Exact outage performance of small-cell network relying device-to-device and non-orthogonal multiple access under perfect and imperfect CSI

Abstract

This paper investigates a non-orthogonal multiple access (NOMA) technique by employing device-to-device (D2D) transmission in small-cell networks. More specifically, the D2D transmission combines the relay selection model from a group of macro base stations (MBS) with the D2D mode in in small-cell network to form a D2D-NOMA system. Such D2D transmission needs assistance from multiple MBS and the small-cell base station (SBS) using decode-and-forward (DF) techniques. In this context, full-duplex (FD) is adopted at intermediate nodes. Furthermore, there are two possible scenarios in practice including perfect channel state information (CSI) and imperfect CSI that need to be studied to examine performance degradation in such considered network. To evaluate system performance, we derive expressions of outage probability for D2D user. Our study results show that the proposed network with more MBSs selection can improve the outage performance compared to the conventional orthogonal multiple access (OMA) scheme. Finally, numerical results confirms the accuracy of the derived expressions.

Appendices

Appendix A

Proof of the Theorem 2

From the expression (28), the first term can be calculated as

$$\begin{aligned} {A_1}&= 1 - \Pr \left( {\gamma _{SD1 \leftarrow 2}^I \geqslant \varepsilon _2^{FD},\gamma _{D12,x2}^I \geqslant \varepsilon _2^{FD}} \right) \\&= 1 - \Pr \left( {X \geqslant \frac{{\varepsilon _2^{FD}\alpha \rho {Z_{SI}} + \varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}} \right) \Pr \left( {{Z_0} \geqslant \frac{{\varepsilon _2^{FD}}}{\rho }} \right) \\&= 1 - \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{h0}}}}} \right) \int _0^\infty \left( {1 - {F_{X}}\left( {\frac{{\varepsilon _2^{FD}\alpha \rho x + \varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}} \right) } \right) \\&\quad {f_{{Z_{SI}}}}\left( x \right) dx \\&= 1 - \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{h0}}}}} \right) \int _0^\infty \exp \left( { - \frac{{\varepsilon _2^{FD}\alpha \rho x + \varepsilon _2^{FD}}}{{{\zeta _1}\rho {\lambda _{g0}}}}} \right) \\&\quad \frac{1}{{{\lambda _{hD1}}}}\exp \left( { - \frac{x}{{{\lambda _{hD1}}}}} \right) dx \\&= 1 - \frac{{{\zeta _1}{\lambda _{g0}}}}{{\varepsilon _2^{FD}\alpha {\lambda _{hD1}} + {\zeta _1}{\lambda _{g0}}}}\exp \left( { - \frac{{\varepsilon _2^{FD}}}{{{\zeta _1}\rho {\lambda _{g0}}}} - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{h0}}}}} \right) , \end{aligned}$$

(52)

and the second term of probability in (28) is calculated as follows

$$\begin{aligned} {A_2}&= \left( {1 - \Pr \left( {\gamma _{SRk^*,x2}^I \geqslant \varepsilon _2^{FD},\gamma _{Rk^*D2,x2}^I \geqslant \varepsilon _2^{FD}} \right) } \right) \\&= 1 - \Pr \left( {{Y_{k^*}} \geqslant \frac{{\varepsilon _2^{FD}\left( {\alpha \rho {Y_{SI}} + {\text { }}1} \right) }}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}} \right) \Pr \left( {{Z_k} \geqslant \frac{{\varepsilon _2^{FD}}}{\rho }} \right) \\&= 1 - \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{hk}}}}} \right) \times \int _0^\infty \left( {1 - {F_{{Y_{k^*}}}}\left( {\frac{{\varepsilon _2^{FD}\left( {\alpha \rho x + {\text { }}1} \right) }}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}} \right) } \right) \\&\quad {f_{{Y_{SI}}}}\left( x \right) dx. \end{aligned}$$

(53)

According to [26], the (CDFs) of the random variable ${{\mathbb {X}}_{{k^*}}}$ is given by

$$\begin{aligned} {F_{{{\mathbb {X}}_{{k^ * }}}}}\left( x \right)&= {\left( {1 - \exp \left( { - \frac{x}{{{\lambda _{{{\mathbb {X}}_{{k^ * }}}}}}}} \right) } \right) ^K} \\&= 1 - \sum \limits _{k = 1}^K {\left( {\begin{array}{*{20}{c}} K \\ k \end{array}} \right) } {( - 1)^{k - 1}}\exp \left( { - \frac{x}{{{\lambda _{{{\mathbb {X}}_{{k^ * }}}}}}}} \right) . \end{aligned}$$

(54)

