})$ $\left[\text{standby} \text{system}\right].$ Therefore, from Equations (15)(15) $$\begin{array}{c}{}_{\mathbb{A}}{\varphi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(1\right)\right]=E\left[s_2-\mathbb{B}|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_{s_1}^{s_2}\frac{F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(15) , (16)(16) $$\begin{array}{c}{}_{\mathbb{A}}{\psi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[s_2-\mathbb{A}|\mathbb{A}<s_1\right]+E\left[\mathbb{B}-s_1|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_0^{s_1}\frac{F_{\mathbb{A}}\left(u\right)du}{F_{\mathbb{A}}\left(s_1\right)}+{\int }_{s_1}^{s_2}\frac{\left(s_2-s_1\right)F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(16) , and (22)(22) $$\begin{array}{c}\mathbb{E}\left[{\mathbb{A}}_{\left(s_1,s_2\right)}^{\mathbb{B}}\left(2\right)\right]={\overline{\phi }}_{\mathbb{A},\mathbb{B}}^{-1}\left(s_1,s_2\right){\int }_0^{\infty }{\int }_{s_1}^{s_2-u-\theta s_1}{F}_{\mathbb{A}}(s_2-u-b)d{F}_{\mathbb{B}}\left(b\right)du\\ +\frac{{\int }_0^{\infty }{F}_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_2-u-\theta s_1\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)},\end{array}$$(22) , we get the average value of ${\mathbb{A}}_{(s_1,s_2)}^{\mathbb{B}}(1)$ as, $$\begin{array}{l}{}_{\mathbb{A}}{\varphi }_{\mathbb{B}}\left(s_1|s_2\right)=\frac{{\int }_{s_1}^{s_2}{\int }_{s_1}^{s_2}.\mathrm{..}{\int }_{s_1}^{s_2}{F}_{\mathbb{B}}(u_1,u_2,\dots ,u_n)d{u}_{1}d{u}_2\dots d{u}_n}{F_{\mathbb{B}}(s_2,s_2,\dots ,s_2)-F_{\mathbb{B}}(s_1,s_1,\dots ,s_1)}\\ =\frac{{\left(s_2-s_1\right)}^{n-1}{\int }_{s_1}^{s_2}\kappa \left(\alpha \left(u\right),\alpha \left(u\right),\dots ,\alpha \left(u\right)\right)du}{\kappa \left(\alpha \left(s_2\right),\alpha \left(s_2\right),\dots ,\alpha \left(s_2\right)\right)-\kappa \left(\alpha \left(s_1\right),\alpha \left(s_1\right),\dots ,\alpha \left(s_1\right)\right)},\end{array}$$ where (23) $$\begin{array}{l}{F}_{\mathbb{B}}\left(s_2\right)=\mathrm{Pr}\left({\mathbb{B}}_{1:n}\le s_2\right)\\ =\mathrm{Pr}\left(T_{1\mathbb{B}}\le s_2,T_{2\mathbb{B}}\le s_2,\dots ,T_{n\mathbb{B}}\le s_2\right)\\ =F_{\mathbb{B}}\left(s_2,s_2,\dots ,s_2\right)\\ ={\kappa }_{\mathbb{B}}\left(\alpha \left(s_2\right),\alpha \left(s_2\right),\dots ,\alpha \left(s_2\right)\right),\end{array}$$(23) where $$\alpha \left(s_2\right)={\mathbb{M}}_{\kappa }\left(F_{1\mathbb{B}}\left(s_2\right),F_{2\mathbb{B}}\left(s_2\right),\dots ,F_{n\mathbb{B}}\left(s_2\right)\right),$$ and ${\mathbb{M}}_{\kappa }$ is the mean function of $\kappa .$

