Buffer-aided cooperative NOMA with partial relay selection

Abstract

The Cooperative Non-orthogonal multiple access (C-NOMA) technique has been considered as a promising solution to improve the coverage extension and transmission reliability. But due to the unavailability of perfect channel state information in dual hope network, transmission link becomes unreliable, and difficulty in decoding also arises in the successive interference cancellation (SIC) process at destination users. To solve this problem, this contribution develops and investigates a joint buffer-aided multi-relay cooperative NOMA scheme (BAN-PRS) with partial relay selection. More specifically, we examine a model, where the source transmits the message to two NOMA users via the best relay node selected from a set of half-duplex (HD) amplify and forward relays. In this model, information from source to destination is transmitted using the power domain NOMA technique. In between the transmission, a cooperative buffer-based relay network is used, where partial relay selection (PRS) is performed, and based on first-hop channel state information, we choose the best buffer-based relay node. To evaluate the system performance, exact and asymptotic analytical expressions of outage probabilities, maximum packet delay, and throughput are derived in integral form for both the destination users. Additionally, relying on these analytical expressions, we accessed the impact of buffer size, the number of relay nodes, and power allocation coefficients between the destinations on the performance of this cooperative NOMA network. Furthermore, we carry out Monte-Carlo simulations to validate the correctness of this analysis.

Appendices

Appendix 1

Proof of proposition 1

In (19), $\mathit{Pr}\left(\frac{{a}_{1}{X}_{1}}{{a}_{2}{X}_{1}+1}<{\Omega }_{th}\right)$ is the CDF of link $BS$ to ${U}_{1}$ and it is equal to the CDF of a random variable (RV) ${X}_{i}$ which is given in (8). Thus, mathematically it can be represented as

$$ F_{{\Omega_{{BSU_{1} }} }} \left( {\Omega_{th} } \right) = F_{{X_{1} }} \left( {\tilde{\Omega }_{th} } \right) = \left[ {1 - exp\left( { - \frac{{\tilde{\Omega }_{th} }}{{\tilde{\theta }_{{BSU_{1} }} }}} \right)} \right] $$

(33)

The second part of (19), $Pr\left(\frac{{a}_{1}{Y}_{m}{Z}_{1m}}{{a}_{2}{Y}_{m}{Z}_{1m}+{Y}_{m}+{Z}_{1m}+1}<{\Omega }_{th}\right)$ represents the CDF of the link ${R}_{m}$ to ${U}_{1}$ which can be solved by calculating the CDF of RV’s ${Y}_{m}$ and ${Z}_{1m}$ as given in (9) and (10).

The CDF of ${Y}_{m}$ is given as

$$ \begin{aligned} F_{{Y_{m} }} \left( {\Omega_{th} } \right) & = \left( {1 - exp\left( { - \frac{{\Omega_{th} }}{{\tilde{\theta }_{{BSR_{m} }} }}} \right)} \right)^{M} \\ &\quad.\left( {1 - exp\left( { - \frac{{\Omega_{th} }}{{\tilde{\theta }_{{BSR_{m} }} }}} \right)} \right)^{N} \\ & = 1 - \mathop \sum \limits_{m = 1}^{M} \left( {\begin{array}{*{20}c} M \\ m \\ \end{array} } \right)\left( { - 1} \right)^{{\left( {m - 1} \right)}} \\ &\quad\mathop \sum \limits_{n = 1}^{N} \left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)\left( { - 1} \right)^{{\left( {n - 1} \right)}} exp\left( { - \frac{{\left( {m + n} \right)\Omega_{th} }}{m}} \right) \\ \end{aligned} $$

(34)

The CDF of ${Z}_{1m}$ is given as

$$ F_{{Z_{1m} { }}} \left( {\Omega_{th} } \right) = 1 - exp\left( { - \frac{{\Omega_{th} }}{{\tilde{\theta }_{{R_{m} U_{1} }} }}} \right) $$

(35)

The CDF of the link ${R}_{m}$ to ${U}_{1}$ can be represented as

$$ \begin{aligned} &F_{{\Omega_{{R_{m} U_{1} }} }} \left( {\Omega_{th} } \right) \\ &\quad =\! F_{{Z_{1m} { }}} \left( {\tilde{\Omega }_{th} } \right) + \mathop \smallint \limits_{{\tilde{\Omega }_{th} }}^{\infty } \!\!F_{{Y_{m} }} \!\left( {\frac{{z\Omega_{th} + \Omega_{th} }}{{z\left( {a_{1} - a_{2} \Omega_{th} } \right)\! -\! \Omega_{th} }}} \right)\!f_{{Z_{1m} { }}} \left( z \right)dz \\ & \quad= F_{{Z_{1m} { }}} \left( {\tilde{\Omega }_{th} } \right) + \mathop \smallint \limits_{{\tilde{\Omega }_{th} }}^{\infty } F_{{Y_{m} }} \left( {\frac{{\tilde{\Omega }_{th} \left( {z + 1} \right)}}{{z - \tilde{\Omega }_{th} }}} \right)f_{{Z_{1m} { }}} \left( z \right)dz \\ \end{aligned} $$

