Appendix A: Expressions of \(\mathcal {J}_i, i \in [1,5]\)
Firstly, using \({\Delta _r^{{j_l},DF}}\) in (12), \(\mathcal {J}_1\) is solved in closed-form as
$$\begin{aligned} \begin{aligned} {{{\mathcal {J}}}_1}&= {\mathbb {P}} \left\{ {\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}}< {\bar{\Delta }} } \right\} \\&\quad = {\mathbb {P}} \left\{ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_u}{k_{tr}} < {\bar{\Delta }} {\sigma _r}} \right\} \\&\quad = \left\{ {\begin{array}{*{20}{c}} {{F_{{k_{tr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{\left[ {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right] {P_u}}}} \right) }&{}{,{\beta _l} > \frac{{{\bar{\Delta }} }}{{1 + {\bar{\Delta }} }}}\\ 1&{}{,{\beta _l} \le \frac{{{\bar{\Delta }} }}{{1 + {\bar{\Delta }} }}} \end{array}} \right. \end{aligned} \end{aligned}$$
(48)
Secondly, using \({\Delta _r^{DF}}\) in (11) and \({\Delta _r^{{j_l},DF}}\) in (12), \(\mathcal {J}_2\) is simplified as
$$\begin{aligned} \begin{aligned} {{{\mathcal {J}}}_2}&= {\mathbb {P}} \left\{ \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}< {\bar{\Delta }},\max \left( {\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}},\frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}} \right) \right. \\&\left. \ge {\bar{\Delta }},\frac{{{k_{tr}}{\beta _u}{P_u}}}{{{\sigma _r}}}< {\bar{\Delta }} \right\} \\&= {\mathbb {P}} \left\{ \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}< {\bar{\Delta }},\frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}} \right. \\&\left. \ge {\bar{\Delta }},\frac{{{k_{tr}}{\beta _u}{P_u}}}{{{\sigma _r}}}< {\bar{\Delta }},\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}}< \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}} \right\} \\&\quad + {\mathbb {P}} \left\{ \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}< {\bar{\Delta }},\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}} \ge {\bar{\Delta }},\frac{{{k_{tr}}{\beta _u}{P_u}}}{{{\sigma _r}}} \right. \\&\quad \left.< {\bar{\Delta }},\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}} \ge \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}} \right\} \\&\quad = {\mathbb {P}} \left\{ \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}< {\bar{\Delta }},\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}} \ge {\bar{\Delta }},\frac{{{k_{tr}}{\beta _u}{P_u}}}{{{\sigma _r}}} \right. \\&\left. \quad< {\bar{\Delta }},\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}} \ge \frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}} \right\} \\&\quad = {\mathbb {P}} \left\{ {\frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}< {\bar{\Delta }},\frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}} \ge {\bar{\Delta }},\frac{{{k_{tr}}{\beta _u}{P_u}}}{{{\sigma _r}}} < {\bar{\Delta }} } \right\} , \end{aligned} \end{aligned}$$
(49)
which reduces to the closed form as
$$\begin{aligned} \begin{aligned} {{{\mathcal {J}}}_2}&= {\mathbb {P}} \left\{ {\frac{{{k_{lr}}{P_l}}}{{{\sigma _r}}}< {\bar{\Delta }} } \right\} {\mathbb {P}} \left\{ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_u}{k_{tr}} \ge {\bar{\Delta }} {\sigma _r},{k_{tr}}< \frac{{{\bar{\Delta }} {\sigma _r}}}{{{\beta _u}{P_u}}}} \right\} \\&\quad = \left\{ {\begin{array}{*{20}{c}} {{F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) {\mathbb {P}} \left\{ {{k_{tr}} \ge \frac{{{\bar{\Delta }} {\sigma _r}}}{{\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_u}}},{k_{tr}}< \frac{{{\bar{\Delta }} {\sigma _r}}}{{{\beta _u}{P_u}}}} \right\} }&{}{,{\beta _l} - {\bar{\Delta }} {\beta _u}> 0}\\ 0&{}{,{\beta _l} - {\bar{\Delta }} {\beta _u} \le 0} \end{array}} \right. \\&\quad = \left\{ {\begin{array}{*{20}{c}} {{F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) {\mathbb {P}} \left\{ {\frac{{{\bar{\Delta }} {\sigma _r}}}{{\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_u}}} \le {k_{tr}}< \frac{{{\bar{\Delta }} {\sigma _r}}}{{{\beta _u}{P_u}}}} \right\} }&{}{,{\beta _l} - {\bar{\Delta }} {\beta _u}> 0,\frac{{{\bar{\Delta }} {\sigma _r}}}{{\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_u}}} < \frac{{{\bar{\Delta }} {\sigma _r}}}{{{\beta _u}{P_u}}}}\\ 0&{}{,\text {otherwise}} \end{array}} \right. \\&\quad = \left\{ {\begin{array}{*{20}{c}} {{F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) \left[ {{F_{{k_{tr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{\beta _u}{P_u}}}} \right) - {F_{{k_{tr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{\left[ {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right] {P_u}}}} \right) } \right] }&{}{,{\beta _l} > \frac{{1 + {\bar{\Delta }} }}{{2 + {\bar{\Delta }} }}}\\ 0&{}{,{\beta _l} \le \frac{{1 + {\bar{\Delta }} }}{{2 + {\bar{\Delta }} }}} \end{array}} \right. \end{aligned} \end{aligned}$$
(50)
Thirdly, using \({\Delta _r^{AF}}\) in (20), \(\mathcal {J}_3\) reduces to
$$\begin{aligned} \begin{aligned} {{{\mathcal {J}}}_3}&= {\mathbb {P}} \left\{ {\frac{{{P_l}{k_{lt}}}}{{{\sigma _t}}}< {\bar{\Delta }},\frac{{{\beta _u}{P_u}{k_{tr}}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) }}{{{\beta _l}{P_u}{k_{tr}}{\sigma _t} + {P_l}{k_{lt}}{\sigma _r} + {\sigma _t}{\sigma _r}}}< {\bar{\Delta }} } \right\} \\&\quad = {\mathbb {P}} \left\{ {k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},\left[ {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}{k_{tr}}\right. \\&\left. \quad< {\bar{\Delta }} {\sigma _r}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) \right\} \\&\quad = {\mathbb {P}} \left\{ {k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}}< \frac{{{\bar{\Delta }} {\sigma _r}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}}\right. \\&\left. \quad ,{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t} > 0 \right\} \\&\quad + {\mathbb {P}} \left\{ {{k_{lt}} < \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t} \le 0} \right\} \end{aligned} \end{aligned}$$
(51)
Since \(\beta _l>0.5\), the case of \({{\bar{\Delta }}}{\beta _l} - {\beta _u} > 0\) or \(\beta _l > \frac{{1}}{{1 + {{{\bar{\Delta }} }}}}\) always holds for \({{\bar{\Delta }} }\ge 1\) (0 dB). Therefore, the rest of this paper considers this case.
