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New look on relay selection strategies for full-duplex multiple-relay NOMA over Nakagami-m fading channels

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Abstract

By removing the orthogonal use of radio-resources, non-orthogonal multiple access (NOMA) has been introduced to improve the spectral efficiency of fifth generation (5G) and beyond networks. This paper studies the system performance in a dual-hop multi-relay NOMA using decode-and-forward (DF) scheme over Nakagami-m fading channels. A group of NOMA users is considered, i.e. the near and far users which are decided by how strong these related channels are. Specifically, we obtain a closed-form expression of the outage probability of the near/far NOMA users when the several relay selection schemes are adopted for selecting the best among M intermediate relays. As main finding, this paper introduces three strategies including two-stage relay selection, max-min and power allocation based relay selection schemes. As main benefit, the NOMA users are considered to employ selection combining technique in order to improve signal transmissions for an increased reliability in the context of massive connections in 5G wireless communications. By conducting numerical simulations, we evaluate the impact of the number of intermediate relays, the NOMA power allocation factor, and the Nakagami-m fading severity parameter on the outage performance of the NOMA users. Finally, the outage probability along with throughout in delay-limited transmission mode are provided via numerical results and the necessary comparisons are provided.

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Authors and Affiliations

  1. Faculty of Electronics Technology, Industrial University of Ho Chi Minh City (IUH), Ho Chi Minh City, Vietnam

    Tu-Trinh Thi Nguyen

  2. Applied Network Technology Laboratory, Department of Computer Science, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

    Chakchai So-In

  3. School of Mathematics and Computer Science, University of Wolverhampton, WV1 1LY, Wolverhampton, United Kingdom

    Md Arafatur Rahman

  4. Department of Computer Science and Information Engineering, College of Information and Electrical Engineering, Asia University, Taichung, 41354, Taiwan

    Dinh-Thuan Do & Yeong-Chin Chen

Authors
  1. Tu-Trinh Thi Nguyen
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  2. Dinh-Thuan Do
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  3. Yeong-Chin Chen
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  4. Chakchai So-In
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Correspondence to Dinh-Thuan Do.

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Appendices

Appendix A: Proof of Lemma 1

It is straight forward to obtain Lemma 1 as below

$$\begin{aligned} \begin{aligned} \varPhi \left( {a,b,k} \right) =&\Pr \left( {a{{\left| {{h_k}} \right| }^2} \ge b{{\left| {{f_k}} \right| }^2} + 1} \right) \\ =&\int \limits _0^\infty {\left( {1 - {F_{{{\left| {{h_k}} \right| }^2}}}\left( {\frac{{bx + 1}}{a}} \right) } \right) {f_{{{\left| {{f_k}} \right| }^2}}}\left( x \right) dx} \\ =&\frac{{\exp \left( {-1} \big / {\left( {a{\beta _h}} \right) } \right) }}{{\varGamma \left( {{m_f}} \right) \beta _f^{{m_f}}}}\sum \limits _{n = 0}^{{m_h} - 1} {\frac{1}{{\beta _h^n{a^n}}}} \int \limits _0^\infty {\frac{{{{\left( {bx + 1} \right) }^n}{x^{{m_f} - 1}}}}{{n!}}{{\mathrm{e}} ^{ - x\left( {\frac{1}{{{\beta _f}}} + \frac{b}{{a{\beta _h}}}} \right) }}dx}. \end{aligned} \end{aligned}$$
(47)

It can be simplified \( \varPhi \left( {a,b,k} \right) \) as below

$$\begin{aligned} \begin{aligned} \varPhi \left( {a,b,k} \right) =&\frac{\exp \left( { - 1} \big / {\left( {a{\beta _h}} \right) } \right) }{\varGamma \left( {{m_f}} \right) \beta _f^{{m_f}}}\sum \limits _{n = 0}^{{m_h} - 1} {\sum \limits _{k = 0}^n {\left( \begin{array}{c} n\\ k \end{array} \right) } \frac{1}{{\beta _h^n{a^n}}}\frac{{{b^k}}}{{n!}}} \int \limits _0^\infty {{x^{k + {m_f} - 1}}{{\mathrm{e}} ^{ - x\left( {\frac{1}{{{\beta _f}}} + \frac{b}{{a{\beta _h}}}} \right) }}dx}\\ =&\frac{\exp \left( - 1 \big / \left( {a{\beta _h}} \right) \right) }{\varGamma \left( {{m_f}} \right) \beta _f^{{m_f}}}\sum \limits _{n = 0}^{{m_h} - 1} {\sum \limits _{k = 0}^n {\left( \begin{array}{c} n\\ k \end{array} \right) } \frac{1}{{\beta _h^n{a^n}}}} \frac{{{b^k}}}{{n!}}{\left( {\frac{1}{{{\beta _f}}} + \frac{b}{{a{\beta _h}}}} \right) ^{ - k - {m_f}}}\varGamma \left( {k + {m_f}} \right) . \end{aligned} \end{aligned}$$
(48)

