Appendix
1.1 Proof of Proposition 1
\(O{P_1}\) can be obtained
$$\begin{aligned} O{P_1}= & {} 1 - \underbrace{\Pr \left( {{{\left| {{h_1}} \right| }^2} \ge \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}}}\left( {{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) } \right) }_{ \buildrel \varDelta \over = {\mathcal{O}_3}}\nonumber \\&\times \underbrace{\left( {1 - {F_{{{\left| {{g_1}} \right| }^2}}}\left( {\frac{{\gamma _0^1}}{{\left( {{\alpha _3} - {\alpha _4}\gamma _0^1} \right) {\rho _r}}}} \right) } \right) }_{ \buildrel \varDelta \over = {\mathcal{O}_4}} \end{aligned}$$
(18)
It need be computed these outage expressions as below
$$\begin{aligned} \begin{aligned} {\mathcal{O}_3} =&\Pr \left( {{{\left| {{h_1}} \right| }^2} \ge \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}}}\left( {{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) } \right) \\ =&\Pr \left( {1 - {F_{{{\left| {{h_1}} \right| }^2}}}\left( {\frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}}}\left( {{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) } \right) } \right) \end{aligned} \end{aligned}$$
(19)
After the implementation of the calculation, we have
$$\begin{aligned} \begin{aligned} {\mathcal{O}_3}\left( {x,y} \right) =&{E_{{{\left| {{h_1}} \right| }^2}}}\left\{ {\Pr \left( {{{\left| {{h_1}} \right| }^2} \ge \frac{{\gamma _0^1}}{{{\upsilon _1}{\rho _s}}}\left( {{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) } \right) } \right\} \\ =&\int \limits _0^\infty {{f_{{{\left| f \right| }^2}}}\left( x \right) dx} \int \limits _0^\infty {\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}}\left( {{\alpha _2}{\rho _s}y + {\rho _r}x + 1} \right) } \right) } {f_{{{\left| {{h_2}} \right| }^2}}}\left( y \right) dy\\ =&\frac{1}{{{\lambda _f}{\lambda _{{h_2}}}}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}}} \right) \int \limits _0^\infty {\exp \left( { - \left( {\frac{{\gamma _0^1{\rho _r}}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _f}}}} \right) x} \right) dx} \\&\times \int \limits _0^\infty {\exp \left( { - \left( {\frac{{\gamma _0^1{\alpha _2}}}{{{\alpha _1}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) y} \right) } dy\\ =&\frac{1}{{{\lambda _f}{\lambda _{{h_2}}}}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}}} \right) {\left( {\frac{{\gamma _0^1{\rho _r}}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _f}}}} \right) ^{ - 1}}{\left( {\frac{{\gamma _0^1{\alpha _2}}}{{{\alpha _1}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}} \end{aligned} \end{aligned}$$
(20)
It is noted that, we have following results
$$\begin{aligned} \int \limits _0^\infty {\exp \left( { - \left( {\frac{{\gamma _0^1{\rho _r}}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _f}}}} \right) x} \right) dx} = {\left( {\frac{{\gamma _0^1{\rho _r}}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _f}}}} \right) ^{ - 1}}, \end{aligned}$$
(21)
$$\begin{aligned} \int \limits _0^\infty {\exp \left( { - \left( {\frac{{\gamma _0^1{\alpha _2}}}{{{\alpha _1}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) y} \right) } dy = {\left( {\frac{{\gamma _0^1{\alpha _2}}}{{{\alpha _1}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}}, \end{aligned}$$
(22)
$$\begin{aligned} {f_{{{\left| {{h_2}} \right| }^2}}}\left( y \right) = \frac{1}{{{\lambda _{{h_2}}}}}\exp \left( { - \frac{y}{{{\lambda _{{h_2}}}}}} \right) , \end{aligned}$$
(23)
$$\begin{aligned} {f_{{{\left| f \right| }^2}}}\left( x \right) = \frac{1}{{{\lambda _f}}}\exp \left( { - \frac{x}{{{\lambda _f}}}} \right) . \end{aligned}$$
(24)
To further computation, we have
$$\begin{aligned} {\mathcal{O}_3} = \frac{1}{{{\lambda _f}{\lambda _{{h_2}}}}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}}} \right) {\left( {\frac{{\gamma _0^1{\rho _r}}}{{{\alpha _1}{\rho _s}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _f}}}} \right) ^{ - 1}}{\left( {\frac{{\gamma _0^1{\alpha _2}}}{{{\alpha _1}{\lambda _{{h_1}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}} \end{aligned}$$