Now, we obtain RL after s₁, by using Equation (16)(16) $$\begin{array}{c}{}_{\mathbb{A}}{\psi }_{\mathbb{B}}\left(s_1|s_2\right)=E\left[s_2-\mathbb{A}|\mathbb{A}<s_1\right]+E\left[\mathbb{B}-s_1|s_1\le \mathbb{B}\le s_2\right]\\ ={\int }_0^{s_1}\frac{F_{\mathbb{A}}\left(u\right)du}{F_{\mathbb{A}}\left(s_1\right)}+{\int }_{s_1}^{s_2}\frac{\left(s_2-s_1\right)F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(u\right)du}{F_{\mathbb{B}}\left(s_2\right)-F_{\mathbb{B}}\left(s_1\right)}.\end{array}$$(16) as follows. $$\begin{array}{l}{}_{\mathbb{A}}{\psi }_{\mathbb{B}}\left(s_1|s_2\right)={\int }_0^{s_1}\frac{{\left(s_2-s_1\right)}^{m-1}{\kappa }_{\mathbb{A}}\left(\beta \left(u\right),\beta \left(u\right),\dots ,\beta \left(u\right)\right)du}{{\kappa }_{\mathbb{A}}\left(\beta \left(s_1\right),\beta \left(s_1\right),\dots ,\beta \left(s_1\right)\right)}\\ -{\int }_{s_1}^{s_2}\frac{{\left(s_2-s_1\right)}^{n-1}{\kappa }_{\mathbb{B}}\left(\alpha \left(u\right),\alpha \left(u\right),\dots ,\alpha \left(u\right)\right)du}{{\kappa }_{\mathbb{B}}\left(\alpha \left(s_2\right),\alpha \left(s_2\right),\dots ,\alpha \left(s_2\right)\right)-{\kappa }_{\mathbb{B}}\left(\alpha \left(s_1\right),\alpha \left(s_1\right),\dots ,\alpha \left(s_1\right)\right)}\\ +\frac{{\left(s_2-s_1\right)}^2{\kappa }_{\mathbb{B}}\left(\alpha \left(s_2\right),\alpha \left(s_2\right),\dots ,\alpha \left(s_2\right)\right)}{{\kappa }_{\mathbb{B}}\left(\alpha \left(s_2\right),\alpha \left(s_2\right),\dots ,\alpha \left(s_2\right)\right)-{\kappa }_{\mathbb{B}}\left(\alpha \left(s_1\right),\alpha \left(s_1\right),\dots ,\alpha \left(s_1\right)\right)},\end{array}$$ where $$\begin{array}{l}{F}_{\mathbb{A}}\left(s_2\right)=\mathrm{Pr}\left({\mathbb{A}}_{1:m}\le s_2\right)\\ =\mathrm{Pr}\left(T_{1\mathbb{A}}\le s_2,T_{2\mathbb{A}}\le s_2,\dots ,T_{m\mathbb{A}}\le s_2\right)\\ =F_{\mathbb{A}}\left(s_2,s_2,\dots ,s_2\right)\\ ={\kappa }_{\mathbb{A}}\left(\beta \left(s_2\right),\beta \left(s_2\right),\dots ,\beta \left(s_2\right)\right),\end{array}$$ and $$\beta \left(s_2\right)={\mathbb{R}}_{\kappa }\left(F_{1\mathbb{A}}\left(s_2\right),F_{2\mathbb{A}}\left(s_2\right),\dots ,F_{n\mathbb{A}}\left(s_2\right)\right),$$ where ${\mathbb{R}}_{\kappa }$ is the mean function of κ while considering random variable $\mathbb{A}$.

As mentioned in Navarro and Spizzichino (2010), a vector ($a_1,a_2,\dots ,a_n$) is said to be majorized by another vector ($b_1,b_2,\dots ,b_n$) ($(a_1,a_2,\dots ,a_n){\le }_M$($b_1,b_2,\dots ,b_n$)) if ${\sum }_{i=1}^n{a}_i={\sum }_{i=1}^n{b}_i$ and ${\sum }_{k=1}^j{a}_{k:n}\ge {\sum }_{k=1}^j{b}_{k:n}$ for $j=1,2,\dots ,n-1$, once $a_{1:n},a_{2:n},\dots ,a_{n:n}$ are order statistics of random variable $\mathbb{A}.$ Analogously, for random variable $\mathbb{B}$, we have $b_{1:n},b_{2:n},\dots ,b_{n:n}$ as order statistics.