The last step can be obtained by applying (Eq. 1.111) in [39], and the integration is derived thank to (Eq. 3.381.4) in [39]. Thus, the lemma is proved.

Appendix B: Proof of Proposition 1

Based on the definition of MM mode using (18), the system outage is given by (49), shown in the top of the next page.

$$\begin{aligned} \begin{aligned} O{P_{MM}} =&\Pr \left( {\min \left( {\frac{{{\gamma _{{r_{{b_{MM}}}} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_{{b_{MM}}}},{d_1}}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_{{b_{MM}}}},{d_2} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_{{b_{MM}}}} \leftarrow 2}}}}{{\delta _2^{\mathrm{th}}}},\frac{{{\gamma _{{r_{{b_{MM}}}},{d_2}}}}}{{\delta _2^{\mathrm{th}}}}} \right)< 1} \right) \\ =&\Pr \left( {{\max }_{k \in {{\mathcal S}_{\mathcal R}}} \left( {\min \left( {\frac{{{\gamma _{{r_k} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_1}}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k} \leftarrow 2}}}}{{\delta _2^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2}}}}}{{\delta _2^{\mathrm{th}}}}} \right) } \right) < 1} \right) . \end{aligned} \end{aligned}$$
(49)

Additionally, due to the independence of channels, (49) can be simplified as

$$\begin{aligned} \begin{aligned} O{P_{MM}} =&\prod \limits _{k = 1}^M {\Pr \left( {\min \left( {\frac{{{\gamma _{{r_k} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_1}}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k} \leftarrow 2}}}}{{\delta _2^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2}}}}}{{\delta _2^{\mathrm{th}}}}} \right) < 1} \right) } \\ =&\prod \limits _{k = 1}^M {1 - \Pr \left( {\min \left( {\frac{{{\gamma _{{r_k} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_1}}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k} \leftarrow 2}}}}{{\delta _2^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2}}}}}{{\delta _2^{\mathrm{th}}}}} \right) \ge 1} \right) } \\ =&\prod \limits _{k = 1}^M {1 - \Pr \left( {\frac{{{\gamma _{{r_k} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}} \ge 1,\frac{{{\gamma _{{r_k},{d_1}}}}}{{\delta _1^{\mathrm{th}}}} \ge 1,\frac{{{\gamma _{{r_k},{d_2} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}} \ge 1,\frac{{{\gamma _{{r_k} \leftarrow 2}}}}{{\delta _2^{\mathrm{th}}}} \ge 1,\frac{{{\gamma _{{r_k},{d_2}}}}}{{\delta _2^{\mathrm{th}}}} \ge 1} \right) }. \end{aligned} \end{aligned}$$
(50)

The expression in multiplication can be further expressed as (51).

$$\begin{aligned} \begin{aligned} \Pr \left( {\min \left( {\frac{{{\gamma _{{r_k} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k} \leftarrow 2}}}}{{\delta _2^{\mathrm{th}}}}} \right) \ge 1} \right) \Pr \left( {\frac{{{\gamma _{{r_k},{d_1}}}}}{{\delta _1^{\mathrm{th}}}} \ge 1} \right) \Pr \left( {\min \left( {\frac{{{\gamma _{{r_k},{d_2} \leftarrow 1}}}}{{\delta _1^{\mathrm{th}}}},\frac{{{\gamma _{{r_k},{d_2}}}}}{{\delta _2^{\mathrm{th}}}}} \right) \ge 1} \right) . \end{aligned} \end{aligned}$$
(51)

Finally, with the same step in (45), the Theorem 1 is derived.

This completes the proof.

