Outage Probability Analysis of Single Energy Constraint Relay NOMA Network

Abstract

In this paper, we investigate energy harvesting decode-and-forward relaying non-orthogonal multiple access (NOMA) networks. Specifically, one source node wishes to transmit two symbols to its two desired destinations directly and via the help of an 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 $D_2$. Our analysis is substantiated via Monte Carlo simulation. The effect of several parameters, such as power allocation factors in both transmissions in two hops, the power splitting ratio, the location of relay node, to the outage performances at two destinations is investigated.

Appendices

A Appendix A: Finding the Closed-Form of Probability $\Pr \left[ {{g_1} \geqslant {u_1},{g_2} < \frac{{{u_2}}}{{{g_1}}}} \right]$

By using the PDF of RV $g_1$ and CDF of RV $g_2$, the probability $\Pr \left[ {{g_1} \geqslant {u_1},{g_2} < \frac{{{u_2}}}{{{g_1}}}} \right]$ can be obtained as

$$\begin{aligned} \begin{gathered} \Pr \left[ {{g_1} \geqslant {u_1},{g_2} < \frac{{{u_2}}}{{{g_1}}}} \right] = \int \limits _{{u_1}}^\infty {{f_{{g_1}}}\left( x \right) } {F_{{g_2}}}\left( {\frac{{{u_2}}}{x}} \right) dx \\ = \int \limits _{{u_1}}^\infty {{\lambda _1}{e^{ - {\lambda _1}x}}\left( {1 - {e^{ - \frac{{{\lambda _2}{u_2}}}{x}}}} \right) dx} \\ = {e^{ - {\lambda _1}{u_1}}} - \underbrace{\int \limits _{{u_1}}^\infty {{\lambda _1}{e^{ - {\lambda _1}x}}{e^{ - \frac{{{\lambda _2}{u_2}}}{x}}}dx} }_{{I_1}} \\ \end{gathered} \end{aligned}$$

(A.1)

To calculate the integral $I_1$, we first apply the Eq. 1.211 of [6]: ${e^x} = \sum \limits _{k = 0}^\infty {\frac{{{x^k}}}{{k!}}}$ to the term ${{e^{ - \frac{{{\lambda _2}{u_2}}}{x}}}}$ to obtain (A.2.1), then using Eq. 3.381.3 of [6]: $\int _u^\infty {{x^{v - 1}}{e^{ - \mu x}}dx} = \frac{1}{{{\mu ^v}}}\varGamma \left( {v,\mu u} \right)$ to obtain (A.2.2) as follows

$$\begin{aligned} \begin{gathered} {I_1}\mathop = \limits ^{{\text {(A.2.1)}}} {\lambda _1}\sum \limits _{k = 0}^\infty {\frac{1}{{k!}}} {\left( { - {\lambda _2}{u_2}} \right) ^k}\int \limits _{{u_1}}^\infty {\frac{{{e^{ - {\lambda _1}x}}}}{{{{\left( x \right) }^k}}}dx} \\ \mathop = \limits ^{{\text {(A.2.2)}}} \sum \limits _{k = 0}^\infty {\frac{1}{{k!}}} {\left( { - {\lambda _1}{\lambda _2}{u_2}} \right) ^k}\varGamma \left( {1 - k,{\lambda _1}{u_1}} \right) \\ \end{gathered} \end{aligned}$$

(A.2)

By substituting (A.3) into (A.1), we obtain:

$$\begin{aligned} \Pr \left[ {{g_1} \geqslant {u_1},{g_2} < \frac{{{u_2}}}{{{g_1}}}} \right] = {e^{ - {\lambda _1}{u_1}}} - \sum \limits _{k = 0}^\infty {\frac{1}{{k!}}} {\left( { - {\lambda _1}{\lambda _2}{u_2}} \right) ^k}\varGamma \left( {1 - k,{\lambda _1}{u_1}} \right) \end{aligned}$$

(A.3)

B Appendix B: Proof of Eq. (28)

First, for the case of ${a_1} < {a_2}\left( {1 + {\gamma _t}} \right)$, the probability $OP_{6.2}$ in (25) can be rewritten as

$$\begin{aligned} \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} \end{aligned}$$

(B.1)

where $OP_{6.2.1}$ and $OP_{6.2.2}$ are given as

$$\begin{aligned} O{P_{6.2.1}} = \left\{ \begin{array}{l} \int \limits _{\frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}}^\infty {{f_{{g_1}}}\left( x \right) \left[ {{F_{{g_3}}}\left( {\frac{{{\omega _3}{\gamma _t}}}{{\left( {{b_1} - {b_2}{\gamma _t}} \right) {\gamma _0}x}}} \right) - {F_{{g_3}}}\left( {\frac{{\left( {{b_1} - {b_2}} \right) {\omega _3}}}{{{{\left( {{b_2}} \right) }^2}{\gamma _0}x}}} \right) } \right] dx,\,\,\,\,\,\,if\,\,{b_1} < {b_2}\left( {1 + {\gamma _t}} \right) } \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,{b_1} \ge {b_2}\left( {1 + {\gamma _t}} \right) \end{array} \right. \end{aligned}$$