Application 2. Suppose ${\mathbb{D}}_1=\phi (\mathbb{A},\mathbb{B})$ is a main system containing lifetimes $\mathbb{A}=\min (T_{1\mathbb{A}},T_{2\mathbb{A}},\dots ,T_{m\mathbb{A}})$ and a standby system $\mathbb{B}=\min (T_{1\mathbb{B}},T_{2\mathbb{B}},\dots ,T_{n\mathbb{B}})$, with joint distribution functions of $T_{1\mathbb{A}},T_{2\mathbb{A}},\dots ,T_{n\mathbb{A}}$ and $T_{1\mathbb{B}},T_{2\mathbb{B}},\dots ,T_{n\mathbb{B}}$ that are written as, $$F_{\mathbb{A}}(s_1,s_2,\dots ,s_n)=\mathrm{Pr}\left(T_{1\mathbb{A}}\le s_1,T_{2\mathbb{A}}\le s_2,\dots ,T_{n\mathbb{A}}\le s_n\right),$$ and $$F_{\mathbb{B}}(s_1,s_2,\dots ,s_n)=\mathrm{Pr}\left(T_{1\mathbb{B}}\le s_1,T_{2\mathbb{B}}\le s_2,\dots ,T_{n\mathbb{B}}\le s_n\right),$$ respectively. Now, let there be another system ${\mathbb{D}}_2=\phi (U,V)$ with lifetimes $U=\min (R_{1U},R_{2U},\dots ,R_{mU})$ and standby system $V=\min (R_{1V},R_{2V},\dots ,R_{nV})$, with joint distribution functions of $R_{1U},R_{2U},\dots ,R_{mU}$ and $R_{1V},R_{2V},\dots ,R_{nV}$ that are written as, $$F_U(r_1,r_2,\dots ,r_n)=\mathrm{Pr}\left(R_{1U}\le r_1,R_{2U}\le r_2,\dots ,R_{nU}\le r_n\right),$$ and $$F_V(r_1,r_2,\dots ,r_n)=\mathrm{Pr}\left(R_{1V}\le r_1,R_{2V}\le r_2,\dots ,R_{nV}\le r_n\right),$$ respectively. Equation (23)(23) $$\begin{array}{l}{F}_{\mathbb{B}}\left(s_2\right)=\mathrm{Pr}\left({\mathbb{B}}_{1:n}\le s_2\right)\\ =\mathrm{Pr}\left(T_{1\mathbb{B}}\le s_2,T_{2\mathbb{B}}\le s_2,\dots ,T_{n\mathbb{B}}\le s_2\right)\\ =F_{\mathbb{B}}\left(s_2,s_2,\dots ,s_2\right)\\ ={\kappa }_{\mathbb{B}}\left(\alpha \left(s_2\right),\alpha \left(s_2\right),\dots ,\alpha \left(s_2\right)\right),\end{array}$$(23) implies that, $$F_{\mathbb{B}}\left(s\right)={\kappa }_{\mathbb{B}}\left({\alpha }_1\left(s\right),{\alpha }_1\left(s\right),\dots ,{\alpha }_1\left(s\right)\right),$$ where $${\alpha }_1\left(s\right)={\mathbb{M}}_{\kappa }\left(F_{1\mathbb{B}}\left(s\right),F_{2\mathbb{B}}\left(s\right),\dots ,F_{n\mathbb{B}}\left(s\right)\right).$$

Similarly, distribution of the second system can be represented as, $$F_V\left(r\right)={\kappa }_V\left({\alpha }_2\left(r\right),{\alpha }_2\left(r\right),\dots ,{\alpha }_2\left(r\right)\right),$$ where $${\alpha }_2\left(s\right)={\mathbb{M}}_{\kappa }\left(F_{1V}\left(r\right),F_{2V}\left(r\right),\dots ,F_{nV}\left(r\right)\right).$$

