Appendix A: Finding the closed-form of integrals I
1 and I
2
First, we calculate I1 and I2 for the case of single relay and transmitter. By substituting the PDF of g1, g2, and g4 into I1, we obtain:
$$\begin{array}{@{}rcl@{}} {I_{1}} &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}^{\infty} {{\lambda_{4}}{e^{- {\lambda_{4}}{x_{4}}}}} d{x_{4}}d{x_{2}}d{x_{1}}\\ &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} {e^{- \frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}d{x_{2}}d{x_{1}} \end{array} $$
(A.1)
With high γmax, we have \(\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{\max }}}} \to 0\). I1 can be approximately expressed by:
$$\begin{array}{@{}rcl@{}} {I_{1}} &\approx& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{0}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} {e^{- \frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}d{x_{2}}d{x_{1}}\\ &=& \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)2s{\lambda_{2}}\sqrt{\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left[ {2\sqrt{\frac{{{\lambda_{2}}{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right] \end{array} $$
(A.2)
where Eq. A.2 is obtained by using the Eq. (3.324.1) of [17].
Similarly, we have the expression of I2 as:
$$\begin{array}{@{}rcl@{}} {I_{2}} &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}^{\infty} {{\lambda_{4}}{e^{- \left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){x_{4}}}}} d{x_{4}}d{x_{2}}d{x_{1}}\\ &\approx& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{0}^{\infty} {\frac{{s{\lambda_{4}}{\lambda_{2}}}}{{{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}{e^{- {\lambda_{2}}{x_{2}} - \left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}} d{x_{2}}d{x_{1}}\\ &=& \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)\frac{{2{s}{\lambda_{4}}{\lambda_{2}}}}{{{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} \\ &&{K_{1}}\left[ {2\sqrt{\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right] \end{array} $$
(A.3)
Next, for the case of the first relay and transmitter selection strategy, with change the PDF of g2 and CDF of g3 as \({f_{{g_{2}}}}\left ({{x_{2}}} \right ) = K{\lambda _{2}}\sum \limits _{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left ({ - 1} \right )}^{k}}{s^{1 + k}}{e^{- \left ({1 + k} \right ){\lambda _{2}}{x_{2}}}}} \) and \({F_{{g_{3}}}}\left ({{x_{3}}} \right ) = 1 - {e^{- N{\lambda _{3}}{x_{3}}}}\), respectively. We obtain the expressions for I1 and I2 as follows:
$$\begin{array}{@{}rcl@{}} {I_{{{1,\text{RTST1}}}}} &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } {e^{- \frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}d{x_{2}}d{x_{1}}\\ &\approx& 2K{\lambda_{2}}\left( {1 - {e^{- \frac{{{{\lambda_{1}}\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}\sqrt{\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {1 + k} \right)\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } {K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}{\lambda_{4}}{\gamma_{t}}\left( {1 + k} \right)\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ \end{array} $$
(A.4)
$$\begin{array}{@{}rcl@{}} I_{{{2,}}{\text{RTST1}}} \!&=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } \\ &&\int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}^{\infty} {{\lambda_{4}}{e^{- \left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){x_{4}}}}} d{x_{4}}d{x_{2}}d{x_{1}}\\ &\approx& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{0}^{\infty} {\frac{{K{\lambda_{2}}{\lambda_{4}}}}{{{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} {e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}} - \left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}} d{x_{2}}d{x_{1}}\\ &=& \left( \! {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)\frac{{2K{\lambda_{2}}{\lambda_{4}}}}{{{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \sqrt{\left( \!{{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \!\right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} \\ &&{K_{1}}\left[ {2\sqrt{\left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right] \end{array} $$
(A.5)
Next, for the case of the second relay and transmitter selection strategy, with change the PDF of g2 as as \({f_{{g_{2}}}}\left ({{x_{2}}} \right ) = KN{\lambda _{2}}\sum \limits _{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left ({ - 1} \right )}^{k}}{s^{1 + k}}{e^{- \left ({1 + k} \right ){\lambda _{2}}{x_{2}}}}} \). We obtain the expressions for I1 and I2 as follows:
$$\begin{array}{@{}rcl@{}} {I_{{{1}},{\text{RTST2}}}} &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {KN{\lambda_{2}}\sum\limits_{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } {e^{- \frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}d{x_{2}}d{x_{1}}\\ &\approx& 2KN{\lambda_{2}}\left( \! {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \!\right)\sum\limits_{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}\sqrt{\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {1 + k} \right)\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } {K_{1}}\!\left( \!{2\sqrt{\frac{{{\lambda_{2}}{\lambda_{4}}{\gamma_{t}}\left( {1 + k} \right)\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ \end{array} $$