Appendix C: Proof of Proposition 3

The above probability \({\eta _1}\) can be calculated as

$$\begin{aligned} \begin{aligned} {\eta _1} =&\Pr \left\{ {\min \left\{ {\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }},\max \left( {0,\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }}} \right) } \right\} \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}:{r_b} \in {{\mathcal B}_{\mathcal R}}} \right\} \\ =&\Pr \left\{ {{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}> \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}},\min \left\{ {\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }},\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }}} \right\} \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}:{r_b} \in {{\mathcal B}_{\mathcal R}}} \right\} \\&+ \underbrace{\Pr \left\{ {{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \le \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}},0 \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}:{r_b} \in {{\mathcal B}_{\mathcal R}}} \right\} }_{ = 0} \\ =&\Pr \left\{ {{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}},\underbrace{\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }} \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}}_{ \buildrel \varDelta \over = {\varpi _1}},\underbrace{\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }} \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}}_{ \buildrel \varDelta \over = {\varpi _2}}:{r_b} \in {{\mathcal B}_{\mathcal R}}} \right\} . \end{aligned} \end{aligned}$$
(52)

With the condition \({\rho _r}{\left| {{g_{{r_b},{d_2}}}} \right| ^2} - \delta _1^{\mathrm{th}} > 0\) \( \Leftrightarrow {\left| {{g_{{r_b},{d_2}}}} \right| ^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}\) \( \Rightarrow \max \left( {0,\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }}} \right) = \frac{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }}\).

We calculate \({\varpi _1}\)

$$\begin{aligned} \begin{aligned} {\varpi _1} =&\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }} \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\\ =&{\left| {{g_{{r_b},{d_1}}}} \right| ^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _2^{\mathrm{th}}\left( {1 + \delta _1^{\mathrm{th}}} \right) }}\\ =&{\left| {{g_{{r_b},{d_1}}}} \right| ^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}}, \end{aligned} \end{aligned}$$
(53)

where \({\rho _r}{\left| {{g_{{r_b},{d_1}}}} \right| ^2} - \delta _1^{\mathrm{th}} > 0\) \( \Leftrightarrow {\left| {{g_{{r_b},{d_1}}}} \right| ^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}\) and \({\varepsilon _1} = \delta _2^{\mathrm{th}}\left( {1 + \delta _1^{\mathrm{th}}} \right) \).

We calculate \({\varpi _2}\)

$$\begin{aligned} \begin{aligned} {\varpi _2} =&\frac{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - \delta _1^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}\left( {1 + \delta _1^{\mathrm{th}}} \right) }} \ge \frac{{\delta _2^{\mathrm{th}}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\\ =&{\left| {{g_{{r_b},{d_2}}}} \right| ^2} \ge \frac{{\delta _2^{\mathrm{th}}\left( {1 + \delta _1^{\mathrm{th}}} \right) + \delta _1^{\mathrm{th}}}}{{{\rho _r}}}\\ =&{\left| {{g_{{r_b},{d_2}}}} \right| ^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}, \end{aligned} \end{aligned}$$
(54)

where \(\left\{ {\begin{array}{*{20}{c}} {{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}> \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}}\\ {{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}} \end{array}} \right. \Leftrightarrow {\left| {{g_{{r_b},{d_2}}}} \right| ^2} > \max \left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}},\frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right) \) \( \Rightarrow {\left| {{g_{{r_b},{d_2}}}} \right| ^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}\) and \({\varepsilon _2} = {\varepsilon _1} + \delta _1^{\mathrm{th}}\).

From \({\varpi _1}\) and \({\varpi _2}\), we have new result as follow

$$\begin{aligned} \begin{aligned} {\eta _1} = \Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}:{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} . \end{aligned} \end{aligned}$$
(55)

We apply the Bayes theorem: \({\mathrm{P}}\left( {{\mathrm{A'}}\left| {\mathrm{C}} \right. } \right) {\mathrm{= }}\frac{{{\mathrm{P}}\left( {{\mathrm{A'}},{\mathrm{C}}} \right) }}{{{\mathrm{P}}\left( {\mathrm{C}} \right) }}\), where \( {\mathrm{P}}\left( {{\mathrm{A',C}}} \right) \) is computed as follow, while \({\mathrm{P}}\left( {\mathrm{C}} \right) = \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} \).