We use the same distribution copula κ with the same mean ${\mathbb{M}}_{\kappa }$ for the first and second main systems. According to Theorem 3.1 in Navarro and Spizzichino (2010), if κ is an increasing function, we get $F_{(1:n)\mathbb{B}}(s)\le F_{(1:n)V}(s)$ if and only if ${\alpha }_1(s)\le {\alpha }_2(s)$ for all s, where ${\mathbb{B}}_{(1:n)}=\min (T_{1\mathbb{B}},T_{2\mathbb{B}},\dots ,T_{n\mathbb{B}})$ with distribution function $F_{(1:n)\mathbb{B}}(s).$ Analogously, $V_{(1:n)}=\min (R_{1V},R_{2V},\dots ,R_{nV})$ with distribution function $F_{(1:n)V}(s).$ Then, we have $${\mathbb{M}}_{\kappa }\left(F_{1\mathbb{B}}\left(s\right),F_{2\mathbb{B}}\left(s\right),\dots ,F_{n\mathbb{B}}\left(s\right)\right)\le {\mathbb{M}}_{\kappa }\left(F_{1V}\left(r\right),F_{2V}\left(r\right),\dots ,F_{nV}\left(r\right)\right),$$ which leads to ${\mathbb{D}}_1{\le }_{ST}{\mathbb{D}}_2.$ By Proposition 2, we have ${\mathbb{D}}_1{\le }_{LR}{\mathbb{D}}_2$, and ${\mathbb{D}}_1{\le }_{hr}{\mathbb{D}}_2.$

Application 3. Suppose Z₁, Z₂ and Z₃ are mutually independent components with the following density functions. $$Z_i\sim GE\left(1,{\alpha }_i\right),i=1,2, \text{and} Z_3\sim \hspace{0.17em}\text{exp}\hspace{0.17em}\left({\alpha }_0\right),$$ where $GE(1,{\alpha }_i)$ denotes the generalized exponential distribution with parameters $(1,{\alpha }_i)$, while $\text{exp}\hspace{0.17em}({\alpha }_0)$ denotes the exponential distribution with parameter α₀. For more details about these distributions see Gupta and Kundu (2001). Let a system have two systems $({\mathbb{B}}_i,i=1,2).$ Assume that all of them are reformed and maintained independently and, $${\mathbb{B}}_i=\min \left(Z_i,Z_3\right),i=1,2.$$

According to Sarhan and Balakrishnan (2007), the joint distribution function of ${\mathbb{B}}_1$ and ${\mathbb{B}}_2$ is, (24) $$G_{{\mathbb{B}}_1,{\mathbb{B}}_2}\left(b_1,b_2\right)=1-\left\{e^{-{\alpha }_{0}m}\left\{1-{\left(1-e^{-b_1}\right)}^{{\alpha }_1}\right\}\left\{1-{\left(1-e^{-b_2}\right)}^{{\alpha }_2}\right\}\right\},$$(24) where $m=\max (b_1,b_2).$ By Lemma (3.2) in Sarhan and Balakrishnan (2007), we have the case of the conditional bivariate exponential density function. Upon setting ${\alpha }_1={\alpha }_2$ in Equation (24)(24) $$G_{{\mathbb{B}}_1,{\mathbb{B}}_2}\left(b_1,b_2\right)=1-\left\{e^{-{\alpha }_{0}m}\left\{1-{\left(1-e^{-b_1}\right)}^{{\alpha }_1}\right\}\left\{1-{\left(1-e^{-b_2}\right)}^{{\alpha }_2}\right\}\right\},$$(24) , we get, (25) $$g_{i|j}\left(b_i|b_j\right)=\{\begin{array}{cc}{e}^{-\left({\alpha }_0+1\right)b_i+{\alpha }_0{b}_j}& \text{if} b_i>b_j,\\ \frac{{\alpha }_0}{{\alpha }_0+1}{e}^{-b_i}& \text{if} b_i=b_j,\\ e^{-b_i}& \text{if} b_i<b_j.\end{array}$$(25) where $i\ne j=1,2,$ and the marginal probability density function of ${\mathbb{B}}_i$ in this case is $$g_i\left({\mathbb{B}}_i\right)=\left({\alpha }_0+1\right)e^{-\left({\alpha }_0+1\right)b_i},b_i>0,i,j=1,2.$$

