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Outage Performance of Energy Harvesting DF Relaying NOMA Networks

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Abstract

In this paper, we investigate energy harvesting decode-and-forward relaying non-orthogonal multiple access (NOMA) networks. We study two cases of single relay and multiple relays with partial relay selection strategy. Specifically, one source node wishes to transmit two symbols to two respective destinations directly and via the help of one selected intermediate energy constraint relay node, and the NOMA technique is applied in the transmission of both hops (from source to relay and from relay to destinations). For performance evaluation, we derive the closed-form expressions for the outage probability (OP) at D 1 and D 2 with both cases of single and multiple relays. Our analysis is substantiated via Monte Carlo simulation. The effect of several parameters, such as power allocation factors in both transmissions in two hops, power splitting ratio, energy harvesting efficiency, and the location of relay nodes to the outage performances at the two destinations is investigated.

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Author information

Authors and Affiliations

  1. Duy Tan University, Da Nang, Vietnam

    Dac-Binh Ha & Sang Quang Nguyen

Authors
  1. Dac-Binh Ha
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  2. Sang Quang Nguyen
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Correspondence to Sang Quang Nguyen.

Appendices

Appendix A: Finding the closed-form of probabilty \(\Pr \left [{g_{1}} \geqslant {u_{1}},{g_{2}} < \frac {u_{2}}{g_{1}} \right ]\)

By using the PDF of RV g 1 and CDF of RV g 2, the probabilty \(\Pr \left [ {g_{1}} \geqslant {u_{1}},{g_{2}} < \frac {u_{2}}{g_{1}} \right ]\) can be obtained as

$$ \begin{array}{ll} \Pr \left[ {{g_{1}} \geqslant {u_{1}},{g_{2}} < \frac{{{u_{2}}}}{{{g_{1}}}}} \right] = \int\limits_{{u_{1}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)} {F_{{g_{2}}}}\left( {\frac{{{u_{2}}}}{x}} \right)dx \hfill \\ = \int\limits_{{u_{1}}}^{\infty} {{\lambda_{1}}{e^{ - {\lambda_{1}}x}}\left( {1 - {e^{ - \frac{{{\lambda_{2}}{u_{2}}}}{x}}}} \right)dx} \hfill \\ = e^ - {\lambda_{1}}{u_{1}} - \underbrace {\int\limits_{{u_{1}}}^{\infty} {{\lambda_{1}}{e^{ - {\lambda_{1}}x}}{e^{ - \frac{{{\lambda_{2}}{u_{2}}}}{x}}}dx}} _{{I_{1}}} \end{array} $$
(39)

To calculate the integral I 1, we first apply the equation 1.211 of [36]: \({e^{x}} = \sum \limits _{k = 0}^{\infty } {\frac {{{x^{k}}}}{{k!}}} \) to the term \({{e^{ - \frac {{{\lambda _{2}}{u_{2}}}}{x}}}}\) to obtain (A.2.1), then using Eq. 3.381.3 of [36]: \(\int _{u}^{\infty } {{x^{v - 1}}{e^{ - \mu x}}dx} = \frac {1}{{{\mu ^{v}}}}\Gamma \left ({v,\mu u} \right )\) to obtain (A.2.2) as follows

$$ \begin{array}{ll} {I_{1}}\mathop = \limits^{{\text{(A.2.1)}}} {\lambda_{1}}\sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - {\lambda_{2}}{u_{2}}} \right)^{k}}\int\limits_{{u_{1}}}^{\infty} {\frac{{{e^{ - {\lambda_{1}}x}}}}{{{{\left( x \right)}^{k}}}}dx} \hfill \\ \mathop = \limits^{{\text{(A.2.2)}}} \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - {\lambda_{1}}{\lambda_{2}}{u_{2}}} \right)^{k}}\Gamma \left( {1 - k,{\lambda_{1}}{u_{1}}} \right) \hfill \end{array} $$
(40)

By substituting (40) into (39), we obtain:

$$ \Pr \left[ {{g_{1}} \geqslant {u_{1}},{g_{2}} < \frac{{{u_{2}}}}{{{g_{1}}}}} \right] = {e^{ - {\lambda_{1}}{u_{1}}}} - \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - {\lambda_{1}}{\lambda_{2}}{u_{2}}} \right)^{k}}\Gamma \left( {1 - k,{\lambda_{1}}{u_{1}}} \right) $$
(41)