$$\begin{aligned} \begin{aligned} {\mathrm{P}}\left( {{\mathrm{A',C}}} \right) = \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}},{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} . \end{aligned} \end{aligned}$$
(56)

It is worth noting that \( {\left| {{g_{{r_b},{d_1}}}} \right| ^2} \ge {\mathrm{}}\frac{{\delta _1^{{\mathrm{th}}}}}{{{\rho _r} - \frac{{{\varepsilon _1}}}{{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}}} \) because \(\frac{{{\varepsilon _1}}}{{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}} \ge 0\).

$$\begin{aligned} \begin{aligned} \Rightarrow {\rho _r} - \frac{{{\varepsilon _1}}}{{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}} \le {\rho _r} \Leftrightarrow \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}\\ \Rightarrow \max \left( {\frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) = \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}}. \end{aligned} \end{aligned}$$
(57)

After the analysis, \({\eta _1}\) is calculated by

$$\begin{aligned} \begin{aligned} {\eta _1} =&\frac{{{\mathrm{P}}\left( {{\mathrm{A'}},{\mathrm{C}}} \right) }}{{{\mathrm{P}}\left( {\mathrm{C}} \right) }}\\ =&\frac{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}},{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}> \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} }}{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}> \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} }}\\ =&\frac{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} \ge \max \left\{ {\frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} ,{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right\} }}{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}> \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} }}\\ =&\frac{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right\} }}{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} }}. \end{aligned} \end{aligned}$$
(58)

We have

$$\begin{aligned} {\eta _1} =&\frac{{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} \ge \frac{{\delta _1^{\mathrm{th}}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}{{{\rho _r}{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} - {\varepsilon _1}}},{{\left| {{g_{{r_b},{d_2}}}} \right| }^2} \ge \frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right\} }}{{1 - {F_{{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) }}. \end{aligned}$$
(59)

Now, we have new variables as \(X = {\left| {{g_{{r_b},{d_1}}}} \right| ^2},Y = {\left| {{g_{{r_b},{d_2}}}} \right| ^2}\). Next, \({\eta _1}\) is calculated as follow

$$\begin{aligned} \begin{aligned} {\eta _1} =&\frac{{\int _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\left( {1 - {F_X}\left( {\frac{{\delta _1^{\mathrm{th}}Y}}{{{\rho _r}Y - {\varepsilon _1}}}} \right) } \right) {f_Y}\left( y \right) dy} }}{{1 - {F_{{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) }}\\ =&\frac{{{\varPhi _1}}}{{{\varPhi _2}}} \end{aligned} \end{aligned}$$
(60)

It is necessary to calculate \({\varPhi _1}\) and \({\varPhi _2}\) respectively as bellows. By using partial integration, we have

$$\begin{aligned} \begin{aligned} {\varPhi _1} =&\int _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\left( {1 - {F_X}\left( {\frac{{\delta _1^{\mathrm{th}}Y}}{{{\rho _r}Y - {\varepsilon _1}}}} \right) } \right) {f_Y}\left( y \right) dy} \\ =&\int _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\left( {1 - {F_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) } \right) {f_{{{\left| {{g_{{r_k},{d_2}}}} \right| }^2}}}\left( y \right) dy}. \end{aligned} \end{aligned}$$
(61)

Then, \( {\varPhi _1} \) is rewritten as

$$\begin{aligned} \begin{aligned} {\varPhi _1} =&\underbrace{\left( {1 - {F_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) } \right) {F_{{{\left| {{g_{{r_k},{d_2}}}} \right| }^2}}}\left( y \right) }_{ \buildrel \varDelta \over = {\varphi _1}}\\&- \underbrace{\int \limits _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\frac{{{\varepsilon _1}\delta _1^{\mathrm{th}}}}{{{{\left( {{\rho _r}y - {\varepsilon _1}} \right) }^2}}}{F_{{{\left| {{g_{{r_k},{d_2}}}} \right| }^2}}}\left( y \right) {f_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) dy} }_{ \buildrel \varDelta \over = {\varphi _2}}, \end{aligned} \end{aligned}$$
(62)

