Fountain Coding Based Two-Way Relaying Cognitive Radio Networks Employing Reconfigurable Intelligent Surface and Energy Harvesting

Abstract

This paper examines two-way relaying cognitive radio networks utilizing fountain coding (FC), reconfigurable intelligent surfaces (RIS), and radio frequency energy harvesting (EH). In the proposed schemes, two secondary sources attempt to exchange data with each other through the assistance of an RIS deployed in the network. Using FC, one source sends its encoded packets to the other source, which must collect enough packets for a successful data recovery. The transmit power of the two sources is adjusted according to an interference constraint given by a primary user and the energy harvested from a power station. In the conventional scheme, one source continuously transmits FC packets to the other, using the maximum number of transmissions allowed. In the modified scheme, as soon as one source collects a sufficient number of FC packets, it notifies the other source to stop transmission. We derive closed-form expressions of outage probability (OP) at each source, system outage probability (SOP), and average number of FC-packet transmissions for the successful data exchange of the considered schemes over Rayleigh fading channels. Simulation results are provided to validate our analysis, to compare the performance of the considered schemes, and to examine the impact of key parameters on performance.

1. Introduction

Two-way relaying $\left(\mathrm{TWR}\right)$ is an effective method for addressing key challenges in wireless networks, including reliability, spectral efficiency, and energy constraints [1,2,3]. In $\mathrm{TWR},$ two source nodes exchange data through one common relay(s), which processes the received data and forwards the processed data to both sources. In [4,5,6], the common relays perform an XOR operation on the packets received from the two sources and then broadcast the XOR-ed packet back to them. Therefore, this scheme is known as the three-phase Digital Network Coding $\left(\mathrm{DNC}\right)$ $\mathrm{TWR}$. Moreover, the schemes introduced in [4,6] operate in cognitive radio $\left(\mathrm{CR}\right)$ environments, where the transmit power of the secondary users is limited by an interference threshold set by the primary users. In [7,8,9], the authors proposed two-phase Analogue Network Coding $\left(\mathrm{ANC}\right)$ $\mathrm{TWR}$ schemes, where the common relays amplify the signals received from two sources during the first phase and then broadcast the amplified signals to both sources in the second phase. Although the $\mathrm{ANC}$ $\mathrm{TWR}$ schemes achieve higher throughput than the DNC TWR ones, the sources in $\mathrm{ANC}$ $\mathrm{TWR}$ must perform interference cancellation, which is too complex to implement in practice. In [10,11,12,13], joint Successive Interference Cancellation $\left(\mathrm{SIC}\right)$ and $\mathrm{DNC}$ are applied at the common relays and these $\mathrm{TWR}$ schemes also use only two phases for the data exchange. In particular, during the first phase, both sources simultaneously transmit their packets to the common relays with different transmit power levels. The common relays then perform $\mathrm{SIC}$ to decode the received packets. Finally, the relays apply $\mathrm{XOR}$ to the decoded packets and transmit the $\mathrm{XOR-ed}$ packet to both sources in the second phase. Recently, many $\mathrm{TWR}$ models utilizing new communication techniques have been developed, proposed, and analyzed. The authors in [14,15] studied the performance of the $\mathrm{TWR}$ schemes using radio-frequency energy harvesting $\left(\mathrm{EH}\right)$, where the transmitters have to harvest energy from the radio signals of the surrounding nodes to transmit data. In [16,17,18], the TWR schemes using full-duplex $\left(\mathrm{FD}\right)$ techniques were proposed, where the source and/or relay nodes were equipped with multi-antennas. Although the $\mathrm{FD}$ $\mathrm{TWR}$ schemes use only one time slot for the data exchange between two sources, they require high synchronization between all nodes as well as complex interference cancellation implementation. However, the related works in [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] did not consider fountain coding $\left(\mathrm{FC}\right)$, which is studied in this paper.

Fountain coding $\left(\mathrm{FC}\right)$ [19,20] can be efficiently used in wireless communication applications due to its simple implementation and environment condition adaptation ability. In fact, an $\mathrm{FC}$ source encodes its data to generate encoded packets (or $\mathrm{FC}$ packets), which are continuously sent to a destination. The destination only needs to collect a sufficient number of $\mathrm{FC}$ packets to recover the original data from the source. Another advantage of $\mathrm{FC}$ is to provide high information security. As proven in [21,22,23], if the destination receives a sufficient number of $\mathrm{FC}$ packets before the eavesdroppers, the source data will be secure. The authors in [24] proposed a cooperative transmission model that employs $\mathrm{FC}$, non-orthogonal multiple access $\left(\mathrm{NOMA}\right)$, cooperative jamming technique, and intelligent reflective surfaces $\left(\mathrm{IRS}\right)$ (or reconfigurable intelligent surface $\left(\mathrm{RIS}\right)$) to enhance secrecy performance.

$\mathrm{IRS}$ or $\mathrm{RIS}$ are flat-structured surfaces whose arrays of passive and programmable components used to reflect incoming signals to the intended receivers for enhancing the quality of the received signals [25,26]. Published works [27] evaluated secrecy performance of $\mathrm{IRS}$-assisted wireless communication networks with presence of an eavesdropper. The authors in [28] proposed a down-link system where a multi-antenna base station uses $\mathrm{NOMA}$ to serve two users with the help of $\mathrm{RIS}$. Additionally, both continuous and discrete phase shifting were considered in [28]. Another report [29] also considered the $\mathrm{IRS}$-assisted wireless communication system using $\mathrm{NOMA}$, and evaluated the performance of the proposed system in an interference-limited environment. In [30], the authors considered a multi-IRS down-link scheme utilizing wirelessly $\mathrm{EH}$. Recently, the $\mathrm{TWR}$ schemes using $\mathrm{RIS}$ have gained much attention of researchers. In [31], the authors evaluated outage probability $\left(\mathrm{OP}\right)$ and average throughput of $\mathrm{IRS}$-aided $\mathrm{TWR}$ schemes, where two sources communicate through the $\mathrm{RIS}$ (instead of common relays in the conventional $\mathrm{TWR}$ schemes). Reference [32] analyzed performance of $\mathrm{IRS}$-aided $\mathrm{TWR}$ networks employing $\mathrm{SIC}$, in terms of $\mathrm{OP}$ and ergodic rate. The authors in [33] proposed three power allocation algorithms for $\mathrm{RIS}$-based Decode-and-forward $\left(\mathrm{DF}\right)$ $\mathrm{TWR}$ models to improve the system sum rate. In [34,35], $\mathrm{IRS}$-aided TWR networks using full-duplex techniques were proposed and analyzed. However, the published works [27,28,29,30,31,32,33,34,35] did not apply $\mathrm{FC}$ into the $\mathrm{RIS}$-based $\mathrm{TWR}$ systems.