Appendix B: Finding the closed-form of probabilty \(\Pr \left [ {{g_{1b}} \geqslant {u_{1}},{g_{2b}} < \frac {{{u_{2}}}}{{{g_{1b}}}}} \right ]\)

By using the PDF of RV g 1b and CDF of RV g 2b as \({f_{{g_{1b}}}}\left (x \right ) = N{\lambda _{1}}\sum \limits _{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left ({ - 1} \right )}^{n}}{e^{ - \left ({n + 1} \right ){\lambda _{1}}x}}} \) and \({F_{{g_{2b}}}}\left (x \right ) = 1 - {e^{ - {\lambda _{2}}x}}\), we obtain:

$$ \begin{array}{ll} \Pr \left[ {{g_{1b}} \geqslant {u_{1}},{g_{2b}} < \frac{{{u_{2}}}}{{{g_{1b}}}}} \right] = N{\lambda_{1}}\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\int\limits_{{u_{1}}}^{\infty} {{e^{ - \left( {n + 1} \right){\lambda_{1}}x}}\left( {1 - {e^{ - \frac{{{\lambda_{2}}{u_{2}}}}{x}}}} \right)dx}} \hfill \\ \mathop = \limits^{{\text{(B.1.1)}}} N{\lambda_{1}}\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\left( {\int\limits_{{u_{1}}}^{\infty} {{e^{ - \left( {n + 1} \right){\lambda_{1}}x}}dx} - \int\limits_{{u_{1}}}^{\infty} {{e^{ - \left( {n + 1} \right){\lambda_{1}}x}}{e^{ - \frac{{{\lambda_{2}}{u_{2}}}}{x}}}dx}} \right)} \hfill \\ \mathop = \limits^{{\text{(B.1.2)}}} {N\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\left( {\frac{{{e^{ - \left( {n + 1} \right){\lambda_{1}}{u_{1}}}}}}{{\left( {n + 1} \right)}} - \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {{\left( { - {\lambda_{1}}{\lambda_{2}}{u_{2}}} \right)}^{k}}{ {\left( {n + 1} \right)}^{k - 1}}\Gamma \left( {1 - k,\left( {n + 1} \right){\lambda_{1}}{u_{1}}} \right)} \right)}} \hfill \end{array} $$
(42)

where (B.1.2) is obtained from (B.1.1) by first using Eq. 1.211 of [36] to the term \({{e^{ - \frac {{{\lambda _{2}}{u_{2}}}}{x}}}}\), then applying Eq. 3.381 of [36].

Appendix C: Proof of Eqs. 34 and 37

First, for the case of a 1 < a 2(1 + γ t ), the probability OP 6.2 in Eq. 31 can be rewritten as

$$ \begin{array}{l} O{P_{6.2}}|_{{a_{1}} < {a_{2}}\left( {1 + {\gamma_{t}}} \right)} = \Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{(1 - \rho + \mu ){\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right)(1 - \rho ){\gamma_{0}}}}\\ \min \left( {\frac{{{b_{1}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}} + 1 + \mu} },\frac{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{1 + \mu} }} \right) < {\gamma_{t}} \end{array} \right]\\ = \Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{(1 - \rho + \mu ){\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right)(1 - \rho ){\gamma_{0}}}}\\ \frac{{{b_{1}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}} + 1 + \mu} } < \frac{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{1 + \mu} },\frac{{{b_{1}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}} + 1 + \mu} } < {\gamma_{t}} \end{array} \right]\\ + \Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{(1 - \rho + \mu ){\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right)(1 - \rho ){\gamma_{0}}}}\\ \frac{{{b_{1}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}} + 1 + \mu} } \ge \frac{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{1 + \mu} },\frac{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}{g_{3}}}}{{1 + \mu} } < {\gamma_{t}} \end{array} \right]\\ = \underbrace {\Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{(1 - \rho + \mu ){\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right)(1 - \rho ){\gamma_{0}}}}\\ {g_{3}} > \frac{{\left( {{b_{1}} - {b_{2}}} \right)\left( {1 + \mu} \right)}}{{{{\left( {{b_{2}}} \right)}^{2}}\eta \rho {\gamma_{0}}{g_{1}}}},{g_{3}} < \frac{{\left( {1 + \mu} \right){\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right)\eta \rho {\gamma_{0}}{g_{1}}}} \end{array} \right]}_{O{P_{6.2.1}}} + \underbrace {\Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{(1 - \rho + \mu ){\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right)(1 - \rho ){\gamma_{0}}}}\\ {g_{3}} \le \frac{{\left( {{b_{1}} - {b_{2}}} \right)\left( {1 + \mu} \right)}}{{{{\left( {{b_{2}}} \right)}^{2}}\eta \rho {\gamma_{0}}{g_{1}}}},{g_{3}} < \frac{{\left( {1 + \mu} \right){\gamma_{t}}}}{{{b_{2}}\eta \rho {\gamma_{0}}{g_{1}}}} \end{array} \right]}_{O{P_{6.2.2}}} \end{array} $$
(43)