Therefore, expression (53) is rewritten as follows

$$\begin{aligned} {A_2}&= 1 - \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{hk}}}}} \right) \\&\quad \times \int _0^\infty \sum \limits _{k = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k - 1}}\exp {{\left( { - \frac{{\varepsilon _2^{FD}\left( {\alpha \rho x + {\text { }}1} \right) }}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) }^k}} \frac{1}{{{\lambda _{hr}}}} \\&\quad \exp \left( { - \frac{x}{{{\lambda _{hr}}}}} \right) dx \\&= 1 - \sum \limits _{k = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k - 1}}\frac{1}{{{\lambda _{hr}}}}} \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{hk}}}} - \frac{{k\varepsilon _2^{FD}}}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) \\&\quad \times \int _0^\infty {\exp \left( { - \frac{{k\varepsilon _2^{FD}\alpha \rho x}}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) \exp \left( { - \frac{x}{{{\lambda _{hr}}}}} \right) dx} \\&= 1 - \sum \limits _{k = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k - 1}}} \frac{{{\zeta _1}{\lambda _{gk}}}}{{k\varepsilon _2^{FD}\alpha {\lambda _{hr}} + {\zeta _1}{\lambda _{gk}}}} \\&\quad \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{hk}}}} - \frac{{k\varepsilon _2^{FD}}}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) . \end{aligned}$$

(55)

It is noted that the expressions (52) and (53) can be obtained by the condition of ${\beta _2} > \varepsilon _2^{FD}{\beta _1}$. The proof is completed. $\square$

Appendix B:

Proof of the Theorem 3

From (37), we can obtained as

$$\begin{aligned}&OP_{1 - II}^{FD - CSI} = 1 - \Pr \left( {\gamma _{SD1 \leftarrow 2}^{CSI} \geqslant \varepsilon _2^{FD},\gamma _{SD1}^{CSI} \geqslant \varepsilon _1^{FD}} \right) \\&\quad = 1 - \Pr \left( {\frac{{{\beta _2}\rho X}}{{{\beta _1}\rho X + {\beta _1}\rho {{{\tilde{X}}}} + {\beta _2}\rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1}} \geqslant \varepsilon _2^{FD},} \right. \\&\qquad \left. {\frac{{{\beta _1}\rho X}}{{{\beta _1}\rho {{{\tilde{X}}}} + {\beta _2}\rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1}} \geqslant \varepsilon _1^{FD}} \right) \\&\quad = 1 - \Pr \left( {X \geqslant \left( {\left( {{\beta _1} + {\beta _2}} \right) \rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1} \right) } \right. \\&\qquad \times \left. {max\left( {\frac{{\varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }},\frac{{\varepsilon _1^{FD}}}{{{\beta _1}\rho }}} \right) } \right) \\&\quad = 1 - \Pr \left( {X \geqslant {\mu ^{FD}}\left( {\tau \rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1} \right) } \right) \\&\quad = 1 - \int \limits _0^\infty {\int \limits _0^\infty {\left( {1 - {F_{X}}\left( {{\mu ^{FD}}\left( {\tau \rho x + \alpha \rho y + {\text { }}1} \right) } \right) } \right) } } {f_{{{{\tilde{X}}}}}}\left( x \right) \\&\qquad dx{f_{{Z_{SI}}}}\left( y \right) dy \\&\quad = 1 - \frac{{{\lambda _{g0}}{\lambda _{g0}}}}{{\left( {{\mu ^{FD}}\tau \rho {\lambda _{{\tilde{g}}0}} + {\lambda _{g0}}} \right) \left( {{\mu ^{FD}}\alpha \rho {\lambda _{hD1}} + {\lambda _{g0}}} \right) }} \\&\qquad \exp \left( { - \frac{{{\mu ^{FD}}}}{{{\lambda _{g0}}}}} \right) , \end{aligned}$$

(56)

where ${\mu ^{FD}} = max\left( {\frac{{\varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }},\frac{{\varepsilon _1^{FD}}}{{{\beta _1}\rho }}} \right) ,\tau = {\beta _1} + {\beta _2}$ and it is under assumption that ${\beta _2} > \varepsilon _2^{FD}{\beta _1}$.

The proof is completed. $\square$

Appendix C:

Proof of the Theorem 4

Considering the expressions in (40), the first term can be rewritten as

$$\begin{aligned} {B_1}&= 1 - \Pr \left( {\gamma _{SD1 \leftarrow 2}^{CSI} \geqslant \varepsilon _2^{FD},\gamma _{D12,x2}^{CSI} \geqslant \varepsilon _2^{FD}} \right) \\&= 1 - \Pr \left( {\frac{{{\beta _2}\rho X}}{{{\beta _1}\rho X + {\beta _1}\rho {{{\tilde{X}}}} + {\beta _2}\rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1}} \geqslant \varepsilon _2^{FD}} \right) \\&\qquad \times \Pr \left( {\frac{{\rho {Z_0}}}{{\rho {{\tilde{Z}}_0} + 1}} \geqslant \varepsilon _2^{FD}} \right) \\&= 1 - \underbrace{\Pr \left( {X \geqslant \frac{{\varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}\left( {{\beta _1}\rho {{{\tilde{X}}}} + {\beta _2}\rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1} \right) } \right) }_{{E_1}} \\&\qquad \times \underbrace{\Pr \left( {{Z_0} \geqslant \frac{{\varepsilon _2^{FD}\rho {{\tilde{Z}}_0} + \varepsilon _2^{FD}}}{\rho }} \right) }_{{E_2}}. \end{aligned}$$