This paper investigates $\mathrm{TWR}$ $\mathrm{CR}$ schemes that incorporate $\mathrm{FC}$, $\mathrm{RIS}$, and wirelessly energy harvesting $\left(\mathrm{EH}\right)$. In our model, two secondary sources aim to exchange data with the help of a RIS deployed in the network. Using $\mathrm{FC}$, one source continuously sends $\mathrm{FC}$ packets until the other source has collected enough to fully recover the original data. Moreover, the transmit power of each source is adjusted based on an interference constraint set by a primary user and the energy harvested from a power station. The new points and main contributions of this work are summarized as follows:

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Firstly, this paper considers two $\mathrm{TWR}$ schemes, i.e., conventional scheme (named Cov-Scm) and modified scheme (named Mod-Scm). The purpose of proposing the Mod-Scm scheme is to enhance the reliability of data transmission and reduce delay time, compared to the Cov-Scm.

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Secondly, we derive closed-form expressions for $\mathrm{OP}$ at each source, system outage probability $\left(\mathrm{SOP}\right)$, and average number of $\mathrm{FC}$ packet transmissions needed for successful data exchange in the proposed schemes over Rayleigh fading channels.

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Next, simulation results are presented to validate our analytical findings and compare the performance of the considered schemes.

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Finally, we examine the effects of key parameters on overall performance. The results also present that the Mod-Scm scheme obtains better performance, as compared with the Cov-Scm scheme, in terms of reliability (OP, SOP) and delay time (average number of FC packet transmission).

The remaining structure of this paper is as follows: Section 2 describes the system model for the considered $\mathrm{TWR}$ $\mathrm{CR}$ schemes along with operational principles. In Section 3, we compute the performance via mathematical expressions. Section 4 validates the analytical findings through simulations. Finally, Section 5 gives important conclusions and insights.

2. System Model

Figure 1 shows the system model of the proposed $\mathrm{FC}$-based $\mathrm{TWR}$$\mathrm{CR}$ model, where secondary sources (${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$) exchange data with each other.

Let $m_1$ and $m_2$ denote the data sent by ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$, respectively. Since direct communication between ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$ is outage due to their far distance, a reconfigurable intelligent surface $\left(\mathrm{RIS}\right)$ is deployed to assist these data exchange. Let $K\left(K\ge 2\right)$ represent the number of small reflective elements in the $\mathrm{RIS}$. Additionally, a power beacon station (denoted by $\mathrm{SB}$) is deployed in the secondary network to provide energy for ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$. To prevent co-channel interference, the frequencies used for energy harvesting differ from those used for the data transmission. The transmit power of ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$ is constrained by an interference threshold established by a primary user ($\mathrm{PU}$). Assume that all channels experience block Rayleigh fading and that all devices are single-antenna nodes.

The proposed scheme can be applied to IoT networks, where ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$ are IoT devices with limited power and energy. Therefore, the station power (SB) is deployed to provide energy for two source nodes. Moreover, due to spectrum scarcity, the underlay cognitive radio technique is employed for the IoT networks.

Following the operational principle of $\mathrm{FC},$ ${\mathrm{SS}}_i\left(i=1,2\right)$ divides $m_i$ into small, equally sized packets, which are then used to create $\mathrm{FC}$ packets. Let $p_i$ denote one FC packet of ${\mathrm{SS}}_i$. To successfully recover $m_i$, ${\mathrm{SS}}_j\left(j=1,2,j\ne i\right)$ must collect at least $H_{\min ,i}$ packets $p_i$. In addition, let $N_{\max ,i}$ as the maximum number of $\mathrm{FC}$ packets sent by ${\mathrm{SS}}_i.$ For simplicity in presentation and analysis, we can assume that $H_{\min ,i}=H_{\min ,j}=H_{\min }$ and $N_{\max ,i}=N_{\max ,j}=N_{\max }$ $\forall i,j$.

In the conventional $\mathrm{FC-TWRCR}$ scheme $\left(\mathrm{Cov-Scm}\right)$, ${\mathrm{SS}}_1$ continuously transmits packets $p_1$ to ${\mathrm{SS}}_2$ through the $\mathrm{RIS}$. After $N_{\max }$ transmission times, ${\mathrm{SS}}_1$ stops the transmission. If ${\mathrm{SS}}_2$ correctly receives at least $H_{\min }$ packets $p_1$, the data transmission is successful, and otherwise, ${\mathrm{SS}}_2$ experiences an outage. Then, ${\mathrm{SS}}_2$ in turn transmits packets $p_2$ to ${\mathrm{SS}}_1$ through the $\mathrm{RIS}$, also using $N_{\max }$ transmission times. Similarly, for the successful recovery of $m_2$, ${\mathrm{SS}}_1$ must receive at least $H_{\min }$ packets $p_2$ by the end of this transmission.