where OP 6.2.1 and OP 6.2.2 are given as

$$ O{P_{6.2.1}} = \left\{ \begin{array}{l} \int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)\left[ {{F_{{g_{3}}}}\left( {\frac{{{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}x}}} \right) - {F_{{g_{3}}}}\left( {\frac{{\left( {{b_{1}} - {b_{2}}} \right){\omega_{3}}}}{{{{\left( {{b_{2}}} \right)}^{2}}{\gamma_{0}}x}}} \right)} \right]dx,\,\,\,\,\,\,if\,\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)} \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,{b_{1}} \ge {b_{2}}\left( {1 + {\gamma_{t}}} \right) \end{array} \right. $$
(44)
$$ \begin{array}{l} O{P_{6.2.2}} = \left\{ \begin{array}{l} \Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}\\ {g_{3}} \le \frac{{\left( {{b_{1}} - {b_{2}}} \right){\omega_{3}}}}{{{{\left( {{b_{2}}} \right)}^{2}}{\gamma_{0}}{g_{1}}}} \end{array} \right]\,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,\\ \Pr \left[ \begin{array}{l} {g_{1}} \ge \frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}\\ {g_{3}} < \frac{{{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}{g_{1}}}} \end{array} \right]\,\,\,\,if\,{b_{1}} \ge {b_{2}}\left( {1 + {\gamma_{t}}} \right)\, \end{array} \right.\\ = \left\{ \begin{array}{l} \int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)\left[ {{F_{{g_{3}}}}\left( {\frac{{\left( {{b_{1}} - {b_{2}}} \right){\omega_{3}}}}{{{{\left( {{b_{2}}} \right)}^{2}}{\gamma_{0}}{g_{1}}}}} \right)} \right]dx,} \,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,\\ \int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)\left[ {{F_{{g_{3}}}}\left( {\frac{{{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}{g_{1}}}}} \right)} \right]dx,} \,\,\,\,if\,{b_{1}} \ge {b_{2}}\left( {1 + {\gamma_{t}}} \right)\, \end{array} \right. \end{array} $$
(45)

By substituting (44) and (45) into (43), and using the result in Appendix A, we obtain

$$ \begin{array}{l} {\left. {O{P_{6.2}}} \right|_{{a_{1}} < {a_{2}}\left( {1 + {\gamma_{t}}} \right)}} = O{P_{6.2.1}} + O{P_{6.2.2}}\\ = \left\{ \begin{array}{l} \int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)\left[ {{F_{{g_{3}}}}\left( {\frac{{{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}{g_{1}}}}} \right)} \right]dx,} \,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,\\ \int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)\left[ {{F_{{g_{3}}}}\left( {\frac{{{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}{g_{1}}}}} \right)} \right]dx,} \,\,\,\,if\,{b_{1}} \ge {b_{2}}\left( {1 + {\gamma_{t}}} \right)\, \end{array} \right.\\ = {e^{ - \frac{{{\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}} - \left\{ \begin{array}{l} \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right)}^{k}}\Gamma \left( {1 - k,\frac{{{\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right),} \,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,\\ \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}}}} \right)}^{k}}\Gamma \left( {1 - k,\frac{{{\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right)} ,\,\,\,\,if\,{b_{1}} \ge {b_{2}}\left( {1 + {\gamma_{t}}} \right)\, \end{array} \right. \end{array} $$
(46)

Next, we can obtain the result for OP 6.2 in the case of a 1 ≥ a 2(1 + γ t ) from Eq. 47 with replacing ′(a 1 − a 2 γ t )′ by \(^{\prime }a_{2}^{\prime }\) as

$$ {\left. {O{P_{6.2}}} \right|_{{a_{1}} \ge {a_{2}}\left( {1 + {\gamma_{t}}} \right)}} = {e^{ - \frac{{{\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}}} - \left\{ \begin{array}{l} \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right)}^{k}}\Gamma \left( {1 - k,\frac{{{\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}} \right),} \,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,\\ \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}}}} \right)}^{k}}\Gamma \left( {1 - k,\frac{{{\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}} \right)} ,\,\,\,\,if\,{b_{1}} \ge {b_{2}}\left( {1 + {\gamma_{t}}} \right)\, \end{array} \right. $$
(47)

By combining (46) and (47), we finish the proof for Eq. 34.