where \( {\varphi _1} \) and \({\varphi _2}\) are computed as follows

$$\begin{aligned}&\begin{aligned} {\varphi _1} =&\left( {1 - {F_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) } \right) {F_{{{\left| {{g_{{r_k},{d_2}}}} \right| }^2}}}\left( y \right) |\begin{array}{*{20}{c}} \infty \\ {\frac{{{\varepsilon _2}}}{{{\rho _r}}}} \end{array}\\ =&\left( {1 - {F_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) } \right) - \left( {1 - {F_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{{\varepsilon _2}\delta _1^{\mathrm{th}}}}{{{\rho _r}\left( {{\varepsilon _2} - {\varepsilon _1}} \right) }}} \right) } \right) \\&\times {\left( {1 - {{\mathrm{e}} ^{ - \frac{{{\varepsilon _2}}}{{{\beta _{{g_{{r_k},{d_2}}}}}{\rho _r}}}}}\sum \limits _{{n_1} = 0}^{{m_{{g_{{r_k},{d_2}}}}} - 1} {\frac{1}{{{n_1}!\beta _{{g_{{r_k},{d_2}}}}^{{n_1}}}}} {{\left( {\frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right) }^{{n_1}}}} \right) ^i}\\ =&{{\mathrm{e}} ^{ - \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_k},{d_1}}}}}}}}}{\sum \limits _{n = 0}^{{m_{{g_{{r_k},{d_1}}}}} - 1} {\frac{1}{{n!\beta _{{g_{{r_k},{d_1}}}}^n}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) } ^n}\\&- {{\mathrm{e}} ^{ - \frac{{{\varepsilon _2}\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_k},{d_1}}}}}\left( {{\varepsilon _2} - {\varepsilon _1}} \right) }}}}\sum \limits _{n = 0}^{{m_{{g_{{r_b},{d_1}}}}} - 1} {\frac{1}{{n!\beta _{{g_{{r_b},{d_1}}}}^n}}} {\left( {\frac{{{\varepsilon _2}\delta _1^{\mathrm{th}}}}{{{\rho _r}\left( {{\varepsilon _2} - {\varepsilon _1}} \right) }}} \right) ^n}\\&\times {\left( {1 - {{\mathrm{e}} ^{ - \frac{{{\varepsilon _2}}}{{{\beta _{{g_{{r_k},{d_2}}}}}{\rho _r}}}}}\sum \limits _{{n_1} = 0}^{{m_{{g_{{r_k},{d_2}}}}} - 1} {\frac{1}{{{n_1}!\beta _{{g_{{r_k},{d_2}}}}^{{n_1}}}}} {{\left( {\frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right) }^{{n_1}}}} \right) ^i}. \end{aligned} \end{aligned}$$
(63)
$$\begin{aligned}&\begin{aligned} {\varphi _2} =&\int \limits _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\frac{{{\varepsilon _1}\delta _1^{\mathrm{th}}}}{{{{\left( {{\rho _r}y - {\varepsilon _1}} \right) }^2}}}{F_{{{\left| {{g_{{r_k},{d_2}}}} \right| }^2}}}\left( y \right) {f_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) dy} \\ =&\frac{1}{{\varGamma \left( {{m_{{g_{{r_k},{d_1}}}}}} \right) \beta _{{g_{{r_k},{d_1}}}}^{{m_{{g_{{r_k},{d_1}}}}}}}}\int \limits _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\frac{{{\varepsilon _1}\delta _1^{\mathrm{th}}}}{{{{\left( {{\rho _r}y - {\varepsilon _1}} \right) }^2}}}{{\left( {1 - {{\mathrm{e}} ^{ - \frac{y}{{{\beta _{{g_{{r_k},{d_2}}}}}}}}}\sum \limits _{{n_2} = 0}^{{m_{{g_{{r_k},{d_2}}}}} - 1} {\frac{1}{{{n_2}!\beta _{{g_{{r_k},{d_2}}}}^{{n_2}}}}} {y^{{n_2}}}} \right) }^i}} \\&\times {\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) ^{{m_{{g_{{r_k},{d_1}}}}} - 1}}{{\mathrm{e}} ^{ - \frac{{\delta _1^{\mathrm{th}}y}}{{{\beta _{{g_{{r_k},{d_1}}}}}\left( {{\rho _r}y - {\varepsilon _1}} \right) }}}}dy. \end{aligned} \end{aligned}$$
(64)