In the modified $\mathrm{FC-TWRCR}$ scheme $\left(\mathrm{Mod-Scm}\right)$, ${\mathrm{SS}}_i\left(i=1,2\right)$ first transmits packets $p_i$ to ${\mathrm{SS}}_j\left(j=1,2,j\ne i\right)$. If ${\mathrm{SS}}_j$ gathers enough $H_{\min }$ packets $p_i$ after $N_i$ transmission times $\left(N_i\le N_{\max }\right)$, ${\mathrm{SS}}_j$ sends an ACK message back to ${\mathrm{SS}}_i$. Upon receiving the ACK message, ${\mathrm{SS}}_i$ ends its transmission, and ${\mathrm{SS}}_j$ begins its transmission. Notably, if $N_i=N_{\max }$, ${\mathrm{SS}}_j$ does not need to send feedback to ${\mathrm{SS}}_i$ because ${\mathrm{SS}}_i$ must stop its transmission, regardless of whether ${\mathrm{SS}}_j$ has received enough $H_{\min }$ packets $p_i$ or not. Otherwise, if $N_i<N_{\max }$, the remaining transmission times $\left(N_{\max }-N_i\right)$ can be allocated to ${\mathrm{SS}}_j$ in the second transmission phase, i.e., allowing ${\mathrm{SS}}_j$ to send at most $N_{\max }+\left(N_{\max }-N_i\right)$ packets $p_j$ to ${\mathrm{SS}}_i$. Similarly, as soon as ${\mathrm{SS}}_i$ receives enough $H_{\min }$ packets $p_j$, it also sends an ACK message to inform ${\mathrm{SS}}_j$.

Remark 1.

In the $\mathrm{Cov-Scm}$ scheme [36], the data transmission of ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$ operates independently, making the transmission order of ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$ irrelevant. However, in the $\mathrm{Mod-Scm}$ scheme, the system performance depends on whether ${\mathrm{SS}}_1$ or ${\mathrm{SS}}_2$ transmits first (this issue will be examined in Section 4). Moreover, the $\mathrm{Cov-Scm}$ scheme always uses a total of $2N_{\max }$ transmission times, while the $\mathrm{Mod-Scm}$ scheme uses fewer, because the transmission stops as soon as each source gathers enough desired packets.

Let $g_{\mathrm{XY}}$ represent the channel gains between nodes X and Y, and its distribution functions expressed as

$$F_{g_{\mathrm{XY}}}\left(x\right)=1-\exp \left(-{\lambda }_{\mathrm{XY}}x\right),f_{g_{\mathrm{XY}}}\left(x\right)={\lambda }_{\mathrm{XY}}\exp \left(-{\lambda }_{\mathrm{XY}}x\right),$$

(1)

where $F_{g_{\mathrm{XY}}}\left(.\right)$ and $f_{g_{\mathrm{XY}}}\left(.\right)$ are cumulative distribution function $\left(\mathrm{CDF}\right)$ and probability density function $\left(\mathrm{PDF}\right)$ of $g_{\mathrm{XY}}$, respectively, ${\lambda }_{\mathrm{XY}}={\left(d_{\mathrm{XY}}\right)}^{\mathrm{PL}}$ [36] with $\mathrm{PL} \left(2\le \mathrm{PL}\le 6\right)$ being the path-loss factor and $d_{\mathrm{XY}}$ being the physical distance between $\mathrm{X}$ and $\mathrm{Y}$.

Let $d_{{\mathrm{XRIS}}_k}$ as the distance between the node $\mathrm{X}$ and the $k-\mathrm{th}$ reflector component of the $\mathrm{RIS}$, where $k=1,2,\dots ,K$,$\mathrm{X}\in \left\{{\mathrm{SS}}_1,{\mathrm{SS}}_2\right\}$. As assumed in [37], we can assume that all the distances $d_{{\mathrm{XRIS}}_k}$ are identical, i.e., $d_{{\mathrm{XRIS}}_k}=d_{\mathrm{XRIS}}$ (${\lambda }_{{\mathrm{XRIS}}_k}={\lambda }_{\mathrm{XRIS}}$) $\forall k$.

Considering the transmission of a packet $p_i$ from ${\mathrm{SS}}_i$ to ${\mathrm{SS}}_j$ via the $\mathrm{RIS}$. Assume that the total delay for each transmission of $p_i$ is normalized to 01 (time unit). During the interval $\alpha \left(0<\alpha <1\right)$, ${\mathrm{SS}}_i$ harvests energy from $\mathrm{SB}$, and its harvested energy can be calculated as

$$Q_i=\eta \alpha P_{\mathrm{SB}}{g}_{{\mathrm{SBSS}}_i},$$

(2)

where $\eta$ represents a conversion efficiency, and $P_{\mathrm{SB}}$ is the transmit power of $\mathrm{SB}$.

The remaining interval $\left(1-\alpha \right)$ is allocated for the transmission of $p_i$. Therefore, the average transmit power of ${\mathrm{SS}}_i$ can be formulated as

$$P_{{\mathrm{SS}}_i}^{(1)}={\displaystyle \frac{Q_i}{1-\alpha }}=\chi P_{\mathrm{SB}}{g}_{{\mathrm{SBSS}}_i},$$

(3)

where $\chi =\eta \alpha /\left(1-\alpha \right).$ Moreover, the transmit power of ${\mathrm{SS}}_i$ must satisfy the interference threshold set by PU, i.e.,

$$P_{{\mathrm{SS}}_i}^{(2)}={\displaystyle \frac{I_{\mathrm{PU}}}{g_{{\mathrm{SS}}_i\mathrm{PU}}}},$$

(4)

where $I_{\mathrm{PU}}$ is the interference threshold.