In the case of N relays, only the PDF of RV g 1 b is changed. Thus, we can obtain OP 6.2 in case of N relays and a 1 < a 2(1 + γ t ) from Eq. 46 as

$$ \begin{array}{ll} {\left. {OP_{6.2}^{\,N\,relays}} \right|_{{a_{1}} < {a_{2}}\left( {1 + {\gamma_{t}}} \right)}} = \left\{ {\begin{array}{*{20}{l}} {\int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1b}}}}\left( x \right)\left[ {{F_{{g_{3b}}}}\left( {\frac{{{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}{g_{1b}}}}} \right)} \right]dx,} \,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,} \\ {\int\limits_{\frac{{{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}^{\infty} {{f_{{g_{1}}}}\left( x \right)\left[ {{F_{{g_{3b}}}}\left( {\frac{{{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}{g_{1b}}}}} \right)} \right]dx,} \,\,\,\,if\,{b_{1}} \geqslant {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,} \end{array}} \right. \\ = \left\{ {\begin{array}{ll} {N\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\left( \begin{array}{ll} \frac{{{e^{ - \frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}}}}{{\left( {n + 1} \right)}} - \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right)^{k}} \hfill \\ {\left( {n + 1} \right)^{k - 1}}\Gamma \left( {1 - k,\frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right) \hfill \end{array} \right)} ,\,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,} \\ {N\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\left( \begin{array}{ll} \frac{{{e^{ - \frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}}}}}{{\left( {n + 1} \right)}} - \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}}}} \right)^{k}} \hfill \\ {\left( {n + 1} \right)^{k - 1}}\Gamma \left( {1 - k,\frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{\left( {{a_{1}} - {a_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right) \hfill \end{array} \right)} ,\,\,\,\,if\,{b_{1}} \geqslant {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,} \end{array}} \right. \end{array} $$
(48)

Similarly, the term OP 6.2 in the case of N relays with a 1 ≥ a 2(1 + γ t ) is expressed as

$$ \begin{array}{ll} {\left. {OP_{6.2}^{\,N\,relays}} \right|_{{a_{1}} \geqslant {a_{2}}\left( {1 + {\gamma_{t}}} \right)}} \hfill \\ = \left\{ {\begin{array}{*{20}{l}} {N\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\left( \begin{array}{ll} \frac{{{e^{ - \frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}}}}}{{\left( {n + 1} \right)}} - \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{\left( {{b_{1}} - {b_{2}}{\gamma_{t}}} \right){\gamma_{0}}}}} \right)^{k}} \hfill \\ {\left( {n + 1} \right)^{k - 1}}\Gamma \left( {1 - k,\frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}} \right) \hfill \end{array} \right)} ,\,\,\,\,if\,{b_{1}} < {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,} \\ {N\sum\limits_{n = 0}^{N - 1} {C_{N - 1}^{n}{{\left( { - 1} \right)}^{n}}\left( \begin{array}{ll} \frac{{{e^{ - \frac{{\left( {n + 1} \right){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}}}}}{{\left( {n + 1} \right)}} - \sum\limits_{k = 0}^{\infty} {\frac{1}{{k!}}} {\left( { - \frac{{{\lambda_{1}}{\lambda_{3}}{\omega_{3}}{\gamma_{t}}}}{{{b_{2}}{\gamma_{0}}}}} \right)^{k}} \hfill \\ {\left( {n + 1} \right)^{k - 1}}\Gamma \left( {1 - k,\frac{{(n + 1){\lambda_{1}}{\omega_{2}}{\gamma_{t}}}}{{{a_{2}}{\gamma_{0}}}}} \right) \hfill \end{array} \right)} ,\,\,\,\,if\,{b_{1}} \geqslant {b_{2}}\left( {1 + {\gamma_{t}}} \right)\,} \end{array}} \right. \hfill \end{array} $$
(49)

By combining (48) and (49), we finish the proof for Eq. 37.

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Ha, DB., Nguyen, S. Outage Performance of Energy Harvesting DF Relaying NOMA Networks. Mobile Netw Appl 23, 1572–1585 (2018). https://doi.org/10.1007/s11036-017-0922-x

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