(A.6)
$$\begin{array}{@{}rcl@{}} {I_{2,{\text{RTST2}}}} &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {KN{\lambda_{2}}\sum\limits_{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } \\ &&\int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}^{\infty} {{\lambda_{4}}{e^{- \left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){x_{4}}}}} d{x_{4}}d{x_{2}}d{x_{1}}\\ &\approx& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{0}^{\infty} {\frac{{KN{\lambda_{2}}{\lambda_{4}}}}{{{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sum\limits_{k = 0}^{KN - 1} {C_{KN - 1}^{k}N{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} {e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}} - \left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}} d{x_{2}}d{x_{1}}\\ &=& \left( \!{1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \!\right)\frac{{2KN{\lambda_{2}}{\lambda_{4}}}}{{{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sum\limits_{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \sqrt{\!\left( \! {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} \\ &&{K_{1}}\left[ {2\sqrt{\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right] \end{array} $$
(A.7)
Appendix B: Finding the closed-form of integrals I
3 and I
4
By substituting the PDF of g1, g2, adn g4 into I3, we obtain:
$$\begin{array}{@{}rcl@{}} {I_{3}} &=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}^{\infty} {{\lambda_{4}}{e^{- {\lambda_{4}}{x_{4}}}}} d{x_{4}}d{x_{2}}d{x_{1}}\\ &=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} {e^{- \frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}}d{x_{2}}d{x_{1}} \end{array} $$
(B.1)
By using Eq. (1.211.1) of [17], we have: \({e^{- \frac {{{\lambda _{4}}{\gamma _{t}}\left ({1 + \mu } \right ){x_{1}}}}{{\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right )\eta \rho {\gamma _{p}}{x_{2}}}}}} = \sum \limits _{t = 0}^{\infty } {\frac {{{{\left ({ - 1} \right )}^{t}}{{\left ({\frac {{{\lambda _{4}}{\gamma _{t}}\left ({1 + \mu } \right )}}{{\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right )\eta \rho {\gamma _{p}}}}} \right )}^{t}}}}{{t!}}{{\left ({\frac {{{x_{1}}}}{{{x_{2}}}}} \right )}^{t}}} \)
Then we obtain:
$$\begin{array}{@{}rcl@{}} {I_{3}} &=& s{\lambda_{1}}{\lambda_{2}}\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}}}{{t!}}} \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{{\left( {{x_{1}}} \right)}^{t}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {\frac{{{e^{- {\lambda_{2}}{x_{2}}}}}}{{{{\left( {{x_{2}}} \right)}^{t}}}}d{x_{2}}d{x_{1}}} \\ &\mathop = \limits^{\left( {{\text{B.2.1}}} \right)}& s{\lambda_{1}}{\lambda_{2}}\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {{\lambda_{2}}} \right)}^{t - 1}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}}}{{t!}}} \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{{\left( {{x_{1}}} \right)}^{t}}{e^{- {\lambda_{1}}{x_{1}}}}{\Gamma} \left( {1 - t,{\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}} \right)} d{x_{1}}\\ &\mathop \approx \limits^{\left( {{\text{B.2.2}}} \right)}& s{\lambda_{1}}{\lambda_{2}}\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {{\lambda_{2}}} \right)}^{t - 1}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}}}{{t!}}\frac{{{{\left( {{\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{1 - t}}{\Gamma} \left( 2 \right)}}{{\left( {1 + t} \right){{\left( {{\lambda_{1}} + {\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}{_{2}}{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + {\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right)} \\ &=& \frac{{s{\lambda_{1}}{\lambda_{2}}}}{{{{\left( {{\lambda_{1}} + {\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}{{\left( {\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{1 - t}}}}{{t!\left( {1 + t} \right)}}{_{2}}{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + {\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right)} \end{array} $$
(B.2)
where (B.2.1) and (B.2.2) are obtained because we have \(\int \limits _{{x_{1}}\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{p}}}}}^{\infty } {\frac {{{e^{- {\lambda _{2}}{x_{2}}}}}}{{{{\left ({{x_{2}}} \right )}^{t}}}}d{x_{2}}} = {\left ({{\lambda _{2}}} \right )^{t - 1}}{\Gamma } \left ({1 - t,{\lambda _{2}}\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{p}}}}{x_{1}}} \right )\) and \(\int \limits _{\frac {{{\gamma _{p}}}}{{{\gamma _{\max }}}}}^{\infty } {{{\left ({{x_{1}}} \right )}^{t}}{e^{- {\lambda _{1}}{x_{1}}}}{\Gamma } \left ({1 - t,{\lambda _{2}}\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{p}}}}{x_{1}}} \right )} = \frac {{{{\left ({{\lambda _{2}}\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{p}}}}} \right )}^{1 - t}}{\Gamma } \left (2 \right )}}{{\left ({1 + t} \right ){{\left ({{\lambda _{1}} + {\lambda _{2}}\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{p}}}}} \right )}^{2}}}}{_{2}}{F_{1}}\left ({1,2;t + 2;\frac {{{\lambda _{1}}}}{{{\lambda _{1}} + {\lambda _{2}}\frac {{{\gamma _{t}}\left ({1 - \rho + \mu } \right )}}{{\left ({1 - \rho } \right ){a_{2}}{\gamma _{p}}}}}}} \right )\) using the Eqs. (3.381.3) and (6.455.1) of [17], respectively.