From Equations (3) and (4), the transmit power of ${\mathrm{SS}}_i$ can be formulated by

$$P_{{\mathrm{SS}}_i}=\min \left(P_{{\mathrm{SS}}_i}^{(1)},P_{{\mathrm{SS}}_i}^{(2)}\right)=\min \left(\chi P_{\mathrm{SB}}{g}_{{\mathrm{SBSS}}_i},{\displaystyle \frac{I_{\mathrm{PU}}}{g_{{\mathrm{SS}}_i\mathrm{PU}}}}\right).$$

(5)

Next, ${\mathrm{SS}}_i$ transmits $p_i$ to ${\mathrm{SS}}_j$, and the received signal at ${\mathrm{SS}}_j$ is given as (see [38,39,40]):

$$y_{{\mathrm{SS}}_j}=\sqrt{P_{{\mathrm{SS}}_i}}{\displaystyle \sum _{k=1}^K{h}_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}}{r}_{{\mathrm{RIS}}_k}{h}_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}{p}_i^{\mathrm{modu}}+n_{{\mathrm{SS}}_j},$$

(6)

where $h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}$ and $h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}$ are channel coefficients of the ${\mathrm{SS}}_i\to {\mathrm{RIS}}_k$ and ${\mathrm{RIS}}_k\to {\mathrm{SS}}_j$ links, respectively, $r_{{\mathrm{RIS}}_k}$ is response of the $k-\mathrm{th}$ component of the $\mathrm{RIS}$, $p_i^{\mathrm{modu}}$ is the modulated signals of $p_i$, and $n_{{\mathrm{SS}}_j}$ is zero-mean Gaussian noise at ${\mathrm{SS}}_j$. We assume that variance of all Gaussian noises is identical and equal to ${\sigma }_0^2$, i.e., $\mathrm{Var}\left\{n_{{\mathrm{SS}}_j}\right\}={\sigma }_0^2$.

We note that $g_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}={\left|h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}\right|}^2$ and $g_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}={\left|h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}\right|}^2$, where $\left|h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}\right|$ and $\left|h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}\right|$ are amplitude of $h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}$ and $h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}$, respectively. Moreover, we can express $h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}$ and $h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}$ in exponential form as follows: $h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}=\left|h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}\right|\exp \left(-j{\varsigma }_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}\right)$ and $h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}=\left|h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}\right|\exp \left(-j{\varsigma }_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}\right)$, where ${\varsigma }_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}$ and ${\varsigma }_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}$ are phases of $h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}$ and $h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}$, respectively. Similarly to [37], we can express $r_{{\mathrm{RIS}}_k}$ as $r_{{\mathrm{RIS}}_k}=\exp \left(j{\varsigma }_{{\mathrm{RIS}}_k}\right)$, where ${\varsigma }_{{\mathrm{RIS}}_k}$ is the phase response of $r_{{\mathrm{RIS}}_k}$, which can be optimally adjusted by ${\varsigma }_{{\mathrm{RIS}}_k}={\varsigma }_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}+{\varsigma }_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}$. Therefore, the maximum $\mathrm{SNR}$ obtained at ${\mathrm{SS}}_j$ for decoding $p_i$ can be given as

$$\begin{array}{ll}{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}& ={\displaystyle \frac{P_{{\mathrm{SS}}_i}{\left|{\displaystyle \sum _{k=1}^K{h}_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}{r}_{{\mathrm{RIS}}_k}{h}_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}}\right|}^2}{{\sigma }_0^2}}={\displaystyle \frac{P_{{\mathrm{SS}}_i}{\left({\displaystyle \sum _{k=1}^K\left|h_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}\right|\left|h_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}\right|}\right)}^2}{{\sigma }_0^2}}\\ & ={\displaystyle \frac{P_{{\mathrm{SS}}_i}{\left({\displaystyle \sum _{k=1}^K\sqrt{g_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}}\sqrt{g_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}}}\right)}^2}{{\sigma }_0^2}}={\displaystyle \frac{P_{{\mathrm{SS}}_i}}{{\sigma }_0^2}}{\left(Z_{i,\mathrm{Sum}}\right)}^2,\end{array}$$

(7)

where $Z_{i,\mathrm{Sum}}={\displaystyle \sum _{k=1}^K\sqrt{g_{{\mathrm{SS}}_i{\mathrm{RIS}}_k}}\sqrt{g_{{\mathrm{RIS}}_k{\mathrm{SS}}_j}}}$.

Using [37], we can obtain $\mathrm{CDF}$ of $Z_{i,\mathrm{Sum}}$ as

$$F_{Z_{i,\mathrm{Sum}}}\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\theta \right)}}\gamma \left(\theta ,{\displaystyle \frac{x}{\varphi }}\right),$$

(8)

where $\mathsf{\Gamma }\left(x\right)={\displaystyle {\int }_0^{+\infty }{t}^{x-1}\exp \left(-t\right)dt}$ and $\gamma \left(x,y\right)={\displaystyle {\int }_0^y{t}^{x-1}\exp \left(-t\right)dt}$ are gamma and lower incomplete gamma functions [41], respectively, and

$$\theta ={\displaystyle \frac{K{\pi }^2}{16-{\pi }^2}},\varphi ={\displaystyle \frac{16-{\pi }^2}{4\pi \sqrt{{\lambda }_{{\mathrm{SS}}_1\mathrm{RIS}}{\lambda }_{{\mathrm{SS}}_2\mathrm{RIS}}}}}.$$

(9)

From Equations (7) and (8), we can see that $Z_{1,\mathrm{sum}}$ and $Z_{2,\mathrm{sum}}$ have the same $\mathrm{CDF}$. Hence, we can omit the subscripts $i$ and $j$, i.e., $F_{Z_{i,\mathrm{Sum}}}\left(x\right)=F_{Z_{j,\mathrm{Sum}}}\left(x\right)=F_{Z_{\mathrm{Sum}}}\left(x\right).$ Then, we can obtain $\mathrm{CDF}$ of ${\left(Z_{1,\mathrm{sum}}\right)}^2$ and ${\left(Z_{2,\mathrm{sum}}\right)}^2$ as

$$F_{{\left(Z_{\mathrm{Sum}}\right)}^2}\left(x\right)=F_{Z_{\mathrm{Sum}}}\left(\sqrt{x}\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\theta \right)}}\gamma \left(\theta ,{\displaystyle \frac{\sqrt{x}}{\varphi }}\right).$$

(10)