Since \(\frac{{{{{\bar{\Delta }} }}{\sigma _t}}}{{{P_l}}} < \frac{{\left( {{{{\bar{\Delta }} }}{\beta _l} - {\beta _u}} \right) {\sigma _t}}}{{{\beta _u}{P_l}}}\) is equivalent to \(\beta _l > \frac{{1 + {{{\bar{\Delta }} }}}}{{1 + 2{{{\bar{\Delta }} }}}}\), \(\mathcal {J}_3\) is further simplified to
$$\begin{aligned} {{{\mathcal {J}}}_3} = \left\{ {\begin{array}{*{20}{c}} {{\mathbb {P}} \left\{ {{k_{lt}} < \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \right\} }&{}{,{\beta _l} > \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}}\\ {{{\mathcal {A}}} + {\mathbb {P}} \left\{ {{k_{lt}} \le \frac{{\left( {{\bar{\Delta }} {\beta _l} - {\beta _u}} \right) {\sigma _t}}}{{{\beta _u}{P_l}}}} \right\} }&{}{,{\beta _l} \le \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}} \end{array}} \right. \end{aligned}$$
(52)
where
$$\begin{aligned}{} & {} {{\mathcal {A}}} = {\mathbb {P}} \left\{ \frac{{\left( {{\bar{\Delta }} {\beta _l} - {\beta _u}} \right) {\sigma _t}}}{{{\beta _u}{P_l}}}< {k_{lt}} \right. \nonumber \\{} & {} \quad \left.< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}} < \frac{{{\bar{\Delta }} {\sigma _r}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}} \right\} \end{aligned}$$
(53)
Using the Gaussian–Chebyshev quadrature in [27] yields the closed form of \({{\mathcal {A}}}\) to be
$$\begin{aligned} \begin{aligned} {{\mathcal {A}}}&= \int \limits _{\frac{{\left( {{\bar{\Delta }} {\beta _l} - {\beta _u}} \right) {\sigma _t}}}{{{\beta _u}{P_l}}}}^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} {{F_{{k_{tr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}\left( {{P_l}x + {\sigma _t}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}x + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}}} \right) {f_{{k_{lt}}}}\left( x \right) dx} \\&\quad = \frac{{\left[ {\left( {1 + {\bar{\Delta }} } \right) {\beta _u} - {\bar{\Delta }} {\beta _l}} \right] {\sigma _t}}}{{2{\beta _u}{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } {F_{{k_{tr}}}}\\&\quad \left( {\frac{{{\bar{\Delta }} {\sigma _r}\left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}}} \right) {f_{{k_{lt}}}}\left( {{\vartheta _m}} \right) \end{aligned} \end{aligned}$$
(54)
where \({\psi _m} = \cos \frac{{\left( {2m - 1} \right) \pi }}{{2M}}\) and \({\vartheta _m} = \frac{{\left( {\left[ {\left( {1 + {\bar{\Delta }} } \right) {\beta _u} - {\bar{\Delta }} {\beta _l}} \right] {\psi _m} + {\bar{\Delta }} - {\beta _u}} \right) {\sigma _t}}}{{2{\beta _u}{P_l}}}\) with M capturing the complexity-accuracy trade-off. Our work opts for \(M=50\) that promises a high exactness as shown in Part 4.