From this, we deduce that, (26) $$\mathrm{Pr}\left({\mathbb{B}}_1\le u-b_2|{\mathbb{B}}_2=b_2\right)=\{\begin{array}{cc}{e}^{{\alpha }_0{b}_j}\left(1-e^{-c\left(u-b_j\right)}\right)/c& \text{if} b_i>b_j,\\ {\alpha }_0\left(1-e^{-u+b_j}\right)/c& \text{if} b_i=b_j,\\ 1-\cosh \left(u-b_j\right)+\sinh \left(u-b_j\right)& \text{if} b_i<b_j,\end{array}$$(26) and (27) $${\int }_{s_1}^{u-s_2}\mathrm{Pr}\left({\mathbb{B}}_1\le u-b_2|{\mathbb{B}}_2=b_2\right)d{G}_{{\mathbb{B}}_2}\left(b_2\right)=\{\begin{array}{cc}{f}_1\left({\alpha }_0,s_2,s_1\right)& \text{if} b_i>b_j,\\ f_2\left({\alpha }_0,s_2,s_1\right)& \text{if} b_i=b_j,\\ f_3\left({\alpha }_0,s_2,s_1\right)& \text{if} b_i<b_j,\end{array}$$(27) where

• $f_1({\alpha }_0,s_2,s_1)=e^{{\alpha }_0{b}_j}(e^{-uc}(s_2-u+s_1)+(e^{-s_{1}c}-e^{(s_2-u)c})/c),$

• $f_2({\alpha }_0,s_2,s_1)=({\alpha }_0(e^{-s_{1}c}-e^{(s_2-u)c})+c{e}^{-u-s_1{\alpha }_0}(e^{(s_2-u+s_1){\alpha }_0}-1))/c^2,$

• $f_3({\alpha }_0,s_2,s_1)=(e^{-c(u+s_1)}({\alpha }_0{e}^{uc}-{\alpha }_0{e}^{c(s_2+s_1)}-c{e}^{s_1+u{\alpha }_0}+c{e}^{s_1+(s_2+s_1){\alpha }_0}))/{\alpha }_{0}c,$ and

• $c=({\alpha }_0+1).$

Similarly, we get, (28) $${\int }_{u-\theta s_2}^{s_2}\mathrm{Pr}\left({\mathbb{B}}_1\le s_1|{\mathbb{B}}_2=b_2\right)d{G}_{{\mathbb{B}}_2}\left(b_2\right)=\{\begin{array}{cc}{f}_1\left({\alpha }_0,s_2,\theta \right)& \text{if} b_i>b_j,\\ f_2\left({\alpha }_0,s_2,\theta \right)& \text{if} b_i=b_j,\\ \frac{{\alpha }_0+1}{{\alpha }_0}{f}_2\left({\alpha }_0,s_2,\theta \right)& \text{if} b_i<b_j,\end{array}$$(28) and (29) $$G_{{\mathbb{B}}_1,{\mathbb{B}}_2}\left(s_2,s_1\right)-G_{{\mathbb{B}}_1,{\mathbb{B}}_2}\left(s_1,s_1\right)=\{\begin{array}{cc}{\phi }_1\left({\alpha }_0,s_2,s_1\right)& \text{if} b_i>b_j,\\ {\phi }_2\left({\alpha }_0,s_2,s_1\right)& \text{if} b_i=b_j,\\ {\phi }_3\left({\alpha }_0,s_2,s_1\right)& \text{if} b_i<b_j,\end{array}$$(29) where $$\begin{array}{c}{f}_1\left({\alpha }_0,s_2,\theta \right)=\left[e^{-s_{2}c-s_{1}c+{\alpha }_0{b}_j-c\left(u-\theta s_1\right)}\left({\mathrm{e}}^{{\mathrm{s}}_1\mathrm{c}}-1\right)\left({\mathrm{e}}^{{\mathrm{s}}_2\mathrm{c}}{-\mathrm{e}}^{\left(\mathrm{u}-\theta {\mathrm{s}}_1\right)\mathrm{c}}\right)\right]/c^2,\\ f_2\left({\alpha }_0,s_2,\theta \right)=\left({\alpha }_0\left(e^{-c\left(u-\theta s_1\right)}-e^{-s_{2}c}\right)\left(1-\cosh \left(s_1\right)+\sinh \left(s_1\right)\right)\right)/c^2,\\ {\phi }_1\left({\alpha }_0,s_2,s_1\right)=\left(s_1-s_2\right)e^{{\alpha }_0{b}_j}\left(e^{-s_{1}c}-1\right)/c,\\ {\phi }_2\left({\alpha }_0,s_2,s_1\right)={\alpha }_0\left(s_2-s_1\right)\left(1-e^{-s_1}\right)/c,\\ {\phi }_3\left({\alpha }_0,s_2,s_1\right)=s_2-s_2{e}^{-s_1}-s_1+s_1{e}^{-s_1}.\end{array}$$