(36)

By substituting (34) and (35) in (36), ${F}_{{\Omega }_{{R}_{m}{U}_{1}}}\left({\Omega }_{th}\right)$ is derived as

$$ \begin{aligned}& F_{{\Omega_{{R_{m} U_{1} }} }} \left( {\Omega_{th} } \right) \\ &\quad= 1 - \mathop \sum \limits_{m = 1}^{M} \left( {\begin{array}{*{20}c} M \\ m \\ \end{array} } \right)\left( { - 1} \right)^{{\left( {m - 1} \right)}} \mathop \sum \limits_{n = 1}^{N} \left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)\left( { - 1} \right)^{{\left( {n - 1} \right)}} \\ & \qquad\times \;exp\left[ { - \frac{{\left( {m + n} \right)\tilde{\Omega }_{th} }}{{\tilde{\theta }_{{BSR_{m} }} }} - \frac{{\tilde{\Omega }_{th} }}{{\tilde{\theta }_{{R_{m} U_{1} }} }}} \right]2\sqrt {\mu_{1} } K_{1} \left( {2\sqrt {\mu_{1} } } \right) \\ \end{aligned} $$

(37)

By substituting (33) and (37) in (19) outage probability of ${U}_{1}$ can be calculated as given in (20).

Appendix 2

Proof of Proposition 2

Similar to Proposition 1, the outage probability of ${U}_{2}$ can be derived as follows,

In (21), $\mathit{Pr}\left(\frac{{a}_{1}{X}_{2}}{{a}_{2}{X}_{2}+1}<{\Omega }_{th}\right)$ denote the CDF of link $BS$ to ${U}_{2}$, which is represented as

$$ F_{{\Omega_{{BSU_{2} }} }} \left( {\Omega_{th} } \right) = F_{{X_{2} }} \left( {\tilde{\Omega }_{th} } \right) = \left[ {1 - exp\left( { - \frac{\emptyset }{{\tilde{\theta }_{{BSU_{2} }} }}} \right)} \right] $$

(38)

The second part of (21), $Pr\left(\frac{{a}_{2}{Y}_{m}{Z}_{2m}}{{Y}_{m}+{Z}_{2m}+1}<{\Omega }_{th}\right)$ represents the CDF of the link ${R}_{m}$ to ${U}_{2}$ which can be solved by calculating the CDF of RV’s: ${Y}_{m}$ and ${Z}_{2m}$ as given in (9) and (10).

The CDF of ${Z}_{2m}$ is given as

$$ F_{{Z_{2m} { }}} \left( {\Omega_{th} } \right) = 1 - exp\left( { - \frac{{\Omega_{th} }}{{\tilde{\theta }_{{R_{m} U_{2} }} }}} \right) $$

(39)

The CDF of the link ${R}_{m}$ to ${U}_{2}$ can be represented as

$$ \begin{aligned}& F_{{\Omega_{{R_{m} U_{2} }} }} \left( {\Omega_{th} } \right) \\ &\quad = F_{{Z_{2m} { }}} \left( \emptyset \right) + \mathop \smallint \limits_{{\tilde{\Omega }_{th} }}^{\infty } F_{{Y_{m} }} \left( {\frac{{z\Omega_{th} + \Omega_{th} }}{{z\left( {a_{1} - a_{2} \Omega_{th} } \right) - \Omega_{th} }}} \right)f_{{Z_{2m} { }}} \left( z \right)dz \\ &\quad = F_{{Z_{2m} { }}} \left( \emptyset \right) + \mathop \smallint \limits_{\emptyset }^{\infty } F_{{Y_{m} }} \left( {\frac{{\emptyset \left( {z + 1} \right)}}{z - \emptyset }} \right)f_{{Z_{2m} { }}} \left( z \right)dz \\ \end{aligned} $$

(40)

By substituting (34) and (39) in (40) ${F}_{{\Omega }_{{R}_{m}{U}_{2}}}\left({\Omega }_{th}\right)$ is derived as

$$ \begin{aligned} &F_{{\Omega_{{R_{m} U_{2} }} }} \left( {\Omega_{th} } \right) \\ &\quad= 1 - \mathop \sum \limits_{m = 1}^{M} \left( {\begin{array}{*{20}c} M \\ m \\ \end{array} } \right)\left( { - 1} \right)^{{\left( {m - 1} \right)}} \mathop \sum \limits_{n = 1}^{N} \left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)\left( { - 1} \right)^{{\left( {n - 1} \right)}} \\ & \qquad\times exp\left[ { - \frac{{\left( {m + n} \right)\emptyset }}{{\tilde{\theta }_{{BSR_{m} }} }} - \frac{\emptyset }{{\tilde{\theta }_{{R_{m} U_{2} }} }}} \right]2\sqrt {\mu_{2} } K_{1} \left( {2\sqrt {\mu_{2} } } \right) \\ \end{aligned} $$

(41)

By substituting (38) and (41) in (21) outage probability of ${U}_{2}$ can be calculated as given in (22).