Inserting \({{\mathcal {A}}}\) into (52), one achieves the closed form of \({{{\mathcal {J}}}_3}\) as
$$\begin{aligned} {{{\mathcal {J}}}_3} = \left\{ {\begin{array}{*{20}{c}} {{F_{{k_{lt}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \right) }&{}{,{\beta _l} > \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}}\\ {{{\mathcal {A}}} + {F_{{k_{lt}}}}\left( {\frac{{\left[ {{\bar{\Delta }} {\beta _l} - {\beta _u}} \right] {\sigma _t}}}{{{\beta _u}{P_l}}}} \right) }&{}{,{\beta _l} \le \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}} \end{array}} \right. \end{aligned}$$
(55)
Fourthly, using \({\Delta _r^{j_l,AF}}\) in (21), \(\mathcal {J}_4\) reduces to
$$\begin{aligned} \begin{aligned} {{{\mathcal {J}}}_4}&= {\mathbb {P}} \left\{ {\frac{{{P_l}{k_{lt}}}}{{{\sigma _t}}}< {\bar{\Delta }},\frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}}< {\bar{\Delta }} } \right\} \\&\quad = {\mathbb {P}} \left\{ {k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}}{P_u}\left[ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}} \right] \right. \\&\left.< {\bar{\Delta }} \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r} \right\} \\&\quad = {\mathbb {P}} \left\{ {k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}}< \frac{{{\bar{\Delta }} \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}}{{{P_u}\left[ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}} \right] }}\right. \\&\left. \quad ,\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}> 0 \right\} \\&\quad + {\mathbb {P}} \left\{ {{k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t} \le 0} \right\} \\&\quad = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\mathbb {P}} \left\{ {{k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}}< \frac{{{\bar{\Delta }} \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}}{{{P_u}\left[ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}} \right] }},{k_{lt}}> \frac{{{\bar{\Delta }} {\sigma _t}}}{{\left[ {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right] {P_l}}}} \right\} \\ \quad + {\mathbb {P}} \left\{ {{k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{lt}} \le \frac{{{\bar{\Delta }} {\sigma _t}}}{{\left[ {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right] {P_l}}}} \right\} \end{array}&{}{,{\beta _l} - {{{\bar{\Delta }} }}{\beta _u} > 0}\\ {{\mathbb {P}} \left\{ {{k_{lt}} < \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \right\} }&{}{,{\beta _l} - {{{\bar{\Delta }} }}{\beta _u} \le 0} \end{array}} \right. \end{aligned} \end{aligned}$$
(56)
Since \(\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}} < \frac{{{\bar{\Delta }} {\sigma _t}}}{{\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}}}\), \(\mathcal {J}_4\) is simplified as
$$\begin{aligned} {{{\mathcal {J}}}_4} = {F_{{k_{lt}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \right) . \end{aligned}$$
(57)
Lastly, using \({\Delta _r^{AF}}\) in (20) and \({\Delta _r^{j_l,AF}}\) in (21), \(\mathcal {J}_5\) reduces to
$$\begin{aligned} \begin{aligned} {{{\mathcal {J}}}_5}&= {\mathbb {P}} \left\{ \frac{{{P_l}{k_{lt}}}}{{{\sigma _t}}}< {\bar{\Delta }},\frac{{{P_l}{k_{lr}}}}{{{\sigma _r}}}< {\bar{\Delta }},\right. \\&\left. \quad \max \left( {\frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}},\frac{{{P_l}{k_{lr}}}}{{{\sigma _r}}}} \right) \ge {\bar{\Delta }}, \right. \\&\quad \left. {\frac{{{\beta _u}{P_u}{k_{tr}}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) }}{{{\beta _l}{P_u}{k_{tr}}{\sigma _t} + {P_l}{k_{lt}}{\sigma _r} + {\sigma _t}{\sigma _r}}}< {\bar{\Delta }} } \right\} \\&\quad = {\mathbb {P}} \left\{ \frac{{{P_l}{k_{lr}}}}{{{\sigma _r}}}< {\bar{\Delta }} \right\} {\mathbb {P}} \left\{ \frac{{{P_l}{k_{lt}}}}{{{\sigma _t}}}< {\bar{\Delta }}\right. \\&\left. \quad ,\frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}} \ge {\bar{\Delta }}, \right. \\&\quad \left. {\frac{{{\beta _u}{P_u}{k_{tr}}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) }}{{{\beta _l}{P_u}{k_{tr}}{\sigma _t} + {P_l}{k_{lt}}{\sigma _r} + {\sigma _t}{\sigma _r}}}< {\bar{\Delta }} } \right\} \\&\quad = {F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) \left[ {\mathbb {P}} \left\{ {k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}} \ge \right. \right. \\&\left. \left. \quad \frac{{{\bar{\Delta }} \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}}{{\left[ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}} \right] {P_u}}}, \right. \right. \\&\quad {k_{tr}}< \frac{{{\bar{\Delta }} {P_l}{k_{lt}}{\sigma _r} + {\bar{\Delta }} {\sigma _t}{\sigma _r}}}{{\left[ {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}},\\&\quad \left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}> 0,\\&\quad \left. {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}> 0} \right\} \\&\quad + {\mathbb {P}}\left\{ {{k_{lt}} < \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{k_{tr}} \ge \frac{{{\bar{\Delta }} \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}}{{\left[ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}} \right] {P_u}}},} \right. \\&\quad \left. {\left. {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t} > 0,{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t} \le 0} \right\} } \right] \\&\quad = 0 \end{aligned} \end{aligned}$$