Differentiating Equation (9) with respect to $x$, we obtain $\mathrm{PDF}$ of $Z_{1,\mathrm{sum}}$ and $Z_{2,\mathrm{sum}}$ as

$$f_{{\left(Z_{\mathrm{Sum}}\right)}^2}\left(x\right)={\displaystyle \frac{1}{2\mathsf{\Gamma }\left(\theta \right){\left(\varphi \right)}^{\theta }}}{x}^{{\displaystyle \frac{\theta }{2}}-1}\exp \left(-{\displaystyle \frac{\sqrt{x}}{\varphi }}\right).$$

(11)

Next, setting $Y_i=P_{{\mathrm{SS}}_i}/{\sigma }_0^2$, we see that $Y_i$ is also a random variable whose $\mathrm{CDF}$ can be formulated as

$$\begin{array}{ll}{F}_{Y_i}\left(x\right)& =\mathrm{Pr}\left(\min \left({\displaystyle \frac{\chi P_{\mathrm{SB}}}{{\sigma }_0^2}}{g}_{{\mathrm{SBSS}}_i},{\displaystyle \frac{I_{\mathrm{PU}}}{{\sigma }_0^2{g}_{{\mathrm{SS}}_i\mathrm{PU}}}}\right)<x\right)\\ & =1-\mathrm{Pr}\left(g_{{\mathrm{SBSS}}_i}\ge {\displaystyle \frac{{\sigma }_0^{2}x}{\chi P_{\mathrm{SB}}}}\right)\mathrm{Pr}\left(g_{{\mathrm{SS}}_i\mathrm{PU}}<{\displaystyle \frac{I_{\mathrm{PU}}}{{\sigma }_0^{2}x}}\right)\\ & =1-\left[1-F_{g_{{\mathrm{SBSS}}_i}}\left({\displaystyle \frac{{\sigma }_0^{2}x}{\chi P_{\mathrm{SB}}}}\right)\right]F_{g_{{\mathrm{SS}}_i\mathrm{PU}}}\left({\displaystyle \frac{I_{\mathrm{PU}}}{{\sigma }_0^{2}x}}\right).\end{array}$$

(12)

Substituting Equation (1) into Equation (12), we obtain

$$\begin{array}{l}{F}_{Y_i}\left(x\right)=1-\exp \left(-{\displaystyle \frac{{\lambda }_{{\mathrm{SBSS}}_i}{\sigma }_0^2}{\chi P_{\mathrm{SB}}}}x\right)\left[1-\exp \left(-{\displaystyle \frac{{\lambda }_{{\mathrm{SS}}_i\mathrm{PU}}{I}_{\mathrm{PU}}}{{\sigma }_0^{2}x}}\right)\right]\\ =1-\exp \left(-{\displaystyle \frac{{\lambda }_{{\mathrm{SBSS}}_i}{\sigma }_0^2}{\chi P_{\mathrm{SB}}}}x\right)+\exp \left(-{\displaystyle \frac{{\lambda }_{{\mathrm{SBSS}}_i}{\sigma }_0^2}{\chi P_{\mathrm{SB}}}}x-{\displaystyle \frac{{\lambda }_{{\mathrm{SS}}_i\mathrm{PU}}{I}_{\mathrm{PU}}}{{\sigma }_0^{2}x}}\right)\\ =1-\exp \left(-{\kappa }_{i}x\right)+\exp \left(-{\kappa }_{i}x-{\displaystyle \frac{{\mu }_i}{x}}\right),\end{array}$$

(13)

where ${\kappa }_i={\displaystyle \frac{{\lambda }_{{\mathrm{SBSS}}_i}{\sigma }_0^2}{\chi P_{\mathrm{SB}}}},{\mu }_i={\displaystyle \frac{{\lambda }_{{\mathrm{SS}}_i\mathrm{PU}}{I}_{\mathrm{PU}}}{{\sigma }_0^2}}$.

Finally, the channel capacity of the ${\mathrm{SS}}_i\to \mathrm{RIS}\to {\mathrm{SS}}_j$ link can be formulated as

$$C_{{\mathrm{SS}}_i{\mathrm{SS}}_j}=\left(1-\alpha \right){\log }_2\left(1+{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}\right).$$

(14)

3. Performance Evaluation

This section derives expressions of $\mathrm{CDF}$ of ${\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}$ $\left(F_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left(x\right)\right)$, where the probability that one $\mathrm{FC}$ packet $p_i$ can be unsuccessfully relayed from ${\mathrm{SS}}_i$ to ${\mathrm{SS}}_j$ $\left({\mathsf{\Phi }}_i\right)$, $\mathrm{OP}$ at each user, the system $\mathrm{OP}$ of the two considered schemes and the average number of transmission times of the $\mathrm{FC}$ packet for a successful data exchange in the $\mathrm{Cov-Scm}$ and $\mathrm{Mod-Scm}$ schemes $\left(T_{\mathrm{avg}}\right)$.

3.1. Derivation of $F_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left(x\right)$ and ${\mathsf{\Phi }}_i$

From Equation (7), we can formulate $F_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left(x\right)$ as

$$\begin{array}{ll}{F}_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left(x\right)& =\mathrm{Pr}\left(Y_i{\left(Z_{i,\mathrm{Sum}}\right)}^2<x\right)=\mathrm{Pr}\left(Y_i<{\displaystyle \frac{x}{{\left(Z_{i,\mathrm{Sum}}\right)}^2}}\right)\\ & ={\displaystyle {\int }_0^{+\infty }{F}_{Y_i}\left(\frac{x}{z}\right)f_{{\left(Z_{\mathrm{Sum}}\right)}^2}}\left(z\right)dz.\end{array}$$

(15)

Substituting Equations (11) and (13) into Equation (15), which yields

$$F_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left(x\right)=1-{\displaystyle \frac{1}{2\mathsf{\Gamma }\left(\theta \right){\left(\varphi \right)}^{\theta }}}{\displaystyle {\int }_0^{+\infty }\exp \left(-\frac{{\kappa }_{i}x}{z}\right)\left[1-\exp \left(-\frac{{\mu }_{i}z}{x}\right)\right]}{z}^{{\displaystyle \frac{\theta }{2}}-1}\exp \left(-{\displaystyle \frac{\sqrt{z}}{\varphi }}\right)dz.$$

(16)

By inter-changing variable $t=\sqrt{z}/\varphi$, we can rewrite (16) as

$$F_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left(x\right)=1-{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\theta \right)}}{\displaystyle {\int }_0^{+\infty }{t}^{\theta -1}\exp \left(-t\right)\exp \left(-\frac{{\kappa }_{i}x}{{\varphi }^2{t}^2}\right)\left[1-\exp \left(-\frac{{\mu }_i{\varphi }^2{t}^2}{x}\right)\right]}dt.$$

(17)

The integral in Equation (17) can be computed by using computer tools.