(B.2)

$$\begin{aligned} \begin{array}{l} O{P_{6.2.2}} = \left\{ \begin{array}{l} \Pr \left[ \begin{array}{l} {g_1} \ge \frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}\\ {g_3} \le \frac{{\left( {{b_1} - {b_2}} \right) {\omega _3}}}{{{{\left( {{b_2}} \right) }^2}{\gamma _0}{g_1}}} \end{array} \right] \,\,\,\,if\,{b_1}< {b_2}\left( {1 + {\gamma _t}} \right) \,\\ \Pr \left[ \begin{array}{l} {g_1} \ge \frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}\\ {g_3}< \frac{{{\omega _3}{\gamma _t}}}{{{b_2}{\gamma _0}{g_1}}} \end{array} \right] \,\,\,\,if\,{b_1} \ge {b_2}\left( {1 + {\gamma _t}} \right) \, \end{array} \right. \\ = \left\{ \begin{array}{l} \int \limits _{\frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}}^\infty {{f_{{g_1}}}\left( x \right) \left[ {{F_{{g_3}}}\left( {\frac{{\left( {{b_1} - {b_2}} \right) {\omega _3}}}{{{{\left( {{b_2}} \right) }^2}{\gamma _0}{g_1}}}} \right) } \right] dx,} \,\,\,\,if\,{b_1} < {b_2}\left( {1 + {\gamma _t}} \right) \,\\ \int \limits _{\frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}}^\infty {{f_{{g_1}}}\left( x \right) \left[ {{F_{{g_3}}}\left( {\frac{{{\omega _3}{\gamma _t}}}{{{b_2}{\gamma _0}{g_1}}}} \right) } \right] dx,} \,\,\,\,if\,{b_1} \ge {b_2}\left( {1 + {\gamma _t}} \right) \, \end{array} \right. \end{array} \end{aligned}$$

(B.3)

By substituting (B.2) and (B.3) into (B.1), and using the result in Appendix A, we obtain

$$\begin{aligned} \begin{array}{l} {\left. {O{P_{6.2}}} \right| _{{a_1}< {a_2}\left( {1 + {\gamma _t}} \right) }} = O{P_{6.2.1}} + O{P_{6.2.2}}\\ = \left\{ \begin{array}{l} \int \limits _{\frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}}^\infty {{f_{{g_1}}}\left( x \right) \left[ {{F_{{g_3}}}\left( {\frac{{{\omega _3}{\gamma _t}}}{{\left( {{b_1} - {b_2}{\gamma _t}} \right) {\gamma _0}{g_1}}}} \right) } \right] dx,} \,\,\,\,if\,{b_1}< {b_2}\left( {1 + {\gamma _t}} \right) \,\\ \int \limits _{\frac{{{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}}^\infty {{f_{{g_1}}}\left( x \right) \left[ {{F_{{g_3}}}\left( {\frac{{{\omega _3}{\gamma _t}}}{{{b_2}{\gamma _0}{g_1}}}} \right) } \right] dx,} \,\,\,\,if\,{b_1} \ge {b_2}\left( {1 + {\gamma _t}} \right) \, \end{array} \right. \\ = {e^{ - \frac{{{\lambda _1}{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}}} - \left\{ \begin{array}{l} \sum \limits _{k = 0}^\infty {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda _1}{\lambda _3}{\omega _3}{\gamma _t}}}{{\left( {{b_1} - {b_2}{\gamma _t}} \right) {\gamma _0}}}} \right) }^k}\varGamma \left( {1 - k,\frac{{{\lambda _1}{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}} \right) ,} \,\,\,\,if\,{b_1} < {b_2}\left( {1 + {\gamma _t}} \right) \,\\ \sum \limits _{k = 0}^\infty {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda _1}{\lambda _3}{\omega _3}{\gamma _t}}}{{{b_2}{\gamma _0}}}} \right) }^k}\varGamma \left( {1 - k,\frac{{{\lambda _1}{\omega _2}{\gamma _t}}}{{\left( {{a_1} - {a_2}{\gamma _t}} \right) {\gamma _0}}}} \right) } ,\,\,\,\,if\,{b_1} \ge {b_2}\left( {1 + {\gamma _t}} \right) \, \end{array} \right. \end{array} \end{aligned}$$

(B.4)

Next, we can obtain the result for $OP_{6.2}$ in the case of ${{a_1} \ge {a_2}\left( {1 + {\gamma _t}} \right) }$ from (B.4) with replacing ‘$\left( {{a_1} - {a_2}{\gamma _t}} \right)$’ by ‘$a_2$’ as

$$\begin{aligned} {\left. {O{P_{6.2}}} \right| _{{a_1} \ge {a_2}\left( {1 + {\gamma _t}} \right) }} = {e^{ - \frac{{{\lambda _1}{\omega _2}{\gamma _t}}}{{{a_2}{\gamma _0}}}}} - \left\{ \begin{array}{l} \sum \limits _{k = 0}^\infty {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda _1}{\lambda _3}{\omega _3}{\gamma _t}}}{{\left( {{b_1} - {b_2}{\gamma _t}} \right) {\gamma _0}}}} \right) }^k}\varGamma \left( {1 - k,\frac{{{\lambda _1}{\omega _2}{\gamma _t}}}{{{a_2}{\gamma _0}}}} \right) ,} \,\,\,\,if\,{b_1} < {b_2}\left( {1 + {\gamma _t}} \right) \,\\ \sum \limits _{k = 0}^\infty {\frac{1}{{k!}}{{\left( { - \frac{{{\lambda _1}{\lambda _3}{\omega _3}{\gamma _t}}}{{{b_2}{\gamma _0}}}} \right) }^k}\varGamma \left( {1 - k,\frac{{{\lambda _1}{\omega _2}{\gamma _t}}}{{{a_2}{\gamma _0}}}} \right) } ,\,\,\,\,if\,{b_1} \ge {b_2}\left( {1 + {\gamma _t}} \right) \, \end{array} \right. \end{aligned}$$

(B.5)

By combining (B.4) and (B.5), we finish the proof for Eq. (28).