(58)
since
$$\begin{aligned} \left\{ {\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t} > 0} \right\} = \emptyset \ \ \ \text {for} \ \ \ {{\beta _l} - {\bar{\Delta }} {\beta _u}}\le 0 \end{aligned}$$
and
$$\begin{aligned}{} & {} \left\{ {{k_{lt}} < \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},\left( {{\beta _l} - {\bar{\Delta }} {\beta _u}} \right) {P_l}{k_{lt}} - {\bar{\Delta }} {\sigma _t}> 0} \right\} = \emptyset \\{} & {} \quad \text {for}\ \ \ {{\beta _l} - {\bar{\Delta }} {\beta _u}} > 0. \end{aligned}$$
Appendix B: Expressions of \(\mathcal {K}_i, i \in [1,4]\)
Firstly, \(\mathcal {K}_1\) is simplified as
$$\begin{aligned} \begin{aligned} {{{\mathcal {K}}}_1}&= {\mathbb {P}} \left\{ {{\Delta _{lr}} \le {\bar{\Delta }},\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\beta _u}} \right] {P_u}{k_{tr}}< \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\sigma _r}} \right\} \\&\quad = {\mathbb {P}} \left\{ {\Delta _{lr}} \le {\bar{\Delta }},{k_{tr}}< \frac{{\left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\beta _u}} \right] {P_u}}},\right. \\&\left. {\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\beta _u} > 0 \right\} \\&\qquad + {\mathbb {P}} \left\{ {{\Delta _{lr}} \le {\bar{\Delta }},{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\beta _u} \le 0} \right\} \\&\quad = \left\{ {\begin{array}{*{20}{c}} {{{\mathcal {Y}}} + {F_{{k_{lr}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) }&{}{,{\beta _l} < \frac{{{{{\bar{\Delta }} }}}}{{1 + {{{\bar{\Delta }} }}}}}\\ {{\mathcal {H}}}&{}{,{\beta _l} \ge \frac{{{{{\bar{\Delta }} }}}}{{1 + {{{\bar{\Delta }} }}}}} \end{array}} \right. \end{aligned} \end{aligned}$$
(59)
where
$$\begin{aligned} \begin{aligned} {{\mathcal {Y}}}&= {\mathbb {P}} \left\{ {{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}< {\Delta _{lr}} \le {\bar{\Delta }},{k_{tr}} < \frac{{\left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\beta _u}} \right] {P_u}}}} \right\} \\&\quad = \int \limits _{{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}}^{{\bar{\Delta }} } {{F_{{k_{tr}}}}\left( {\frac{{\left( {{\bar{\Delta }} - x} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - x} \right) {\beta _u}} \right] {P_u}}}} \right) } \frac{{{\sigma _r}}}{{{P_l}}}{f_{{k_{lr}}}}\left( {\frac{{{\sigma _r}}}{{{P_l}}}x} \right) dx\\&\quad = \frac{{{\beta _l}{\sigma _r}}}{{2{\beta _u}{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } {F_{{k_{tr}}}}\left( {\frac{{\left( {{\bar{\Delta }} - {\mu _m}} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\mu _m}} \right) {\beta _u}} \right] {P_u}}}} \right) \\&\qquad {f_{{k_{lr}}}}\left( {\frac{{{\sigma _r}{\mu _m}}}{{{P_l}}}} \right) \end{aligned} \end{aligned}$$
(60)
and
$$\begin{aligned} \begin{aligned} {{\mathcal {H}}}&= {\mathbb {P}} \left\{ {{\Delta _{lr}} \le {\bar{\Delta }},{k_{tr}} < \frac{{\left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{lr}}} \right) {\beta _u}} \right] {P_u}}}} \right\} \\&\quad = \int \limits _0^{{\bar{\Delta }} } {{F_{{k_{tr}}}}\left( {\frac{{\left( {{\bar{\Delta }} - x} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - x} \right) {\beta _u}} \right] {P_u}}}} \right) } \frac{{{\sigma _r}}}{{{P_l}}}{f_{{k_{lr}}}}\left( {\frac{{{\sigma _r}}}{{{P_l}}}x} \right) dx\\&\quad = \frac{{{\bar{\Delta }} {\sigma _l}}}{{2{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } {F_{{k_{tr}}}}\left( {\frac{{\left( {{\bar{\Delta }} - {\tau _m}} \right) {\sigma _r}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\tau _m}} \right) {\beta _u}} \right] {P_u}}}} \right) \\&\qquad {f_{{k_{lr}}}}\left( {\frac{{{\sigma _r}{\tau _m}}}{{{P_l}}}} \right) \end{aligned} \end{aligned}$$
(61)
with \({\mu _m} = \frac{{{\beta _l}}}{{2{\beta _u}}}\left( {{\psi _m} - 1} \right) + {{\bar{\Delta }}}\) and \({\tau _m} = \frac{{{\bar{\Delta }} }}{2}\left( {{\psi _m} + 1} \right) \).
The last equalities in (60) and (61) come from the Gaussian–Chebyshev quadrature.
Secondly, \(\mathcal {K}_2\) is expressed in closed-form as
$$\begin{aligned} \begin{aligned}&{{{\mathcal {K}}}_2} = {\mathbb {P}} \left\{ {{\bar{\Delta }} - \frac{{{\beta _l}{P_u}{k_{tr}}}}{{{k_{tr}}{\beta _u}{P_u} + {\sigma _r}}}< {\Delta _{lr}} \le {\bar{\Delta }},{k_{tr}} < \frac{{{\sigma _r}{\bar{\Delta }} }}{{{\beta _u}{P_u}}}} \right\} \\&\quad = \int \limits _0^{\frac{{{\sigma _r}{\bar{\Delta }} }}{{{\beta _u}{P_u}}}} \left[ {{F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) - {F_{{k_{lr}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_u}x}}{{x{\beta _u}{P_u} + {\sigma _r}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) } \right] \\&\qquad {f_{{k_{tr}}}}\left( x \right) dx \\&\quad = \frac{{{\sigma _r}{\bar{\Delta }} }}{{2{\beta _u}{P_u}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } \\&\qquad \left[ {{F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) - {F_{{k_{lr}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_u}{\zeta _m}}}{{{\zeta _m}{\beta _u}{P_u} + {\sigma _r}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) } \right] {f_{{k_{tr}}}}\left( {{\zeta _m}} \right) , \end{aligned} \end{aligned}$$
(62)
where \({\zeta _m} = \frac{{{\sigma _r}{\bar{\Delta }} }}{{2{\beta _u}{P_u}}}\left( {{\psi _m} + 1} \right) \).