(25)
and
$$\begin{aligned} {\mathcal{O}_4} = 1 - {F_{{{\left| {{g_1}} \right| }^2}}}\left( {\frac{{\gamma _0^1}}{{\left( {{\alpha _3} - {\alpha _4}\gamma _0^1} \right) {\rho _r}}}} \right) = \exp \left( { - \frac{{\gamma _0^1}}{{\left( {{\alpha _3} - {\alpha _4}\gamma _0^1} \right) {\rho _r}{\lambda _{{g_1}}}}}} \right) \end{aligned}$$
(26)
1.2 Proof of Proposition 2
We have following equation as
$$\begin{aligned} \begin{aligned} {\mathcal{O}_1} =&\Pr \left( {\frac{{{\alpha _1}{\rho _s}{{\left| {{h_1}} \right| }^2}}}{{{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1}} \ge \gamma _0^1,\frac{{{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2}}}{{{\alpha _1}{\rho _s}{{\left| {{k_1}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1}} \ge \gamma _0^2} \right) \\ =&\Pr \left( {{{\left| {{h_1}} \right| }^2} \ge \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}}}\left( {{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) ,{{\left| {{h_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {{\alpha _1}{\rho _s}{{\left| {{k_1}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) } \right) \end{aligned} \end{aligned}$$
(27)
After placing the following variables \(x = {\left| f \right| ^2},y = {\left| {{h_2}} \right| ^2},z = {\left| {{k_1}} \right| ^2}\) and performing calculations, we have
$$\begin{aligned} \begin{aligned} {\mathcal{O}_1} =&\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}}} \right) \int \limits _0^\infty {\exp \left( { - \frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}}x} \right) {f_{{{\left| f \right| }^2}}}\left( x \right) } dx\\&\times \int \limits _{\frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {{\alpha _1}{\rho _s}z + {\rho _r}x + 1} \right) }^\infty {\exp \left( { - \frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}}y} \right) {f_{{{\left| {{h_2}} \right| }^2}}}\left( y \right) dy} \int \limits _0^\infty {{f_{{{\left| {{k_1}} \right| }^2}}}\left( z \right) } dz\\ =&\frac{1}{{{\lambda _{{h_2}}}}}{\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} - \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) \\&\times \int \limits _0^\infty {\exp \left( { - \left( {\frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} + \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) x} \right) {f_{{{\left| f \right| }^2}}}\left( x \right) } dx\\&\times \int \limits _0^\infty {\exp \left( { - \frac{{{\alpha _1}\gamma _0^2}}{{{\alpha _2}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) z} \right) {f_{{{\left| {{k_1}} \right| }^2}}}\left( z \right) } dz\\ =&\frac{1}{{{\lambda _{{h_2}}}}}{\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} - \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) \\&\times \frac{1}{{{\lambda _f}}}{\left( {\frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} + \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) + \frac{1}{{{\lambda _f}}}} \right) ^{ - 1}}\\&\times \frac{1}{{{\lambda _{{k_1}}}}}{\left( {\frac{{{\alpha _1}\gamma _0^2}}{{{\alpha _2}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) + \frac{1}{{{\lambda _{{k_1}}}}}} \right) ^{ - 1}} \end{aligned} \end{aligned}$$
(28)
The following results can be given as
$$\begin{aligned} \begin{aligned}&\int \limits _{\frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {{\alpha _1}{\rho _s}z + {\rho _r}x + 1} \right) }^\infty {\exp \left( { - \frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}}y} \right) {f_{{{\left| {{h_2}} \right| }^2}}}\left( y \right) dy} = \frac{1}{{{\lambda _{{h_2}}}}}{\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}}\\&\quad \times \exp \left( { - \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) \\&\quad \times \exp \left( { - \frac{{{\alpha _1}\gamma _0^2}}{{{\alpha _2}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) z} \right) \\&\quad \times \exp \left( { - \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) x} \right) , \end{aligned} \end{aligned}$$