(57)

Furthermore, the expressions $E_1$ and $E_2$ can be calculated as, respectively.

$$\begin{aligned} {E_1}&= \Pr \left( {X \geqslant \frac{{\varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}\left( {{\beta _1}\rho {{{\tilde{X}}}} + {\beta _2}\rho {{{\tilde{X}}}} + \alpha \rho {Z_{SI}} + {\text { }}1} \right) } \right) \\&= \int _0^\infty \int _0^\infty \left( {1 - {F_{X}}\left( {\frac{{\varepsilon _2^{FD}}}{{{\zeta _1}\rho }}\left( {\tau \rho x + \alpha \rho y + {\text { }}1} \right) } \right) } \right) {f_{{{{\tilde{X}}}}}}\left( x \right) \\&\quad dx{f_{{Z_{SI}}}}\left( y \right) dy \\&= \frac{{\zeta _1^2\lambda _{g0}^2}}{{\left( {\varepsilon _2^{FD}\tau {\lambda _{{\tilde{g}}0}} + {\zeta _1}{\lambda _{g0}}} \right) \left( {\varepsilon _2^{FD}\alpha {\lambda _{hD1}} + {\zeta _1}{\lambda _{g0}}} \right) }} \\&\quad \exp \left( { - \frac{{\varepsilon _2^{FD}}}{{{\zeta _1}\rho {\lambda _{g0}}}}} \right) , \end{aligned}$$

(58)

and

$$\begin{aligned} {E_2}&= \Pr \left( {{Z_0} \geqslant \frac{{\varepsilon _2^{FD}\rho {{\tilde{Z}}_0} + \varepsilon _2^{FD}}}{\rho }} \right) \\&= \int _0^\infty {\left( {1 - {F_{{Z_0}}}\left( {\frac{{\varepsilon _2^{FD}\rho x + \varepsilon _2^{FD}}}{\rho }} \right) } \right) {f_{{{\tilde{Z}}_0}}}\left( x \right) dx} \\&= \int _0^\infty {\exp \left( { - \frac{{\varepsilon _2^{FD}\rho x + \varepsilon _2^{FD}}}{{\rho {\lambda _{h0}}}}} \right) \frac{1}{{{\lambda _{{\tilde{h}}0}}}}\exp \left( { - \frac{x}{{{\lambda _{{\tilde{h}}0}}}}} \right) dx} \\&= \frac{{{\lambda _{h0}}}}{{\varepsilon _2^{FD}{\lambda _{{\tilde{h}}0}} + {\lambda _{h0}}}}\exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{h0}}}}} \right) . \end{aligned}$$

(59)

Besides, the second term in (40) can be calculated as

$$\begin{aligned} {B_2}&= \left( {1 - \Pr \left( {\gamma _{SRk^*,x2}^{CSI} \geqslant \varepsilon _2^{FD},\gamma _{Rk^*D2,x2}^{CSI} \geqslant \varepsilon _2^{FD}} \right) } \right) \\&= 1 - \underbrace{\Pr \left( {{Y_{k^*}} \geqslant \frac{{\varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}\left( {\left( {{\beta _1} + {\beta _2}} \right) \rho {{\tilde{Y}}_{k^*}} + \alpha \rho {Y_{SI}} + {\text { }}1} \right) } \right) }_{{F_1}} \\&\quad \times \underbrace{\Pr \left( {{Z_k} \geqslant \frac{{\varepsilon _2^{FD}\rho {{\tilde{Z}}_k} + \varepsilon _2^{FD}}}{\rho }} \right) }_{{F_2}}. \end{aligned}$$

(60)

Consequently, the probability expressions in (60), i.e., $F_1$ and $F_2$, is determined as, respectively.