Thirdly, \(\mathcal {K}_3\) is rewritten as
$$\begin{aligned} \begin{aligned}&{{{\mathcal {K}}}_3} = {\mathbb {P}} \left\{ {\Delta _{lt}}< {\bar{\Delta }},{\Delta _{lr}} < {\bar{\Delta }} \right. \\&\qquad \left. -\frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}} \right\} \\&\quad = \int \limits _0^\infty \int \limits _0^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} {F_{{k_{lr}}}}\\&\quad \left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}y{P_u}x}}{{y{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) + \left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) \\&\qquad {f_{{k_{lt}}}}\left( x \right) {f_{{k_{tr}}}}\left( y \right) dxdy \\&\quad = {F_{{k_{lt}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \right) - \frac{{{\mathcal {V}}}}{{{\varphi _{tr}}}}{e^{ - \frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}{\varphi _{lr}}}}}} \end{aligned} \end{aligned}$$
(63)
where
$$\begin{aligned} {{\mathcal {V}}} = \int \limits _0^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} {\left[ {\int \limits _0^\infty {{e^{\frac{{y{\beta _l}{P_u}x}}{{y{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) + \left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}\frac{{{\sigma _r}}}{{{\varphi _{lr}}}} - \frac{y}{{{\varphi _{tr}}}}}}dy} } \right] {f_{{k_{lt}}}}\left( x \right) dx}. \end{aligned}$$
(64)
Performing two variable changes consecutively (\(z = y + \frac{{\left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) }}\) and \(w = \frac{1}{z}\)), \({{\mathcal {V}}}\) is expressed in closed-form as
$$\begin{aligned} \begin{aligned} {{\mathcal {V}}}&= \int \limits _0^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} {e^{\frac{{{\beta _l}x{\sigma _r}}}{{{\varphi _{lr}}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) }} + \frac{{\left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) {\varphi _{tr}}}}}}\\&\qquad \left[ {\int \limits _0^{\frac{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) }}{{\left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}} {{w^{ - 2}}{e^{ - \frac{{\left( {{P_l}x + {\sigma _t}} \right) {\beta _l}xw\sigma _r^2}}{{{\varphi _{lr}}{P_u}{{\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) }^2}}} - \frac{1}{{{\varphi _{tr}}w}}}}dw} } \right] {f_{{k_{lt}}}}\left( x \right) dx \\&\quad = \frac{{{\bar{\Delta }} {\sigma _t}}}{{2{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } \\&\qquad \left[ {{e^{\frac{{{\beta _l}{\lambda _m}{\sigma _r}}}{{{\varphi _{lr}}\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) }} + \frac{{\left( {{P_l}{\lambda _m} + {\sigma _t}} \right) {\sigma _r}}}{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) {\varphi _{tr}}}}}}\frac{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) }}{{2\left( {{P_l}{\lambda _m} + {\sigma _t}} \right) {\sigma _r}{\varphi _{tr}}}} \times } \right. \\&\qquad \left. {\sum \limits _{v = 1}^V {\frac{{\pi \sqrt{1 - \eta _v^2} }}{{V\alpha _{vm}^2}}} {e^{ - \frac{{\left( {{P_l}{\lambda _m} + {\sigma _t}} \right) {\beta _l}{\lambda _m}\sigma _r^2{\alpha _{vm}}}}{{{\varphi _{lr}}{P_u}{{\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) }^2}}} - \frac{1}{{{\varphi _{tr}}{\alpha _{vm}}}}}}} \right] {f_{{k_{lt}}}}\left( {{\lambda _m}} \right) , \end{aligned} \end{aligned}$$
(65)
where \({\eta _v} = \cos \frac{{\left( {2v - 1} \right) \pi }}{{2V}}\), \({\lambda _m} = \frac{{{\bar{\Delta }} {\sigma _t}}}{{2{P_l}}}\left( {{\psi _m} + 1} \right) \), and \({\alpha _{vm}} = \frac{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) }}{{2\left( {{P_l}{\lambda _m} + {\sigma _t}} \right) {\sigma _r}}}\left( {{\eta _v} + 1} \right) \) with V capturing the complexity-accuracy trade-off of the Gaussian–Chebyshev quadrature. This work opts for \(V = 50\) that promises a high exactness as shown in Part 4.
Finally, \({{\mathcal {K}}}_4\) is rewritten as
$$\begin{aligned} \begin{aligned} {{{\mathcal {K}}}_4}&= {\mathbb {P}} \left\{ {\Delta _{lt}}< {\bar{\Delta }},{\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}} \right. \\&\quad \left.< {\Delta _{lr}} \le {\bar{\Delta }}, \frac{{{\beta _u}{P_u}{k_{tr}}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) }}{{{\beta _l}{P_u}{k_{tr}}{\sigma _t} + {P_l}{k_{lt}}{\sigma _r} + {\sigma _t}{\sigma _r}}} \right. \\&\quad \left.< {\bar{\Delta }} \right\} \\&\quad = {\mathbb {P}} \left\{ {{k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}}< {\Delta _{lr}} \le {\bar{\Delta }},} \right. \\&\quad \left. {{k_{tr}}< \frac{{{\bar{\Delta }} \left( {{P_l}{k_{lt}}{\sigma _r} + {\sigma _t}{\sigma _r}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}},{k_{lt}} > \left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}}} \right\} \\&\qquad + {\mathbb {P}} \left\{ {{k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},} \right. \\&\quad \left. {\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}} < {\Delta _{lr}} \le {\bar{\Delta }},{k_{lt}} \right. \\&\left. \le \left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}} \right\} \end{aligned} \end{aligned}$$
(66)