Similarly, we obtain the expression for I4 as
$$\begin{array}{@{}rcl@{}} {I_{4}} \!&=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \frac{{{\lambda_{4}}}}{{{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}{e^{- \left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}}d{x_{2}}d{x_{1}}\\ &=& \frac{{s{\lambda_{1}}{\lambda_{4}}{\lambda_{2}}}}{{\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)}}\!\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}}}{{t!}}{{\!\left[ {\left( \! {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \!\right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right]}^{t}}} \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{{\!\left( {{x_{1}}} \right)}^{t}}{e^{- {\lambda_{1}}{x_{1}}}}} \!\int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {\frac{{{e^{- {\lambda_{2}}{x_{2}}}}}}{{{{\left( {{x_{2}}} \right)}^{t}}}}d{x_{2}}d{x_{1}}} \\ &=& \frac{{s{\lambda_{1}}{\lambda_{4}}\lambda_{2}}}{{\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){{\left( {{\lambda_{1}} + {\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}\\ &&\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left[ {\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right]}^{t}}{{\left( {\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{1 - t}}}}{{t!\left( {1 + t} \right)}}} {_{2}}{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + {\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right) \end{array} $$
(B.3)
Next, for the case of the first relay and transmitter selection strategy, with change the PDF of g2 and CDF of g3 as \({f_{{g_{2}}}}\left ({{x_{2}}} \right ) = K{\lambda _{2}}\sum \limits _{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left ({ - 1} \right )}^{k}}{s^{1 + k}}{e^{- \left ({1 + k} \right ){\lambda _{2}}{x_{2}}}}} \) and \({F_{{g_{3}}}}\left ({{x_{3}}} \right ) = 1 - {e^{- N{\lambda _{3}}{x_{3}}}}\), respectively. We obtain the expressions for I3 and I4 as follows:
$$\begin{array}{@{}rcl@{}} {I_{{{3,}}{\text{RTST1}}{}}} &=& K{\lambda_{1}}{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}}}{{t!}}} \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{{\left( {{x_{1}}} \right)}^{t}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {\frac{{{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}}}{{{{\left( {{x_{2}}} \right)}^{t}}}}d{x_{2}}d{x_{1}}} \\ &\approx& K{\lambda_{1}}{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left[ {\left( {1 + k} \right){\lambda_{2}}} \right]}^{t - 1}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}}}{{t!}}} \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{\left( x_{1} \right)}^{t}}e^{- {\lambda_{1}}{x_{1}}}\\ &&{\Gamma} \left( 1 - t,\frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}} \right) d{x_{1}}\\ &=& K{\lambda_{1}}{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {\frac{{C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{{{\left( {{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}} \sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}{{\left( {\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{1 - t}}}}{{t!\left( {1 + t} \right)}}} {_{2}}\\ &&{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right) \end{array} $$
(B.4)