Next, we define ${\mathsf{\Phi }}_i$ as the probability that $C_{{\mathrm{SS}}_i{\mathrm{SS}}_j}$ is below a pre-determined data rate $C_{\mathrm{th}}$. Indeed, ${\mathsf{\Phi }}_i$ can be calculated as

$${\mathsf{\Phi }}_i=\mathrm{Pr}\left(C_{{\mathrm{SS}}_i{\mathrm{SS}}_j}<C_{\mathrm{th}}\right)=F_{{\gamma }_{{\mathrm{SS}}_i{\mathrm{SS}}_j}}\left({\gamma }_{\mathrm{th}}\right),$$

(18)

where ${\gamma }_{\mathrm{th}}=2^{C_{\mathrm{th}}/\left(1-\alpha \right)}-1$.

We note that the probability that one packet $p_i$ is correctly relayed from ${\mathrm{SS}}_i$ to ${\mathrm{SS}}_j$ and is computed as $\overline{{\mathsf{\Phi }}_i}=1-{\mathsf{\Phi }}_i.$

3.2. OP at Each User

In the $\mathrm{Cov-Scm}$ scheme, we first denote $r_2$ as the number of packet $p_1$ that ${\mathrm{SS}}_2$ successfully receives after the transmission from ${\mathrm{SS}}_1$ is complete. As mentioned above, if $r_2<H_{\min }$, ${\mathrm{SS}}_2$ cannot recover the data $m_1$, resulting in an outage at ${\mathrm{SS}}_2$. Therefore, we can formulate $\mathrm{OP}$ at ${\mathrm{SS}}_2$ as

$${\mathrm{OP}}_2^{\mathrm{Cov-Scm}}={\displaystyle \sum _{r_2=0}^{H_{\min }-1}{C}_{N_{\max }}^{r_2}}{\left(1-{\mathsf{\Phi }}_1\right)}^{r_2}{\left({\mathsf{\Phi }}_1\right)}^{N_{\max }-r_2},$$

(19)

where $C_{N_{\max }}^{r_2}\left(C_{N_{\max }}^{r_2}={\displaystyle \frac{N_{\max }!}{r_2!\left(N_{\max }-r_2\right)!}}\right)$ is the binomial coefficient in NewTon expansion, and also represents the total number of possible cases in which ${\mathrm{SS}}_2$ can decode correctly the $p_1$ packet. Then, substituting Equation (18) into Equation (19), we obtain expression of $\mathrm{OP}$ at ${\mathrm{SS}}_2$.

Similarly, $\mathrm{OP}$ at ${\mathrm{SS}}_1$ can be given as

$${\mathrm{OP}}_1^{\mathrm{Cov-Scm}}={\displaystyle \sum _{r_1=0}^{H_{\min }-1}{C}_{N_{\max }}^{r_1}}{\left(1-{\mathsf{\Phi }}_2\right)}^{r_1}{\left({\mathsf{\Phi }}_2\right)}^{N_{\max }-r_1}.$$

(20)

In the $\mathrm{Mod-Scm}$ scheme; assume that the source ${\mathrm{SS}}_i$ transmits its data first. Similarly to Equations (19) and (20), $\mathrm{OP}$ of ${\mathrm{SS}}_j$ can be calculated as

$${\mathrm{OP}}_j^{\mathrm{Mod-Scm}}={\displaystyle \sum _{r_j=0}^{H_{\min }-1}{C}_{N_{\max }}^{r_j}}{\left(1-{\mathsf{\Phi }}_i\right)}^{r_j}{\left({\mathsf{\Phi }}_i\right)}^{N_{\max }-r_j}.$$

(21)

As mentioned above, let $N_i$ denote the number of transmission of ${\mathrm{SS}}_i$, where $H_{\min }\le N_i\le N_{\max }$. If $N_i<N_{\max }$, this indicates the ${\mathrm{SS}}_j$ source can gather enough $H_{\min }$ packets $p_i$ before ${\mathrm{SS}}_i$ reaches its maximum transmission times $\left(N_{\max }\right)$. The probability that ${\mathrm{SS}}_j$ obtains enough $H_{\min }$ packets $p_i$ after $N_i$ attempts of ${\mathrm{SS}}_i$ is computed as

$$\mathrm{Pr}\left(\left.r_j=H_{\min }\right|N_i<N_{\max }\right)=C_{N_i-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_i\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_i\right)}^{N_i-H_{\min }}.$$

(22)

Equation (22) can be explained that after the $\left(N_i-1\right)-\mathrm{th}$ transmission of ${\mathrm{SS}}_i$, ${\mathrm{SS}}_j$ must receive successfully $\left(H_{\min }-1\right)$ packets $p_i$, and the probability of this event is given as $C_{N_i-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_i\right)}^{H_{\min }-1}{\left({\mathsf{\Phi }}_i\right)}^{N_i-H_{\min }}.$ Then, at the $N_i-\mathrm{th}$ transmission of ${\mathrm{SS}}_i$, ${\mathrm{SS}}_j$ decodes correctly $p_i$, and hence, we obtain Equation (22).