Since \(\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}} < \left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}}\) is equivalent to \({\beta _l} > \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}\), \({{\mathcal {K}}}_4\) is decomposed as
$$\begin{aligned} {{{\mathcal {K}}}_4} = \left\{ {\begin{array}{*{20}{c}} {{\mathcal {M}}}&{}{,{\beta _l} > \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}}\\ {{{\mathcal {Q}}} + {{\mathcal {L}}}}&{}{,{\beta _l} \le \frac{{1 + {\bar{\Delta }} }}{{1 + 2{\bar{\Delta }} }}} \end{array}} \right. \end{aligned}$$
(67)
where
$$\begin{aligned}{} & {} {{\mathcal {M}}} = {\mathbb {P}} \left\{ {k_{lt}}< \frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}},{\bar{\Delta }}\right. \nonumber \\ -{} & {} \qquad \left. \frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}} < {\Delta _{lr}} \le {\bar{\Delta }} \right\} \end{aligned}$$
(68)
$$\begin{aligned}{} & {} {{\mathcal {Q}}} = {\mathbb {P}} \left\{ \left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}}< {k_{lt}} \right. \nonumber \\{} & {} \quad \left. \quad<\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}, \right. {k_{tr}}< \frac{{{\bar{\Delta }} \left( {{P_l}{k_{lt}}{\sigma _r} + {\sigma _t}{\sigma _r}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}{k_{lt}} + {\sigma _t}} \right) - {{{\bar{\Delta }} }}{\beta _l}{\sigma _t}} \right] {P_u}}},\nonumber \\{} & {} \quad \left. {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _l}}} < {\Delta _{lr}} \le {\bar{\Delta }} } \right\} \end{aligned}$$
(69)
$$\begin{aligned}{} & {} {{\mathcal {L}}} = {\mathbb {P}} \left\{ {\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{tr}}{P_u}{k_{lt}}}}{{{k_{tr}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) + \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _r}}} \right. \nonumber \\{} & {} \quad \left. < {\Delta _{lr}} \le {\bar{\Delta }},{k_{lt}} \le \left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}} \right\} . \end{aligned}$$
(70)
By following the derivation of \(\mathcal {K}_3\), \(\mathcal {M}\) and \(\mathcal {L}\) are expressed in closed-form to be
$$\begin{aligned} \begin{aligned}&{{\mathcal {M}}} = \int \limits _0^\infty \int \limits _0^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \left[ {F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) \right. \\&\left. \qquad - {F_{{k_{lr}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}y{P_u}x}}{{y{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) + \left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) \right] \times \\&\quad {f_{{k_{lt}}}}\left( x \right) {f_{{k_{tr}}}}\left( y \right) dxdy\\&\quad = {e^{ - \frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}{\varphi _{lr}}}}}}\left[ {\frac{{{\mathcal {V}}}}{{{\varphi _{tr}}}} - {F_{{k_{lt}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \right) } \right] \end{aligned} \end{aligned}$$
(71)
and
$$\begin{aligned} \begin{aligned}&{{\mathcal {L}}} = \int \limits _0^\infty \int \limits _0^{\left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}}} \left[ {F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) \right. \\&\left. \qquad -{F_{{k_{lr}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}y{P_u}x}}{{y{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) + \left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) \right] \times \\&\quad {f_{{k_{lt}}}}\left( x \right) {f_{{k_{tr}}}}\left( y \right) dxdy\\&\quad = {e^{ - \frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}{\varphi _{lr}}}}}}\left[ {\frac{{{\mathcal {U}}}}{{{\varphi _{tr}}}} - {F_{{k_{lt}}}}\left( {\left[ {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right] \frac{{{\sigma _t}}}{{{P_l}}}} \right) } \right] \end{aligned} \end{aligned}$$
(72)
where
$$\begin{aligned} \begin{aligned} {{\mathcal {U}}}&= \frac{{{\sigma _t}}}{{2{P_l}}}\left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} }\\&\quad \left[ {{e^{\frac{{{\beta _l}{\Psi _m}{\sigma _r}}}{{{\varphi _{pd}}\left( {{\sigma _t} + {\beta _u}{P_l}{\Psi _m}} \right) }} + \frac{{\left( {{P_l}{\Psi _m} + {\sigma _t}} \right) {\sigma _r}}}{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\Psi _m}} \right) {\varphi _{tr}}}}}}\frac{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\Psi _m}} \right) }}{{2\left( {{P_l}{\Psi _m} + {\sigma _t}} \right) {\sigma _r}{\varphi _{tr}}}} \times } \right. \\&\quad \left. {\sum \limits _{v = 1}^V {\frac{{\pi \sqrt{1 - \eta _v^2} }}{{V\Lambda _{vm}^2}}} {e^{ - \frac{{\left( {{P_l}{\Psi _m} + {\sigma _t}} \right) {\beta _l}{\Psi _m}\sigma _r^2{\Lambda _{vm}}}}{{{\varphi _{lr}}{P_u}{{\left( {{\sigma _t} + {\beta _u}{P_l}{\Psi _m}} \right) }^2}}} - \frac{1}{{{\varphi _{tr}}{\Lambda _{vm}}}}}}} \right] {f_{{k_{lt}}}}\left( {{\Psi _m}} \right) \end{aligned} \end{aligned}$$
(73)
with \({\Psi _m} = \frac{{{\sigma _t}}}{{2{P_l}}}\left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \left( {{\psi _m} + 1} \right) \) and \({\Lambda _{vm}} = \frac{{{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\Psi _m}} \right) }}{{2\left( {{P_l}{\Psi _m} + {\sigma _t}} \right) {\sigma _r}}}\left( {{\eta _v} + 1} \right) \).