$$\begin{array}{@{}rcl@{}} {I_{{{4,}}{\text{RTST1}}{}}} &=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{\lambda_{1}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } \\ &&\frac{{{\lambda_{4}}}}{{{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}{e^{- \left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}}d{x_{2}}d{x_{1}}\\ &=& \frac{{K{\lambda_{1}}{\lambda_{4}}{\lambda_{2}}}}{{\left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \\ &&\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}}}{{t!}}{{\left[ {\left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right]}^{t}}} \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{{\left( {{x_{1}}} \right)}^{t}}{e^{- {\lambda_{1}}{x_{1}}}}} \int\limits_{{x_{1}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}^{\infty} {\frac{{{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}}}{{{{\left( {{x_{2}}} \right)}^{t}}}}d{x_{2}}d{x_{1}}} \\ &=& \begin{array}{l} \frac{{K{\lambda_{1}}{\lambda_{4}}{\lambda_{2}}}}{{\left( {{\lambda_{4}} + {N\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)}}\sum\limits_{k = 0}^{K - 1} {\frac{{C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{{{\left( {{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}} \\ \sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left[ {\left( {{\lambda_{4}} + N{\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right]}^{t}}{{\left( {\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{1 - t}}}}{{t!\left( {1 + t} \right)}}} {_{2}}{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right) \end{array} \end{array} $$
(B.5)
Next, for the case of the second relay and transmitter selection strategy, with change the PDF of g2 as as \({f_{{g_{2}}}}\left ({{x_{2}}} \right ) = KN{\lambda _{2}}\sum \limits _{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left ({ - 1} \right )}^{k}}{s^{1 + k}}{e^{- \left ({1 + k} \right ){\lambda _{2}}{x_{2}}}}} \). We obtain the expressions for I3 and I4 as follows:
$$\begin{array}{@{}rcl@{}} {I_{3{{,}}{\text{RTST2}}}} &=& KN{\lambda_{1}}{\lambda_{2}}\sum\limits_{k = 0}^{KN - 1} {\frac{{C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{{{\left( {{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}} \\ &&\sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left( {\frac{{{\lambda_{4}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right)}^{t}}{{\left( {\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{1 - t}}}}{{t!\left( {1 + t} \right)}}} {_{2}}{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right) \end{array} $$
(B.6)
$$\begin{array}{@{}rcl@{}} {I_{{{4}},{\text{RTST2}}}} &=& \frac{{KN{\lambda_{1}}{\lambda_{4}}{\lambda_{2}}}}{{\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)}}\sum\limits_{k = 0}^{KN - 1} {\frac{{C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{{{\left( {{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)}^{2}}}}} \\ &&\times\left( \sum\limits_{t = 0}^{\infty} {\frac{{{{\left( { - 1} \right)}^{t}}{{\left[ {\left( {{\lambda_{4}} + {\lambda_{3}}\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}}}} \right]}^{t}}{{\left( \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}} \right)}^{1 - t}}}}{{t!\left( {1 + t} \right)}}} {_{2}}\right.\\ &&\left.{F_{1}}\left( {1,2;t + 2;\frac{{{\lambda_{1}}}}{{{\lambda_{1}} + \left( {1 + k} \right){\lambda_{2}}\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}} \right) \right) \end{array} $$
(B.7)
Appendix C: Finding the closed-form for Pr(3.1) with signle relay, single transmitter and two relay and transmitter selection strategies
1.1 With single relay (N = 1), single transmitter (K = 1)
The PDF of five RVs g1, g2, g3, g4, and g5 are \(f_{g_{1}}\left (x_{1} \right ) = \lambda _{1}e^{- \lambda _{1}x_{1}}\), \(f_{g_{2}}\left ({{x_{2}}} \right )\)\(= s{\lambda _{2}}{e^{- {\lambda _{2}}{x_{2}}}}\), \({f_{{g_{3}}}}\left ({{x_{3}}} \right ) = {\lambda _{3}}{e^{- {\lambda _{3}}{x_{3}}}}\), \({f_{{g_{4}}}}\left ({{x_{4}}} \right ) = {\lambda _{4}}{e^{- {\lambda _{4}}{x_{4}}}}\), and \({f_{{g_{5}}}}\left ({{x_{5}}} \right ) = {\lambda _{5}}{e^{- {\lambda _{5}}{x_{5}}}}\), respectively.