Next, we consider the case where $N_i=N_{\max }$, and this case occurs when ${\mathrm{SS}}_j$ correctly receives at most $\left(H_{\min }-1\right)$ packets $p_i$ after $\left(N_{\max }-1\right)$ transmission times of ${\mathrm{SS}}_i.$ As a result, we can calculate the probability of this case as

$$\mathrm{Pr}\left(N_i=N_{\max }\right)={\displaystyle \sum _{r_j=0}^{H_{\min }-1}{C}_{N_{\max }-1}^{r_j}{\left(1-{\mathsf{\Phi }}_i\right)}^{r_j}{\left({\mathsf{\Phi }}_i\right)}^{N_{\max }-1-r_j}}.$$

(23)

We now consider $\mathrm{OP}$ at ${\mathrm{SS}}_i$ under condition that the maximum number of transmissions of ${\mathrm{SS}}_j$ is $2N_{\max }-N_i$; similar to (19)–(21), we have

$${\mathrm{OP}}_{i|N_i}^{\mathrm{Mod-Scm}}={\displaystyle \sum _{r_i=0}^{H_{\min }-1}{C}_{2N_{\max }-N_i}^{r_i}}{\left(1-{\mathsf{\Phi }}_j\right)}^{r_i}{\left({\mathsf{\Phi }}_j\right)}^{2N_{\max }-N_i-r_i}.$$

(24)

From Equations (22)–(24), $\mathrm{OP}$ at ${\mathrm{SS}}_i$ can be expressed as follows:

$$\begin{array}{l}{\mathrm{OP}}_i^{\mathrm{Mod-Scm}}={\displaystyle \sum _{N_i=H_{\min }}^{N_{\max }}\left[\mathrm{Pr}\left(\left.r_j=H_{\min }\right|N_i<N_{\max }\right)+\mathrm{Pr}\left(N_i=N_{\max }\right)\right]\times {\mathrm{OP}}_{i|N_i}^{\mathrm{Mod-Scm}}}\\ ={\displaystyle \sum _{N_i=H_{\min }}^{N_{\max }}\left[\begin{array}{l}{C}_{N_i-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_i\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_i\right)}^{N_i-H_{\min }}u\left(N_{\max }-1-N_i\right)\\ +{\displaystyle \sum _{r_j=0}^{H_{\min }-1}{C}_{N_{\max }-1}^{r_j}{\left(1-{\mathsf{\Phi }}_i\right)}^{r_j}{\left({\mathsf{\Phi }}_i\right)}^{N_{\max }-1-r_j}u\left(N_i-N_{\max }\right)}\end{array}\right]}\\ \times \left[{\displaystyle \sum _{r_i=0}^{H_{\min }-1}{C}_{2N_{\max }-N_i}^{r_i}}{\left(1-{\mathsf{\Phi }}_j\right)}^{r_i}{\left({\mathsf{\Phi }}_j\right)}^{2N_{\max }-N_i-r_i}\right],\end{array}$$

(25)

where

$$u\left(x\right)=\left\{\begin{array}{l}1,\mathrm{if}x\ge 0\\ 0,\mathrm{if}x<0\end{array}\right.$$

(26)

Remark 2.

From Equations (21) and (24), we can see that $\mathrm{OP}$ at  ${\mathrm{SS}}_j$  in the  $\mathrm{Cov-Scm}$  and $\mathrm{Mod-Scm}$ schemes is same. However, $\mathrm{OP}$ at  ${\mathrm{SS}}_i$  in  $\mathrm{Mod-Scm}$  is lower than that in $\mathrm{Cov-Scm}$ due to the higher number of transmission times of ${\mathrm{SS}}_j$ in $\mathrm{Mod-Scm}$.

3.3. System OP (SOP)

Since the proposed schemes involve two sources, this paper evaluates the combined performance of both sources, referred to as system performance or $\mathrm{SOP}$. Indeed, $\mathrm{SOP}$ is defined as the probability that one of two sources is outage. Therefore, we can express $\mathrm{SOP}$ of the scheme X, $\mathrm{X}\in \left\{\mathrm{Cov-Scm},\mathrm{Mod-Scm}\right\}$, as follows:

$${\mathrm{SOP}}^{\mathrm{X}}=1-\left({1-\mathrm{OP}}_1^{\mathrm{X}}\right)\left({1-\mathrm{OP}}_2^{\mathrm{X}}\right).$$

(27)

In Equation (27), $\left({1-\mathrm{OP}}_1^{\mathrm{X}}\right)\left({1-\mathrm{OP}}_2^{\mathrm{X}}\right)$ is the probability that both sources in the X scheme successfully recover their desired data.

3.4. Average Number of Transmission Times ($T_{avg}$)

At first, we can see that the $\mathrm{Cov-Scm}$ scheme always uses $2N_{\max }$ transmission times of the $\mathrm{FC}$ packets for each successful data exchange, i.e., $T_{\mathrm{avg}}^{\mathrm{Cov-Scm}}=2N_{\max }$.

For $\mathrm{Mod-Scm}$, we again assume that ${\mathrm{SS}}_i$ transmits its data first, and it stops its transmission after $N_i\left(N_i\le N_{\max }\right)$ attempts. By re-using Equation (22), we obtain the probability that ${\mathrm{SS}}_j$ sufficiently collects $H_{\min }$ packets $p_i$ after $N_i$ transmission times of ${\mathrm{SS}}_i$ as

$$\mathrm{Pr}\left(\left.r_j=H_{\min }\right|N_i\right)=C_{N_i-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_i\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_i\right)}^{N_i-H_{\min }}.$$

(28)

Next, let $N_j$ denote the number of transmission times of ${\mathrm{SS}}_j$, where $H_{\min }\le N_j\le 2N_{\max }-N_i$. Also, the probability that ${\mathrm{SS}}_i$ collects enough $H_{\min }$ packets $p_j$ after $N_j$ transmission times of ${\mathrm{SS}}_j$ is written as

$$\mathrm{Pr}\left(\left.r_i=H_{\min }\right|N_j\right)=C_{N_j-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_j\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_j\right)}^{N_j-H_{\min }}.$$

(29)