In the meantime, \(\mathcal {Q}\) is expressed in closed-form as
$$\begin{aligned} \begin{aligned}&{{\mathcal {Q}}} = \int \limits _{\left( {\frac{{{\bar{\Delta }} {\beta _l}}}{{{\beta _u}}} - 1} \right) \frac{{{\sigma _t}}}{{{P_l}}}}^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} {\int \limits _0^{\frac{{{\bar{\Delta }} \left( {{P_l}x{\sigma _r} + {\sigma _t}{\sigma _r}} \right) }}{{\left[ {{\beta _u}\left( {{P_l}x + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}}} {\left[ {{F_{{k_{lr}}}}\left( {\frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}}}} \right) - } \right. } } \\&\quad \left. {{F_{{k_{lr}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}y{P_u}x}}{{y{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}x} \right) + \left( {{P_l}x + {\sigma _t}} \right) {\sigma _r}}}} \right] \frac{{{\sigma _r}}}{{{P_l}}}} \right) } \right] \\&\quad {f_{{k_{tr}}}}\left( y \right) {f_{{k_{lt}}}}\left( x \right) dxdy\\&\quad = {e^{ - \frac{{{\bar{\Delta }} {\sigma _r}}}{{{P_l}{\varphi _{lr}}}}}}\frac{{\left[ {\left( {1 + {\bar{\Delta }} } \right) {\beta _u} - {\bar{\Delta }} {\beta _l}} \right] {\sigma _t}}}{{2{\beta _u}{P_l}}}\\&\quad \sum \limits _{m = 1}^M {\frac{{\pi \sqrt{1 - \psi _m^2} {\bar{\Delta }} \left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) {\sigma _r}}}{{2M\left[ {{\beta _u}\left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}}} \times \\&\quad \sum \limits _{v = 1}^V {\frac{\pi }{V}\sqrt{1 - \eta _v^2} } \left( {{e^{\frac{{{\beta _l}{\Upsilon _{vm}}{P_u}{\vartheta _m}{\sigma _r}}}{{\left[ {{\Upsilon _{vm}}{P_u}\left( {{\sigma _t} + {\beta _u}{P_l}{\vartheta _m}} \right) + \left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) {\sigma _r}} \right] {\varphi _{lr}}}}}} - 1} \right) \\&\quad {f_{{k_{tr}}}}\left( {{\Upsilon _{vm}}} \right) {f_{{k_{lt}}}}\left( {{\vartheta _m}} \right) , \end{aligned} \end{aligned}$$
(74)
where \({\Upsilon _{vm}} = \frac{{{\bar{\Delta }} \left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) {\sigma _r}\left( {{\eta _v} + 1} \right) }}{{2\left[ {{\beta _u}\left( {{P_l}{\vartheta _m} + {\sigma _t}} \right) - {\bar{\Delta }} {\beta _l}{\sigma _t}} \right] {P_u}}}.\)
Appendix C: Expressions of \(\mathcal {G}_i, i \in [1,2]\)
Firstly, \({{{\mathcal {G}}}_1}\) is rewritten as
$$\begin{aligned} \begin{aligned} {{{\mathcal {G}}}_1}&= {\mathbb {P}} \left\{ {\Delta _{ll}}< {\bar{\Delta }},{k_{tl}}< \frac{{\left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\beta _u}} \right] {P_u}}},\right. \\&\left. \quad {\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\beta _u} > 0 \right\} \\&\quad + {\mathbb {P}} \left\{ {{\Delta _{ll}}< {\bar{\Delta }},{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\beta _u} \le 0} \right\} \\&\quad = \left\{ {\begin{array}{*{20}{c}} {{{{\mathcal {G}}}_{11}}}&{}{,{\beta _l} < \frac{{{\bar{\Delta }} }}{{1 + {\bar{\Delta }} }}}\\ {{{{\mathcal {G}}}_{12}}}&{}{,{\beta _l} \ge \frac{{{\bar{\Delta }} }}{{1 + {\bar{\Delta }} }}} \end{array}} \right. \end{aligned} \end{aligned}$$
(75)
where
$$\begin{aligned} \begin{aligned} {{{\mathcal {G}}}_{11}}&= {\mathbb {P}} \left\{ {{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}< {\Delta _{ll}}< {\bar{\Delta }},{k_{tl}} < \frac{{\left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\beta _u}} \right] {P_u}}}} \right\} \\&\quad +{\mathbb {P}} \left\{ {{\Delta _{ll}} \le {\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}} \right\} \\&\quad = \int \limits _{{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}}^{{\bar{\Delta }} } {{F_{{k_{tl}}}}\left( {\frac{{\left( {{\bar{\Delta }} - x} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - x} \right) {\beta _u}} \right] {P_u}}}} \right) } \frac{{{\sigma _l}}}{{{P_l}}}{f_{{k_{ll}}}}\left( {\frac{{{\sigma _l}}}{{{P_l}}}x} \right) dx \\&\quad + {F_{{k_{ll}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}} \right] \frac{{{\sigma _l}}}{{{P_l}}}} \right) \\&\quad = \frac{{{\beta _l}{\sigma _l}}}{{2{\beta _u}{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } {F_{{k_{tl}}}}\left( {\frac{{\left( {{\bar{\Delta }} - {\mu _m}} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\mu _m}} \right) {\beta _u}} \right] {P_u}}}} \right) \\&\quad {f_{{k_{ll}}}}\left( {\frac{{{\sigma _l}{\mu _m}}}{{{P_l}}}} \right) + {F_{{k_{ll}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}}}{{{\beta _u}}}} \right] \frac{{{\sigma _l}}}{{{P_l}}}} \right) \end{aligned} \end{aligned}$$
(76)
and
$$\begin{aligned} \begin{aligned}&{{{\mathcal {G}}}_{12}} = {{\mathbb {P}}}\left\{ {{\Delta _{ll}}< {\bar{\Delta }},{k_{tl}} < \frac{{\left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) {\beta _u}} \right] {P_u}}}} \right\} \\&\quad = \int \limits _0^{{\bar{\Delta }} } {{F_{{k_{tl}}}}\left( {\frac{{\left( {{\bar{\Delta }} - x} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - x} \right) {\beta _u}} \right] {P_u}}}} \right) \frac{{{\sigma _l}}}{{{P_l}}}{f_{{k_{ll}}}}\left( {\frac{{{\sigma _l}x}}{{{P_l}}}} \right) dx} \\&\quad = \frac{{{\bar{\Delta }} {\sigma _l}}}{{2{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } {F_{{k_{tl}}}}\left( {\frac{{\left( {{\bar{\Delta }} - {\tau _m}} \right) {\sigma _l}}}{{\left[ {{\beta _l} - \left( {{\bar{\Delta }} - {\tau _m}} \right) {\beta _u}} \right] {P_u}}}} \right) \\&\quad {f_{{k_{ll}}}}\left( {\frac{{{\sigma _l}{\tau _m}}}{{{P_l}}}} \right) . \end{aligned} \end{aligned}$$
(77)
The last equalities in (76) and (77) come from the Gaussian–Chebyshev quadrature.