The term Pr(3.1) in (30) can be expressed into two case of \(\frac {{{\gamma _{p}}}}{{{\gamma _{\max }}}} \geq {g_{1}}\) and \(\frac {{{\gamma _{p}}}}{{{\gamma _{\max }}}} < {g_{1}}\) as follows
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1) \!&=& \underbrace {\text{Pr} \!\left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} \geq {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}\\ {g_{4}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{4}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}}\\ {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{({{b_{1}} - {b_{2}}{\gamma_{t}}})\eta \rho {\gamma_{\max}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]}_{\text{Pr} (3.1.1)} \\ &&+ \underbrace {\text{Pr} \!\left[\! \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} < {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{g_{1}},\\ {g_{4}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{4}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}}\\ {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{{({{b_{1}} - {b_{2}}{\gamma_{t}}})}\eta \rho {\gamma_{p}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \!\right]}_{\text{Pr} (3.1.2)}\\ \end{array} $$
(C.1)
The term Pr(3.1.1) in Eq. C.1 is express as
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.1) &=& \text{Pr} \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} \geq {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}\\ {g_{4}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{4}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}}\\ {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]\\ &=& \underbrace {\text{Pr}\! \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} \geq {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}},\\ {g_{4}} \geq {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]}_{\text{Pr} (3.1.1.1)}\! + \underbrace {\text{Pr} \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} \geq {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}},\\ \frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{g_{2}}}} \le {g_{4}} < {g_{5}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{4}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]}_{\text{Pr} (3.1.1.2)}\\ \end{array} $$
(C.2)
where Pr(3.1.1) can be obtained as follows
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.1.1) &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {{f_{{g_{2}}}}\left( {{x_{2}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}^{\infty} {{f_{{g_{5}}}}\left( {{x_{5}}} \right)} \int\limits_{{x_{5}}}^{\infty} {{f_{{g_{4}}}}\left( {{x_{4}}} \right)} \int\limits_{0}^{\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{x_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} {{f_{{g_{3}}}}\left( {{x_{3}}} \right)} d{x_{3}}d{x_{4}}d{x_{5}}d{x_{2}}d{x_{1}}\\ &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {{f_{{g_{2}}}}\left( {{x_{2}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}^{\infty} {{\lambda_{5}}{e^{- {\lambda_{5}}{x_{5}}}}} {e^{- {\lambda_{4}}{x_{5}}}}\left( {1 - {e^{- \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{x_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{5}}d{x_{2}}d{x_{1}}\\ &=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \left( {\frac{{{\lambda_{5}}{e^{- \frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}{e^{- \frac{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{2}}d{x_{1}}\\ &\approx& \left( \!{1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \!\right)\!\left( \! \begin{array}{l} \frac{{2s{\lambda_{5}}{\lambda_{2}}}}{{{\lambda_{5}} + {\lambda_{4}}}}\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ - \frac{{2s{\lambda_{5}}{\lambda_{2}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right) \end{array} \right)\\ \end{array} $$
(C.3)
The result for Pr(3.1.1.2) is from that of Pr(3.1.1.1) by replace 2 RVs g4 and g5 together, with their PDFs as \({f_{{g_{4}}}}\left ({{x_{4}}} \right ) = {\lambda _{4}}{e^{- {\lambda _{4}}{x_{4}}}}\) and \({f_{{g_{5}}}}\left ({{x_{5}}} \right ) = {\lambda _{5}}{e^{- {\lambda _{5}}{x_{5}}}}\):
$$\begin{array}{@{}rcl@{}} &&\text{Pr} (3.1.1.2) \approx \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)\\ &&\left( {\begin{array}{*{20}{l}} {\frac{{2s{\lambda_{4}}{\lambda_{2}}}}{{{\lambda_{4}} + {\lambda_{5}}}}\sqrt{\frac{{\left( {{\lambda_{4}} + {\lambda_{5}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}\left( {{\lambda_{4}} + {\lambda_{5}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)}\\ { - \frac{{2s{\lambda_{4}}{\lambda_{2}}}}{{{\lambda_{4}} + {\lambda_{5}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\frac{{\left( {{\lambda_{4}} + {\lambda_{5}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{{\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}\left( {{\lambda_{4}} + {\lambda_{5}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)} \end{array}} \right) \end{array} $$