(29)
$$\begin{aligned} \begin{array}{c} \int \limits _0^\infty {\exp \left( { - \left( {\frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} + \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) x} \right) {f_{{{\left| f \right| }^2}}}\left( x \right) } dx\\ = \frac{1}{{{\lambda _f}}}{\left( {\frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} + \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) + \frac{1}{{{\lambda _f}}}} \right) ^{ - 1}}, \end{array} \end{aligned}$$
(30)
$$\begin{aligned} \begin{array}{c} \int \limits _0^\infty {\exp \left( { - \frac{{{\alpha _1}\gamma _0^2}}{{{\alpha _2}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) z} \right) {f_{{{\left| {{k_1}} \right| }^2}}}\left( z \right) } dz\\ = \frac{1}{{{\lambda _{{k_1}}}}}{\left( {\frac{{{\alpha _1}\gamma _0^2}}{{{\alpha _2}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) + \frac{1}{{{\lambda _{{k_1}}}}}} \right) ^{ - 1}} \end{array} \end{aligned}$$
(31)
To further compute \({\mathcal{O}_2}\), we have
$$\begin{aligned} \begin{aligned} {\mathcal{O}_2} =&\Pr \left( {\frac{{{\alpha _3}{\rho _r}{{\left| {{g_2}} \right| }^2}}}{{{\alpha _4}{\rho _r}{{\left| {{g_2}} \right| }^2} + 1}} \ge \gamma _0^1,\frac{{{\alpha _4}{\rho _r}{{\left| {{g_2}} \right| }^2}}}{{{\alpha _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1}} \ge \gamma _0^2} \right) \\ =&\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}\left( {{\alpha _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) } \right) \\ =&\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \max \left( {\frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},\frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}\left( {{\alpha _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) } \right) } \right) \\ =&\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}\left( {{\alpha _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) ,{{\left| {{k_2}} \right| }^2}> \frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) \\&+ \Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},{{\left| {{k_2}} \right| }^2}< \frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) \\ =&\underbrace{\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\upsilon _4}{\rho _r}}}\left( {{\upsilon _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) ,{{\left| {{k_2}} \right| }^2} > \max \left( {0,\frac{1}{{{\upsilon _3}{\rho _r}}}\left( {\frac{{{\upsilon _4}\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) }_{ \buildrel \varDelta \over = {\varDelta _1}}\\&+ \underbrace{\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}}},{{\left| {{k_2}} \right| }^2} < \max \left( {0,\frac{1}{{{\upsilon _3}{\rho _r}}}\left( {\frac{{{\upsilon _4}\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) }_{ \buildrel \varDelta \over = {\varDelta _2}} \end{aligned} \end{aligned}$$
(32)
If \({\left| {{k_2}} \right| ^2} > \frac{1}{{{\upsilon _3}{\rho _r}}}\left( {\frac{{{\upsilon _4}\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) \gamma _0^2}} - 1} \right) \) \( \Rightarrow \) \(\max \left( {\frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}}},\frac{{\gamma _0^2}}{{{\upsilon _4}{\rho _r}}}\left( {{\upsilon _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) } \right) = \frac{{\gamma _0^2}}{{{\upsilon _4}{\rho _r}}}\left( {{\upsilon _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) \). Else \({\left| {{k_2}} \right| ^2} < \frac{1}{{{\upsilon _3}{\rho _r}}}\left( {\frac{{{\upsilon _4}\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) \gamma _0^2}} - 1} \right) \)
\(\Rightarrow \) \(\max \left( {\frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}}},\frac{{\gamma _0^2}}{{{\upsilon _4}{\rho _r}}}\left( {{\upsilon _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) } \right) = \frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}}}\).