Secondly, \(\mathcal {G}_2\) is computed as
$$\begin{aligned} \begin{aligned} {{{\mathcal {G}}}_2}&= {\mathbb {P}} \left\{ {\Delta _{ll}}< {\bar{\Delta }},{\Delta _{lt}}< {\bar{\Delta }},{k_{tl}} \right. \\&\quad \left.<\frac{{\left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _l}}}{{\left[ {{\beta _l}{P_l}{k_{lt}} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) \left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) } \right] {P_u}}}, \right. \\&\quad \left. {{\beta _l}{P_l}{k_{lt}} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) \left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) > 0} \right\} \\&\quad + {\mathbb {P}} \left\{ {{\Delta _{ll}}< {\bar{\Delta }},{\Delta _{lt}}< {\bar{\Delta }},{\beta _l}{P_l}{k_{lt}} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) \left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) \le 0} \right\} \\&\quad = {\mathbb {P}} \left\{ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{lt}}}}{{{\sigma _t} + {\beta _u}{P_l}{k_{lt}}}}< {\Delta _{ll}}< {\bar{\Delta }},{\Delta _{lt}}< {\bar{\Delta }},} \right. \\&\quad \left. {{k_{tl}}< \frac{{\left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) \left( {{P_l}{k_{lt}} + {\sigma _t}} \right) {\sigma _l}}}{{\left[ {{\beta _l}{P_l}{k_{lt}} - \left( {{\bar{\Delta }} - {\Delta _{ll}}} \right) \left( {{\sigma _t} + {\beta _u}{P_l}{k_{lt}}} \right) } \right] {P_u}}}} \right\} \\&\quad + {\mathbb {P}} \left\{ {{\Delta _{lt}} < {\bar{\Delta }},{\Delta _{ll}} \le {\bar{\Delta }} - \frac{{{\beta _l}{P_l}{k_{lt}}}}{{{\sigma _t} + {\beta _u}{P_l}{k_{lt}}}}} \right\} \\&\quad = {{\mathcal {C}}} + {{\mathcal {D}}}, \end{aligned} \end{aligned}$$
(78)
where
$$\begin{aligned} \begin{aligned} {{\mathcal {C}}}&= \int \limits _0^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} \left[ \int \limits _{{\bar{\Delta }} - \frac{{{\beta _l}{P_l}y}}{{\sigma _t}+{\beta _u}{P_l}y}} ^{{\bar{\Delta }} } {F_{{k_{tl}}}}\right. \left. \left( {\frac{{\left( {{\bar{\Delta }} - x} \right) \left( {{P_l}y + {\sigma _t}} \right) {\sigma _l}}}{{\left[ {{\beta _l}{P_l}y - \left( {{\bar{\Delta }} - x} \right) \left( {{\sigma _t} + {\beta _u}{P_l}y} \right) } \right] {P_u}}}} \right) \right. \left. \frac{{{\sigma _l}}}{{{P_l}}}{f_{{k_{ll}}}}\left( {\frac{{{\sigma _l}}}{{{P_l}}}x} \right) dx \right] {f_{{k_{lt}}}}\left( y \right) dy \\& = \frac{{{\bar{\Delta }} {\sigma _t}{\sigma _l}}}{{2P_l^2}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } \left[ {\frac{{{\beta _l}{P_l}{\lambda _m}{f_{{k_{lt}}}}\left( {{\lambda _m}} \right) }}{{2\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) }}\sum \limits _{v = 1}^V {\frac{\pi }{V}\sqrt{1 - \eta _v^2} } \times } \right. \\&\quad \left. {{F_{{k_{tl}}}}\left( {\frac{{\left( {{\bar{\Delta }} - {\phi _{vm}}} \right) \left( {{P_l}{\lambda _m} + {\sigma _t}} \right) {\sigma _l}}}{{\left[ {{\beta _l}{P_l}{\lambda _m} - \left( {{\bar{\Delta }} - {\phi _{vm}}} \right) \left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) } \right] {P_u}}}} \right) {f_{{k_{ll}}}}\left( {\frac{{{\sigma _l}{\phi _{vm}}}}{{{P_l}}}} \right) } \right] \end{aligned} \end{aligned}$$
(79)
and
$$\begin{aligned} \begin{aligned} {{\mathcal {D}}}&= \int \limits _0^{\frac{{{\bar{\Delta }} {\sigma _t}}}{{{P_l}}}} {{F_{{k_{ll}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}x}}{{{\sigma _t} + {\beta _u}{P_l}x}}} \right] \frac{{{\sigma _l}}}{{{P_l}}}} \right) {f_{{k_{lt}}}}\left( x \right) dx} \\&\quad = \frac{{{\bar{\Delta }} {\sigma _t}}}{{2{P_l}}}\sum \limits _{m = 1}^M {\frac{\pi }{M}\sqrt{1 - \psi _m^2} } {F_{{k_{ll}}}}\left( {\left[ {{\bar{\Delta }} - \frac{{{\beta _l}{P_l}{\lambda _m}}}{{{\sigma _t} + {\beta _u}{P_l}{\lambda _m}}}} \right] \frac{{{\sigma _l}}}{{{P_l}}}} \right) \\&\quad {f_{{k_{lt}}}}\left( {{\lambda _m}} \right) \end{aligned} \end{aligned}$$
(80)
with \({\phi _{vm}} = {{{\bar{\Delta }} }} + \frac{{{\beta _l}{P_l}{\lambda _m}\left( {{\eta _v} - 1} \right) }}{{2\left( {{\sigma _t} + {\beta _u}{P_l}{\lambda _m}} \right) }}\).
The last equalities in (79) and (80) come from the Gaussian–Chebyshev quadrature.