(C.4)
Substituting Eqs. C.3 and C.4 into C.2, we obtain:
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.1) \approx 2s\left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)\sqrt{\frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} \left( {\begin{array}{*{20}{l}} {\sqrt{\left( {{\lambda_{5}} + {\lambda_{4}}} \right)} {K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)}\\ {\begin{array}{*{20}{l}} { - \left( {{\lambda_{5}} + {\lambda_{4}}} \right)\sqrt{\frac{1}{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)}}} }\\ {{K_{1}}\left( {2\sqrt{\frac{{{\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)} \end{array}} \end{array}} \right) \end{array} $$
(C.5)
Next, Pr(3.1.2) is also expressed into sum of two elements Pr(3.1.2.1) and Pr(3.1.2.2) as follows
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.2) &=& \text{Pr} \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} < {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{g_{1}},\\ {g_{4}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{4}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}}\\ {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]\\ &=& \underbrace {\text{Pr} \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} < {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{g_{1}},\\ {g_{4}} \geq {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]}_{\text{Pr} (3.1.2.1)} + \underbrace {{\mathop\mathrm{P}\nolimits} \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} < {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{g_{1}},\\ \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{g_{2}}}} \le {g_{4}} < {g_{5}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{4}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]}_{\text{Pr} (3.1.2.2)}\\ \end{array} $$
(C.6)
where Pr(3.1.2.1) is obtained as
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.2.1) &=& \text{Pr} \left[ \begin{array}{l} \frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}} < {g_{1}},{g_{2}} \geq \frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{g_{1}},\\ {g_{4}} \geq {g_{5}} \geq \frac{{{\gamma_{t}}\left( {1 + \mu } \right){g_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{g_{2}}}},\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{g_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}} \geq {g_{3}} \end{array} \right]\\ &=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}^{\infty} {{f_{{g_{2}}}}\left( {{x_{2}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}},}^{\infty} {{f_{{g_{5}}}}\left( {{x_{5}}} \right)} \int\limits_{{x_{5}}}^{\infty} {{f_{{g_{4}}}}\left( {{x_{4}}} \right)} \int\limits_{0}^{\frac{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{x_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} {{f_{{g_{3}}}}\left( {{x_{3}}} \right)} d{x_{3}}d{x_{4}}d{x_{5}}d{x_{2}}d{x_{1}}\\ &=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}^{\infty} {{f_{{g_{2}}}}\left( {{x_{2}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}^{\infty} {{\lambda_{5}}{e^{- {\lambda_{5}}{x_{5}}}}} {e^{- {\lambda_{4}}{x_{5}}}}\left( {1 - {e^{- \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}{x_{5}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{5}}d{x_{2}}d{x_{1}}\\ &=& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \left( {\frac{{{\lambda_{5}}{e^{- \frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}{e^{- \frac{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right){x_{1}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{p}}{x_{2}}}}}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{2}}d{x_{1}} \end{array} $$
(C.7)
For large γp, we can approximate \({e^{- \frac {{\left ({{\lambda _{5}} + {\lambda _{4}}} \right ){\gamma _{t}}\left ({1 + \mu } \right ){x_{1}}}}{{\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right )\eta \rho {\gamma _{p}}{x_{2}}}}}} \approx 1 - \frac {{\left ({{\lambda _{5}} + {\lambda _{4}}} \right ){\gamma _{t}}\left ({1 + \mu } \right ){x_{1}}}}{{\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right )\eta \rho {\gamma _{p}}{x_{2}}}}\) and \({e^{- \frac {{\left ({{\lambda _{5}} + {\lambda _{4}} + \frac {{{\lambda _{3}}\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right ){\gamma _{p}}}}{{{\gamma _{t}}\left ({1 + \mu } \right )}}} \right ){\gamma _{t}}\left ({1 + \mu } \right ){x_{1}}}}{{\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right )\eta \rho {\gamma _{p}}{x_{2}}}}}} \approx 1 - \frac {{\left ({{\lambda _{5}} + {\lambda _{4}} + \frac {{{\lambda _{3}}\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right ){\gamma _{p}}}}{{{\gamma _{t}}\left ({1 + \mu } \right )}}} \right ){\gamma _{t}}\left ({1 + \mu } \right ){x_{1}}}}{{\left ({{b_{1}} - {b_{2}}{\gamma _{t}}} \right )\eta \rho {\gamma _{p}}{x_{2}}}}\).