It worth noting that, some important results can be achieved as
$$\begin{aligned} \begin{aligned} {\varDelta _1} =&\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}\left( {{\alpha _3}{\rho _r}{{\left| {{k_2}} \right| }^2} + 1} \right) ,{{\left| {{k_2}} \right| }^2} > \underbrace{\max \left( {0,\frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) }_{ \buildrel \varDelta \over = \Phi }} \right) \\ =&\int \limits _\Phi ^\infty {\left( {1 - {F_{{{\left| {{g_2}} \right| }^2}}}\left( {\frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}\left( {{\alpha _3}{\rho _r}x + 1} \right) } \right) } \right) {f_{{{\left| {{k_2}} \right| }^2}}}\left( x \right) dx} \\ =&\frac{1}{{{\lambda _{{k_2}}}}}\exp \left( { - \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}{\lambda _{{g_2}}}}}} \right) \int \limits _\Phi ^\infty {\exp \left( { - \left( {\frac{{{\alpha _3}\gamma _0^2}}{{{\alpha _4}{\lambda _{{g_2}}}}} + \frac{1}{{{\lambda _{{k_2}}}}}} \right) x} \right) dx} \\ =&\frac{1}{{{\lambda _{{k_2}}}}}{\left( {\frac{{{\alpha _3}\gamma _0^2}}{{{\alpha _4}{\lambda _{{g_2}}}}} + \frac{1}{{{\lambda _{{k_2}}}}}} \right) ^{ - 1}}\exp \left( { - \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}{\lambda _{{g_2}}}}} - \Phi \left( {\frac{{{\alpha _3}\gamma _0^2}}{{{\alpha _4}{\lambda _{{g_2}}}}} + \frac{1}{{{\lambda _{{k_2}}}}}} \right) } \right) \end{aligned} \end{aligned}$$
(33)
and
$$\begin{aligned} {\varDelta _2}&= \Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},{{\left| {{k_2}} \right| }^2}< \max \left( {0,\frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) \nonumber \\&= \Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}}} \right) \Pr \left( {{{\left| {{k_2}} \right| }^2} < \max \left( {0,\frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) \nonumber \\&= \left( {1 - {F_{{{\left| {{g_2}} \right| }^2}}}\left( {\frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}}}} \right) } \right) \left( {{F_{{{\left| {{k_2}} \right| }^2}}}\left( {\max \left( {0,\frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) } \right) \nonumber \\&= \exp \left( { - \frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}{\lambda _{{g_2}}}}}} \right) \left( {1 - \exp \left( { - \frac{1}{{{\lambda _{{k_2}}}}}\max \left( {0,\frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) } \right) \end{aligned}$$
(34)
Substituting (33) and (34) into (32), we obtain final formula of
$$\begin{aligned} {\mathcal{O}_2}&= {\varDelta _1} + {\varDelta _2}\nonumber \\&= \frac{1}{{{\lambda _{{k_2}}}}}{\left( {\frac{{{\alpha _3}\gamma _0^2}}{{{\alpha _4}{\lambda _{{g_2}}}}} + \frac{1}{{{\lambda _{{k_2}}}}}} \right) ^{ - 1}}\exp \left( { - \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}{\lambda _{{g_2}}}}} - \Phi \left( {\frac{{{\alpha _3}\gamma _0^2}}{{{\alpha _4}{\lambda _{{g_2}}}}} + \frac{1}{{{\lambda _{{k_2}}}}}} \right) } \right) \nonumber \\&+ \exp \left( { - \frac{{\gamma _0^1}}{{\left( {{\upsilon _3} - \gamma _0^1{\upsilon _4}} \right) {\rho _r}{\lambda _{{g_2}}}}}} \right) \nonumber \\&\quad \left( {1 - \exp \left( { - \frac{1}{{{\lambda _{{k_2}}}}}\max \left( {0,\frac{1}{{{\alpha _3}{\rho _r}}}\left( {\frac{{{\alpha _4}\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) \gamma _0^2}} - 1} \right) } \right) } \right) } \right) \end{aligned}$$
(35)
It completes Proposition 2.