$$\begin{aligned} {F_1}&= \Pr \left( {{Y_{k^*}} \geqslant \frac{{\varepsilon _2^{FD}}}{{{\beta _2}\rho - \varepsilon _2^{FD}{\beta _1}\rho }}\left( {\left( {{\beta _1} + {\beta _2}} \right) \rho {{\tilde{Y}}_{k^*}} + \alpha \rho {Y_{SI}} + {\text { }}1} \right) } \right) \\&= \int \limits _0^\infty \int \limits _0^\infty \left( {1 - {F_{{Y_{k^*}}}}\left( {\frac{{\varepsilon _2^{FD}}}{{{\zeta _1}\rho }}\left( {\tau \rho x + \alpha \rho y + {\text { }}1} \right) } \right) } \right) {f_{{\tilde{Y}}_{k^*}}}\left( x \right) \\&\quad dx{f_{{Y_{SI}}}}\left( y \right) dy \\&= \int \limits _0^\infty {\int \limits _0^\infty {\left( {\sum \limits _{k = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k - 1}}\exp \left( { - \frac{{\varepsilon _2^{FD}k}}{{{\zeta _1}\rho {\lambda _{gk}}}}\left( {\tau \rho x + \alpha \rho y + {\text { }}1} \right) } \right) } } \right. } } \\&\quad \left. { \times \sum \limits _{k = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k - 1}}\frac{k}{{{\lambda _{{\tilde{g}}k}}}}\exp \left( { - \frac{{kx}}{{{\lambda _{{\tilde{g}}k}}}}} \right) } dx\frac{1}{{{\lambda _{hr}}}}\exp \left( { - \frac{y}{{{\lambda _{hr}}}}} \right) dy} \right) \\&= \sum \limits _{k = 1}^K {\sum \limits _{j = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) \left( \begin{aligned} K \\ j \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k + j - 2}}\frac{j}{{{\lambda _{{\tilde{g}}k}}}}\frac{1}{{{\lambda _{hr}}}}} \exp \left( { - \frac{{\varepsilon _2^{FD}k}}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) } \\&\quad \times \int \limits _0^\infty \exp \left( { - \frac{{\varepsilon _2^{FD}k\tau x}}{{{\zeta _1}{\lambda _{gk}}}}} \right) \exp \left( { - \frac{{jx}}{{{\lambda _{{\tilde{g}}k}}}}} \right) dx\int _0^\infty \\&\quad {\exp \left( { - \frac{{\varepsilon _2^{FD}k\alpha y}}{{{\zeta _1}{\lambda _{gk}}}}} \right) \exp \left( { - \frac{y}{{{\lambda _{hr}}}}} \right) dy} \\&= \sum \limits _{k = 1}^K {\sum \limits _{j = 1}^K {\left( \begin{aligned} K \\ k \\ \end{aligned} \right) \left( \begin{aligned} K \\ j \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k +j-2}}\frac{j}{{{\lambda _{{\tilde{g}}k}}}}\frac{1}{{{\lambda _{hr}}}}} \exp \left( { - \frac{{\varepsilon _2^{FD}k}}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) } \\&\quad \times \int _0^\infty {\exp \left( { - \left( {\frac{{\varepsilon _2^{FD}k\tau }}{{{\zeta _1}{\lambda _{gk}}}} + \frac{j}{{{\lambda _{{\tilde{g}}k}}}}} \right) x} \right) dx} \\&\quad \int _0^\infty {\exp \left( { - \left( {\frac{{\varepsilon _2^{FD}k\alpha }}{{{\zeta _1}{\lambda _{gk}}}} + \frac{1}{{{\lambda _{hr}}}}} \right) y} \right) dy} \\&= \sum \limits _{k = 1}^K \sum \limits _{j = 1}^K \left( \begin{aligned} K \\ k \\ \end{aligned} \right) \left( \begin{aligned} K \\ j \\ \end{aligned} \right) {{\left( { - 1} \right) }^{k + j - 2}} \\&\quad \frac{{\zeta _1^2\lambda _{gk}^2}}{{\left( {\varepsilon _2^{FD}k\tau {\lambda _{{\tilde{g}}k}} + {\zeta _1}{\lambda _{gk}}} \right) \left( {\varepsilon _2^{FD}k\alpha {\lambda _{hr}} + {\zeta _1}{\lambda _{gk}}} \right) }} \\&\quad \times \exp \left( { - \frac{{\varepsilon _2^{FD}k}}{{{\zeta _1}\rho {\lambda _{gk}}}}} \right) , \end{aligned}$$

(61)

and it is similar to (59), we have that

$$\begin{aligned} {F_2} = \frac{{{\lambda _{hk}}}}{{\varepsilon _2^{FD}{\lambda _{{\tilde{h}}k}} + {\lambda _{hk}}}}\exp \left( { - \frac{{\varepsilon _2^{FD}}}{{\rho {\lambda _{hk}}}}} \right) . \end{aligned}$$

(62)

Note that the condition ${\beta _1} > \varepsilon _1^{FD}{\beta _2}$ has maintained so far. Substituting (58) and (59) into (57) and replacing (60) by (61) and (62), then combining (57) with (60), the proof is completed. $\square$