Then, we obtain:
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.2.1) &\approx& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}^{\infty} {s{\lambda_{2}}{e^{- {\lambda_{2}}{x_{2}}}}} \left( {\frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{2}}d{x_{1}}\\ &=& \left( {\frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)\int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {s{\lambda_{1}}} {e^{- {\lambda_{1}}{x_{1}}}}{e^{- \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}}d{x_{1}}\\ &=& \left( {\frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)\frac{{s{\lambda_{1}}}}{{{\lambda_{1}} + \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}{e^{- \frac{{\left( {{\lambda_{1}} + \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right){\gamma_{p}}}}{{{\gamma_{\max}}}}}} \end{array} $$
(C.8)
Similarly, we can obtain the term Pr(3.1.2.2) from (C.8) as follows
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.2.2) \approx \left( {\frac{{{\lambda_{4}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{4}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)\frac{{s{\lambda_{1}}}}{{{\lambda_{1}} + \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}{e^{- \frac{{\left( {{\lambda_{1}} + \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right){\gamma_{p}}}}{{{\gamma_{\max}}}}}} \end{array} $$
(C.9)
Substituting Eq. C.8 and C.9 into C.6, we obtain:
$$\begin{array}{@{}rcl@{}} \text{Pr} (3.1.2) \approx \left( {1 - \frac{{{\lambda_{4}} + {\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)\frac{{s{\lambda_{1}}}}{{{\lambda_{1}} + \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}}{e^{- \frac{{\left( {{\lambda_{1}} + \frac{{{\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right){\gamma_{p}}}}{{{\gamma_{\max}}}}}} \end{array} $$
(C.10)
Finally, Pr(3.1) is derived by combining Eq. C.1, C.5, and C.10.
With RTST1
The PDF of g2 and CDF of g3 are chaged as \({f_{{g_{2}}}}\left ({{x_{2}}} \right ) = K{\lambda _{2}}\sum \limits _{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left ({ - 1} \right )}^{k}}{s^{1 + k}}{e^{- \left ({1 + k} \right ){\lambda _{2}}{x_{2}}}}} \) and \({F_{{g_{3}}}}\left ({{x_{3}}} \right ) = 1 - {e^{- N{\lambda _{3}}{x_{3}}}}\), respectively.
We can obtain Pr(3.1.1.1) and Pr(3.1.1.2) with this strategy from C.3 and C.4 as follows
$$\begin{array}{@{}rcl@{}} \text{Pr} {(3.1.1.1)_{{\text{RTST1}}}} \!&=& \int\limits_{0}^{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{\max}}}}}^{\infty} {K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } \\ &&\left( {\frac{{{\lambda_{5}}{e^{- \frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}{e^{- \frac{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}{x_{2}}}}}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{2}}d{x_{1}}\\ &\approx& \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \\ &&\left( \!\!\begin{array}{l} \frac{{2{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}}\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ - \frac{{2{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( \! {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ \end{array} \!\right)\\ \end{array} $$
(C.11)
$$\begin{array}{@{}rcl@{}} \text{Pr} {(3.1.1.2)_{{\text{RTST1}}}} &\approx& \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \\ &&\left( \!\! \begin{array}{l} \frac{{2{\lambda_{4}}}}{{{\lambda_{5}} + {\lambda_{4}}}}\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ - \frac{{2{\lambda_{4}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( \!{2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right) \end{array} \!\right)\!\\ \end{array} $$
(C.12)
Then Pr(3.1.1)RTST1 is obtain by Pr(3.1.1)RTST1 = Pr(3.1.1.1)RTST1 + Pr(3.1.1.2)RTST1:
$$\begin{array}{@{}rcl@{}} &&\text{Pr} {(3.1.1)_{{\text{RTST1}}}} \approx \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} \\ &&\left( \begin{array}{l} 2\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)\\ - \frac{{2\left( {{\lambda_{5}} + {\lambda_{4}}} \right)}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\eta \rho {\gamma_{\max}}}}} \\ {K_{1}}\left( {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right) \end{array} \right) \end{array} $$
(C.13)
From Eq. C.8, we can obtain Pr(3.1.2.2)RTST1:
$$\begin{array}{@{}rcl@{}} \text{Pr} {(3.1.2.1)_{{\text{RTST1}}}} &\approx& \int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {{f_{{g_{1}}}}\left( {{x_{1}}} \right)} \int\limits_{\frac{{{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}^{\infty} {K{\lambda_{2}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}{e^{- \left( {1 + k} \right){\lambda_{2}}{x_{2}}}}} } \\ &&\left( {\frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)d{x_{2}}d{x_{1}}\\ &=& \left( {\frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)\int\limits_{\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}^{\infty} {K{\lambda_{1}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}\frac{{{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{\left( {1 + k} \right)}}} } {e^{- {\lambda_{1}}{x_{1}}}}{e^{- \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}{x_{1}}}}d{x_{1}}\\ &=& \left( {\frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)K{\lambda_{1}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}\frac{{{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{\left( {1 + k} \right)}}} \frac{{{e^{- \left( {{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}}}}{{{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}} \end{array} $$
(C.14)
And Pr(3.1.2.2)RTST1 is obtained from Pr(3.1.2.1)RTST1 with change the PDF of RVs g4 and g5 together:
$$\begin{array}{@{}rcl@{}} \text{Pr} {(3.1.2.2)_{{\text{RTST1}}}} \approx \left( {\frac{{{\lambda_{4}}}}{{{\lambda_{5}} + {\lambda_{4}}}} - \frac{{{\lambda_{4}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)K{\lambda_{1}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}\frac{{{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{\left( {1 + k} \right)}}} \frac{{{e^{- \left( {{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}}}}{{{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}} \end{array} $$
(C.15)
Then, Pr(3.1.2)RTST1 is obtained by Pr(3.1.2)RTST1 = Pr(3.1.2.1)RTST1 + Pr(3.1.2.2)RTST1:
$$\begin{array}{@{}rcl@{}} \text{Pr} {(3.1.2)_{{\text{RTST1}}}} = \left( {1 - \frac{{{\lambda_{4}} + {\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)K{\lambda_{1}}\sum\limits_{k = 0}^{K - 1} {C_{K - 1}^{k}\frac{{{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{\left( {1 + k} \right)}}} \frac{{{e^{- \left( {{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}}}}{{{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}} \end{array} $$
(C.16)
Finally, Pr(3.1)RTST1 is obtained by summing of Pr(3.1.1)RTST1 in (C.13) and Pr(3.1.2)RTST1 in (C.16).
With RTST2
only the PDF of g2 is changed as \({f_{{g_{2}}}}\left ({{x_{2}}} \right ) = KN{\lambda _{2}}\sum \limits _{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left ({ - 1} \right )}^{k}}{s^{1 + k}}{e^{- \left ({1 + k} \right ){\lambda _{2}}{x_{2}}}}} \). Wealso obtain Pr(3.1)RTST2 by summing of Pr(3.1.1)RTST2 in Eq. C.13 and Pr(3.1.2)RTST2 in Eq. C.16, where Pr(3.1.1)RTST2 and Pr(3.1.2)RTST2 are derived from.
We can obtain the results for
$$\begin{array}{@{}rcl@{}} &&{\text{Pr} {{(3.1.1)}_{{\text{RTST2}}}} \approx \left( {1 - {e^{- \frac{{{\lambda_{1}}{\gamma_{p}}}}{{{\gamma_{\max}}}}}}} \right)KN{\lambda_{2}}\sum\limits_{k = 0}^{KN - 1} {C_{KN - 1}^{k}{{\left( { - 1} \right)}^{k}}{s^{1 + k}}} }\\ &&{\left( {\begin{array}{l} {2\sqrt{\frac{{\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} {K_{1}}\left( {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)}\\ { - \frac{{2\left( {{\lambda_{5}} + {\lambda_{4}}} \right)}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{N{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}\sqrt{\frac{{{\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {1 + k} \right){\lambda_{2}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right)\eta \rho {\gamma_{\max}}}}} }\\ {{K_{1}}\left( {2\sqrt{\frac{{\left( {1 + k} \right){\lambda_{2}}\left( {{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}} \right){\gamma_{t}}\left( {1 + \mu } \right)}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{\max}}}}} } \right)} \end{array}} \right)} \end{array} $$
(C.17)
$$\begin{array}{@{}rcl@{}} \text{Pr} {(3.1.2)_{{\text{RTST2}}}} = \left( {1 - \frac{{{\lambda_{4}} + {\lambda_{5}}}}{{{\lambda_{5}} + {\lambda_{4}} + \frac{{{\lambda_{3}}\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{p}}}}{{{\gamma_{t}}\left( {1 + \mu } \right)}}}}} \right)KN{\lambda_{1}}\sum\limits_{k = 0}^{KN - 1} {C_{K - 1}^{k}\frac{{{{\left( { - 1} \right)}^{k}}{s^{1 + k}}}}{{\left( {1 + k} \right)}}} \frac{{{e^{- \left( {{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}} \right)\frac{{{\gamma_{p}}}}{{{\gamma_{\max}}}}}}}}{{{\lambda_{1}} + \frac{{\left( {1 + k} \right){\lambda_{2}}{\gamma_{t}}\left( {1 - \rho + \mu } \right)}}{{\left( {1 - \rho } \right){a_{2}}{\gamma_{p}}}}}} \end{array} $$
(C.18)
And Pr(3.1)RTST2 is obtained by Pr(3.1)RTST2 = Pr(3.1.1)RTST2 + Pr(3.1.2)RTST2.