Since the CDF of \({\left| {{g_{{r_b},{d_2}}}} \right| ^2}\), e.g. , \({F_Y}\left( y \right) = {\left( {{F_{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\left( y \right) } \right) ^i}\), then it can be achieved PDF as follow

$$\begin{aligned} \begin{aligned} {f_{{{\left| {{g_{{r_k},{d_2}}}} \right| }^2}}}\left( y \right) =&{\left( {{{\left( {{F_{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\left( y \right) } \right) }^i}} \right) ^\prime }\\ =&i \times {\left( {{F_{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\left( y \right) } \right) ^{i - 1}}{\left( {{F_{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\left( y \right) } \right) ^\prime }\\ =&i \times {\left( {{F_{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\left( y \right) } \right) ^{i - 1}}{f_{{{\left| {{g_{{r_b},{d_2}}}} \right| }^2}}}\left( y \right) \\ =&i {\left( {1 - {{\mathrm{e}} ^{ - \frac{y}{{{\beta _{{g_{{r_k},{d_2}}}}}}}}}\sum \limits _{n = 0}^{{m_{{g_{{r_k},{d_2}}}}} - 1} {\frac{1}{{n!\beta _{{g_{{r_k},{d_2}}}}^n}}} {y^n}} \right) ^{i - 1}}\frac{{{y^{{m_{{g_{{r_b},{d_2}}}}} - 1}}}}{{\varGamma \left( {{m_{{g_{{r_b},{d_2}}}}}} \right) \beta _{{g_{{r_b},{d_2}}}}^{{m_{{g_{{r_b},{d_2}}}}}}}}{{\mathrm{e}} ^{ - \frac{y}{{{\beta _{{g_{{r_b},{d_2}}}}}}}}}. \end{aligned} \end{aligned}$$
(65)

Finally, we have

$$\begin{aligned} \begin{aligned} {\varPhi _1} =&{\varphi _1} - {\varphi _2}\\ =&\exp \left( { - \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_k},{d_1}}}}}}}} \right) {\sum \limits _{n = 0}^{{m_{{g_{{r_k},{d_1}}}}} - 1} {\frac{1}{{n!\beta _{{g_{{r_k},{d_1}}}}^n}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) } ^n}\\&- \exp \left( { - \frac{{{\varepsilon _2}\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_k},{d_1}}}}}\left( {{\varepsilon _2} - {\varepsilon _1}} \right) }}} \right) \sum \limits _{n = 0}^{{m_{{g_{{r_b},{d_1}}}}} - 1} {\frac{1}{{n!\beta _{{g_{{r_b},{d_1}}}}^n}}} {\left( {\frac{{{\varepsilon _2}\delta _1^{\mathrm{th}}}}{{{\rho _r}\left( {{\varepsilon _2} - {\varepsilon _1}} \right) }}} \right) ^n}\\&\times {\left( {1 - \exp \left( { - \frac{1}{{{\beta _{{g_{{r_k},{d_2}}}}}}}\frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right) \sum \limits _{{n_1} = 0}^{{m_{{g_{{r_k},{d_2}}}}} - 1} {\frac{1}{{{n_1}!\beta _{{g_{{r_k},{d_2}}}}^{{n_1}}}}} {{\left( {\frac{{{\varepsilon _2}}}{{{\rho _r}}}} \right) }^{{n_1}}}} \right) ^i}\\&- \frac{1}{{\varGamma \left( {{m_{{g_{{r_k},{d_1}}}}}} \right) \beta _{{g_{{r_k},{d_1}}}}^{{m_{{g_{{r_k},{d_1}}}}}}}}\int \limits _{\frac{{{\varepsilon _2}}}{{{\rho _r}}}}^\infty {\frac{{{\varepsilon _1}\delta _1^{\mathrm{th}}}}{{{{\left( {{\rho _r}y - {\varepsilon _1}} \right) }^2}}}{{\left( {1 - {{\mathrm{e}} ^{ - \frac{y}{{{\beta _{{g_{{r_k},{d_2}}}}}}}}}\sum \limits _{{n_2} = 0}^{{m_{{g_{{r_k},{d_2}}}}} - 1} {\frac{1}{{{n_2}!\beta _{{g_{{r_k},{d_2}}}}^{{n_2}}}}} {y^{{n_2}}}} \right) }^i}} \\&\times {\left( {\frac{{\delta _1^{\mathrm{th}}y}}{{{\rho _r}y - {\varepsilon _1}}}} \right) ^{{m_{{g_{{r_k},{d_1}}}}} - 1}}\exp \left( { - \frac{{\delta _1^{\mathrm{th}}y}}{{{\beta _{{g_{{r_k},{d_1}}}}}\left( {{\rho _r}y - {\varepsilon _1}} \right) }}} \right) dy. \end{aligned} \end{aligned}$$
(66)