Equations (25)–(29) demonstrate that the COD of ${\mathbb{B}}_1+{\mathbb{B}}_2$ given $\left\{{\mathbb{B}}_1<s_1,s_1\le {\mathbb{B}}_2\le s_2\right\}$ for $s_1\le u<2s_1$ is as follows. $$\begin{array}{l}\mathrm{Pr}\left({\mathbb{B}}_1+{\mathbb{B}}_2\le u|{\mathbb{B}}_1<s_1,s_1\le {\mathbb{B}}_2\le s_2\right)\\ =\{\begin{array}{cc}{\phi }_1^{-1}\left({\alpha }_0,s_2,s_1\right)\left(f_1\left({\alpha }_0,s_2,\theta \right)+f_1\left({\alpha }_0,s_2,s_1\right)\right)& \text{if} b_i>b_j,\\ {\phi }_2^{-1}\left({\alpha }_0,s_2,s_1\right)\left(f_2\left({\alpha }_0,s_2,\theta \right)+f_2\left({\alpha }_0,s_2,s_1\right)\right)& \text{if} b_i=b_j,\\ {\phi }_3^{-1}\left({\alpha }_0,s_2,s_1\right)\left(f_3\left({\alpha }_0,s_2,\theta \right)+f_3\left({\alpha }_0,s_2,s_1\right)\right)& \text{if} b_i<b_j.\end{array}\end{array}$$

Similarly, for $u\ge 2s_1,\theta \in (0,1)$ and $2s_1<s_2$, we get the COD of ${\mathbb{B}}_1+{\mathbb{B}}_2$ given $\left\{{\mathbb{B}}_1<s_1,s_1\le {\mathbb{B}}_2\le s_2\right\}$ as, $$\begin{array}{l}\mathrm{Pr}\left({\mathbb{B}}_1+{\mathbb{B}}_2\le u|{\mathbb{B}}_1<s_1,s_1\le {\mathbb{B}}_2\le s_2\right)\\ =\{\begin{array}{cc}{\phi }_1^{-1}\left({\alpha }_0,s_2,s_1\right)\left(f_1\left({\alpha }_0,s_2,\theta \right)+{\psi }_1\left(s_2,s_1\right)\right)& \text{if} b_i>b_j,\\ {\phi }_2^{-1}\left({\alpha }_0,s_2,s_1\right)\left(f_2\left({\alpha }_0,s_2,\theta \right)+{\psi }_2\left(s_2,s_1\right)\right)& \text{if} b_i=b_j,\\ {\phi }_3^{-1}\left({\alpha }_0,s_2,s_1\right)\left(f_3\left({\alpha }_0,s_2,\theta \right)+{\psi }_3\left(s_2,s_1\right)\right)& \text{if} b_i<b_j,\end{array}\end{array}$$ where $$\begin{array}{l}{\psi }_1\left(s_2,s_1\right)=\left(e^{{\alpha }_0{b}_j}\left(e^{s_{1}c}-e^{-c\left(u-\theta s_1\right)}+c\left(s_1-u+\theta s_1\right)e^{-uc}\right)\right)/c^2,\\ {\psi }_2\left(s_2,s_1\right)=\left({\alpha }_0\left(e^{-s_{1}c}-e^{-\left(u-\theta s_1\right)c}\right)+c{e}^{-u-s_1{\alpha }_0}(e^{{\alpha }_0(s_1-u+\theta s_1)}-1)\right)/c^2,\\ {\psi }_3\left(s_2,s_1\right)=e^{-c\left(u+s_1\right)}\left(\left({\alpha }_0\left(e^{uc}-e^{s_1\left(1-\theta \right)c}\right)+c\left(e^{s_1+u{\alpha }_0}+e^{s_1{\alpha }_0(1+\theta )+s_1}\right)\right)\right)/{\alpha }_{0}c.\end{array}$$

Acknowledgements

The author is deeply thankful to the editor and reviewers for their time and efforts.