1.3 Proof of Proposition 3
We have following equation as
$$\begin{aligned} \begin{aligned} {\zeta _1} =&\Pr \left( {\frac{{{\alpha _1}{\rho _s}{{\left| {{h_1}} \right| }^2}}}{{{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1}} \ge \gamma _0^1,\frac{{{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2}}}{{{\rho _r}{{\left| f \right| }^2} + 1}} \ge \gamma _0^2} \right) \\ =&\Pr \left( {{{\left| {{h_1}} \right| }^2} \ge \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}}}\left( {{\alpha _2}{\rho _s}{{\left| {{h_2}} \right| }^2} + {\rho _r}{{\left| f \right| }^2} + 1} \right) ,{{\left| {{h_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {{\rho _r}{{\left| f \right| }^2} + 1} \right) } \right) \end{aligned} \end{aligned}$$
(36)
After placing the following variables \(x = {\left| f \right| ^2},y = {\left| {{h_2}} \right| ^2}\) and performing calculations, we have
$$\begin{aligned} \begin{aligned} {\zeta _1} =&\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}}} \right) \int \limits _0^\infty {\exp \left( { - \frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}}x} \right) {f_{{{\left| f \right| }^2}}}\left( x \right) } dx\\&\times \int \limits _{\frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {{\rho _r}x + 1} \right) }^\infty {\exp \left( { - \frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}}y} \right) {f_{{{\left| {{h_2}} \right| }^2}}}\left( y \right) dy} \\ =&\frac{1}{{{\lambda _{{h_2}}}}}{\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} - \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) \\&\times \int \limits _0^\infty {\exp \left( { - \left( {\frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} + \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) x} \right) {f_{{{\left| f \right| }^2}}}\left( x \right) } dx\\ =&\frac{1}{{{\lambda _{{h_2}}}}}{\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) ^{ - 1}}\exp \left( { - \frac{{\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} - \frac{{\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) } \right) \\&\times \frac{1}{{{\lambda _f}}}{\left( {\frac{{{\rho _r}\gamma _0^1}}{{{\alpha _1}{\rho _s}{\lambda _{_{{h_1}}}}}} + \frac{{{\rho _r}\gamma _0^2}}{{{\alpha _2}{\rho _s}}}\left( {\frac{{{\alpha _2}\gamma _0^1}}{{{\alpha _1}{\lambda _{_{{h_1}}}}}} + \frac{1}{{{\lambda _{{h_2}}}}}} \right) + \frac{1}{{{\lambda _f}}}} \right) ^{ - 1}} \end{aligned} \end{aligned}$$
(37)
And to further examine \({\zeta _2}\), it can be given by
$$\begin{aligned} {\zeta _2} =&\Pr \left( {\frac{{{\alpha _3}{\rho _r}{{\left| {{g_2}} \right| }^2}}}{{{\alpha _4}{\rho _r}{{\left| {{g_2}} \right| }^2} + 1}} \ge \gamma _0^1,{\alpha _4}{\rho _r}{{\left| {{g_2}} \right| }^2} \ge \gamma _0^2} \right) \nonumber \\ =&\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},{{\left| {{g_2}} \right| }^2} \ge \frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}} \right) \nonumber \\ =&\Pr \left( {{{\left| {{g_2}} \right| }^2} \ge \max \left( {\frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},\frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}} \right) } \right) \nonumber \\ =&\exp \left( { - \frac{1}{{{\lambda _{{g_2}}}}}\max \left( {\frac{{\gamma _0^1}}{{\left( {{\alpha _3} - \gamma _0^1{\alpha _4}} \right) {\rho _r}}},\frac{{\gamma _0^2}}{{{\alpha _4}{\rho _r}}}} \right) } \right) \nonumber \\ \end{aligned}$$
(38)
It completes Proposition 3.