and

$$\begin{aligned} \begin{aligned} {\varPhi _2} =&1 - {F_{{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) \\ =&1 - 1 - \frac{1}{{\varGamma \left( {{m_{{g_{{r_b},{d_1}}}}}} \right) }}\varGamma \left( {{m_{{g_{{r_b},{d_1}}}}},\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_b},{d_1}}}}}}}} \right) \\ =&\frac{1}{{\varGamma \left( {{m_{{g_{{r_b},{d_1}}}}}} \right) }}\varGamma \left( {{m_{{g_{{r_b},{d_1}}}}},\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_b},{d_1}}}}}}}} \right) \\ =&{{\mathrm{e}} ^{ - \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_k},{d_1}}}}}}}}}\sum \limits _{{n_3} = 0}^{{m_{{g_{{r_k},{d_1}}}}} - 1} {\frac{{{{\left( {\delta _1^{\mathrm{th}}} \right) }^{{n_3}}}}}{{{n_3}!\rho _r^{{n_3}}\beta _{{g_{{r_k},{d_1}}}}^{{n_3}}}}}. \end{aligned} \end{aligned}$$
(67)

Substituting (66) and (67) into (60), \( \eta _1 \) can be computed.

Defining \({\varOmega _\theta }\) as the probability that the relay \({r_i}\) is in the active relay set \({{\mathcal B}_{\mathcal R}}\), the \({\varOmega _\theta }\) can be mathematically formulated as

$$\begin{aligned} \begin{aligned} {\varOmega _\theta } \buildrel \varDelta \over =&\Pr \left( {{r_i} \in {{\mathcal B}_{\mathcal R}}} \right) \\ =&\Pr \left\{ {{{\left| {{h_i}} \right| }^2} \ge \chi ,{{\left| {{g_{{r_b},{d_1}}}} \right| }^2}> \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} \\ =&\underbrace{\Pr \left\{ {{{\left| {{h_i}} \right| }^2} \ge \chi } \right\} }_{ \buildrel \varDelta \over = \varphi _1^*} \times \underbrace{\Pr \left\{ {{{\left| {{g_{{r_b},{d_1}}}} \right| }^2} > \frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right\} }_{ \buildrel \varDelta \over = \varphi _2^*}. \end{aligned} \end{aligned}$$
(68)

We have

$$\begin{aligned} \begin{aligned} \varphi _1^* =&\Pr \left\{ {{{\left| {{h_i}} \right| }^2} \ge \chi } \right\} \\ =&\Pr \left\{ {{{\left| {{h_i}} \right| }^2} \ge \frac{\mu }{{{\rho _s}}}\left( {{\rho _r}{{\left| {{f_k}} \right| }^2} + 1} \right) } \right\} \\ =&\int \limits _0^\infty {\left( {1 - {F_{{{\left| {{h_i}} \right| }^2}}}\left( {\frac{\mu }{{{\rho _s}}}\left( {{\rho _r}x + 1} \right) } \right) } \right) } {f_{{{\left| {{f_k}} \right| }^2}}}\left( x \right) dx\\ =&\int \limits _0^\infty {\left( {1 - \left( {1 - \exp \left( { - \frac{\mu }{{{\beta _h}{\rho _s}}}\left( {{\rho _r}x + 1} \right) } \right) \sum \limits _{n = 0}^{{m_h} - 1} {\frac{{{\mu ^n}{{\left( {{\rho _r}x + 1} \right) }^n}}}{{n!\rho _s^n\beta _h^n}}} } \right) } \right) } \frac{{{x^{{m_{{f_k}}} - 1}}}}{{\varGamma \left( {{m_{{f_k}}}} \right) \beta _{{f_k}}^{{m_{{f_k}}}}}}\exp \left( { - \frac{x}{{{\beta _{{f_k}}}}}} \right) dx\\ =&\exp \left( { - \frac{\mu }{{{\beta _h}{\rho _s}}}} \right) \int \limits _0^\infty {\sum \limits _{n = 0}^{{m_h} - 1} {\frac{{{\mu ^n}{{\left( {{\rho _r}x + 1} \right) }^n}}}{{n!\rho _s^n\beta _h^n}}} } \frac{{{x^{{m_{{f_k}}} - 1}}}}{{\varGamma \left( {{m_{{f_k}}}} \right) \beta _{{f_k}}^{{m_{{f_k}}}}}}\exp \left( { - \left( {\frac{1}{{{\beta _{{f_k}}}}} + \frac{{\mu {\rho _r}}}{{{\beta _h}{\rho _s}}}} \right) x} \right) dx\\ =&\frac{\exp \left( - \mu \big / \beta _h{\rho _s} \right) }{\varGamma \left( {{m_{{f_k}}}} \right) \beta _{{f_k}}^{{m_{{f_k}}}}}\int \limits _0^\infty {\sum \limits _{n = 0}^{{m_h} - 1} {\sum \limits _{k = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right) } \rho _r^k\frac{{{\mu ^n}}}{{n!\rho _s^n\beta _h^n}}} } {x^{k + {m_{{f_k}}} - 1}}\exp \left( { - \left( {\frac{1}{{{\beta _{{f_k}}}}} + \frac{{\mu {\rho _r}}}{{{\beta _h}{\rho _s}}}} \right) x} \right) dx\\ =&\frac{\exp \left( - \mu \big / {\beta _h}{\rho _s} \right) }{\varGamma \left( {{m_{{f_k}}}} \right) \beta _{{f_k}}^{{m_{{f_k}}}}}\sum \limits _{n = 0}^{{m_h} - 1} {\sum \limits _{k = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right) } \rho _r^k\frac{{{\mu ^n}}}{{n!\rho _s^n\beta _h^n}}} {\left( {\frac{1}{{{\beta _{{f_k}}}}} + \frac{{\mu {\rho _r}}}{{{\beta _h}{\rho _s}}}} \right) ^{ - k - {m_{{f_k}}}}}\varGamma \left( {k + {m_{{f_k}}}} \right) . \end{aligned} \end{aligned}$$
(69)

where \(\mu = \max \left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\upsilon _1} - {\upsilon _2}\delta _1^{\mathrm{th}}}},\frac{{\delta _2^{\mathrm{th}}}}{{{\upsilon _2}}}} \right) \), \(\chi = \max \left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\upsilon _1} - {\upsilon _2}\delta _1^{\mathrm{th}}}},\frac{{\delta _2^{\mathrm{th}}}}{{{\upsilon _2}}}} \right) \left( {\frac{{{\rho _r}{{\left| {{f_k}} \right| }^2} + 1}}{{{\rho _s}}}} \right) \).

It is further to have \( \varphi _2^* \) computed by

$$\begin{aligned} \begin{aligned} \varphi _2^* =&1 - {F_{{{\left| {{g_{{r_k},{d_1}}}} \right| }^2}}}\left( {\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}}}} \right) \\ =&\frac{1}{{\varGamma \left( {{m_{{g_{{r_k},{d_1}}}}}} \right) }}\varGamma \left( {{m_{{g_{{r_k},{d_1}}}}},\frac{{\delta _1^{\mathrm{th}}}}{{{\rho _r}{\beta _{{g_{{r_k},{d_1}}}}}}}} \right) . \end{aligned} \end{aligned}$$
(70)

Then, the probability for the event that there are \({r_i}\) active relays in \({{\mathcal B}_{\mathcal R}}\) can be evaluated as follows:

$$\begin{aligned} \begin{aligned} \Pr \left( {\left| {{{\mathcal B}_{\mathcal R}}} \right| = i} \right) = \left( {\begin{array}{*{20}{c}} M\\ i \end{array}} \right) {\left( {{\varOmega _\theta }} \right) ^i}{\left( {1 - {\varOmega _\theta }} \right) ^{M - i}}. \end{aligned} \end{aligned}$$
(71)

Substituting result of (60) and (71) into (44), the outage probability of considered system can be computed as in Proposition 3. This completes the proof.

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Thi Nguyen, TT., Do, DT., Chen, YC. et al. New look on relay selection strategies for full-duplex multiple-relay NOMA over Nakagami-m fading channels. Wireless Netw 27, 3827–3843 (2021). https://doi.org/10.1007/s11276-021-02676-1

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