Now, we can formulate the average number of transmission times of two sources per successful data exchange in $\mathrm{Mod-Scm}$ as

$$T_{\mathrm{avg}}^{\mathrm{Mod-Scm}}={\displaystyle \frac{{\displaystyle \sum _{N_i=H_{\min }}^{N_{\max }}{\displaystyle \sum _{N_j=H_{\min }}^{2N_{\max }-N_i}\left(N_i+N_j\right)}}\mathrm{Pr}\left(\left.r_j=H_{\min }\right|N_i\right)\mathrm{Pr}\left(\left.r_i=H_{\min }\right|N_j\right)}{\left(1-{\mathrm{OP}}_1^{\mathrm{Mod-Scm}}\right)\left(1-{\mathrm{OP}}_2^{\mathrm{Mod-Scm}}\right)}},$$

(30)

where $\left(N_i+N_j\right)$ is the total number of transmission times of two sources in $\mathrm{Mod-Scm},$ and $\left(1-{\mathrm{OP}}_1^{\mathrm{Mod-Scm}}\right)\left(1-{\mathrm{OP}}_2^{\mathrm{Mod-Scm}}\right)$ is the probability that the data exchange between two sources is successful.

Substituting Equations (27)–(29) into Equation (30), we obtain an expression of $T_{\mathrm{avg}}^{\mathrm{Mod-Scm}}$ as $T_{\mathrm{avg}}^{\mathrm{Mod-Scm}}=A/B,$ where

$$A={\displaystyle \sum _{N_i=H_{\min }}^{N_{\max }}{\displaystyle \sum _{N_j=H_{\min }}^{2N_{\max }-N_i}\left(N_i+N_j\right)\left[\begin{array}{l}{C}_{N_i-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_i\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_i\right)}^{N_i-H_{\min }}\\ \times C_{N_j-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_j\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_j\right)}^{N_j-H_{\min }}\end{array}\right]}}.$$

(31)

$$\begin{array}{l}B=1-\left[1-{\displaystyle \sum _{r_j=0}^{H_{\min }-1}{C}_{N_{\max }}^{r_j}}{\left(1-{\mathsf{\Phi }}_i\right)}^{r_j}{\left({\mathsf{\Phi }}_i\right)}^{N_{\max }-r_j}\right]\\ \times \left\{\begin{array}{l}1-{\displaystyle \sum _{N_i=H_{\min }}^{N_{\max }}\left[\begin{array}{l}{C}_{N_i-1}^{H_{\min }-1}{\left(1-{\mathsf{\Phi }}_i\right)}^{H_{\min }}{\left({\mathsf{\Phi }}_i\right)}^{N_i-H_{\min }}u\left(N_{\max }-1-N_i\right)\\ +{\displaystyle \sum _{r_j=0}^{H_{\min }-1}{C}_{N_{\max }-1}^{r_j}{\left(1-{\mathsf{\Phi }}_i\right)}^{r_j}{\left({\mathsf{\Phi }}_i\right)}^{N_{\max }-1-r_j}u\left(N_i-N_{\max }\right)}\end{array}\right]}\\ \times \left[{\displaystyle \sum _{r_i=0}^{H_{\min }-1}{C}_{2N_{\max }-N_i}^{r_i}}{\left(1-{\mathsf{\Phi }}_j\right)}^{r_i}{\left({\mathsf{\Phi }}_j\right)}^{2N_{\max }-N_i-r_i}\right]\end{array}\right\}.\end{array}$$

(32)

4. Simulation and Analytical Results

Section 4 validates the formulas derived in Section 3 using Monte Carlo simulations. Throughout this section, we fix positions of two sources and the $\mathrm{PU}$ node at ${\mathrm{SS}}_1\left(0,0\right)$, ${\mathrm{SS}}_2\left(1,0\right)$, $\mathrm{PU}\left(0.5,-0.75\right)$, and $\mathrm{RIS}\left(0.5,0.75\right)$, while the $\mathrm{SB}$ station is positioned at $\mathrm{SB}\left(x_{\mathrm{SB}},0.5\right)$, where $0<x_{\mathrm{SB}}<1$. We also assign values to the following parameters as follows: $\mathrm{PL}=3$, ${\sigma }_0^2=1,$ $H_{\min }=5,$ $C_{\mathrm{th}}=1$, and $\eta =0.5$ (see

Table 1. Mathematical notations and their values.

Notations	Meaning	Value
$\mathrm{PL}$	Path-loss exponential	3
${\sigma }_0^2$	Variance of Gaussian noise	1
$H_{\min }$	Minimum number of FC packets required for data recovery	5
$N_{\max }$	Maximum number of FC packets exchanged between two sources	Change
$C_{\mathrm{th}}$	Target rate	1
$\eta$	Conversion efficiency	0.5
$\mathsf{\Delta }$	$\mathsf{\Delta }=P_{\mathrm{SB}}/{\sigma }_0^2$: Transmit SNR	Change
$x_{\mathrm{SB}}$	x-coordinate of the SB node	$0<x_{\mathrm{SB}}<1$
$\alpha$	Fraction of time allocated for EH	$0<\alpha <1$
${\mathsf{\Phi }}_i$	$\mathrm{Probability}\mathrm{that}\mathrm{one}\mathrm{FC}$$\mathrm{packet}{p}_i$$\mathrm{is}\mathrm{unsuccessfully}\mathrm{relayed}\mathrm{from}{\mathrm{SS}}_i$$\mathrm{to}{\mathrm{SS}}_j,$$\mathrm{where}i,j=1,2,i\ne j$	$0<{\mathsf{\Phi }}_i<1$

). In all results, we fix $I_{\mathrm{PU}}=0.5P_{\mathrm{SB}}$, and denote the simulation and analytical results by $\mathrm{SIM}$ and $\mathrm{ANA}$, respectively. It is noted that our derived expressions are applicable to all parameter values in practice. The reason we fix the values of these parameters is to focus on analyzing the impact of the key parameters (i.e., $\mathsf{\Delta }\left(\mathsf{\Delta }=P_{\mathrm{SB}}/{\sigma }_0^2\right),$ $\alpha ,$ $x_{\mathrm{SB}},$ $K,$ $N_{\max }$) on the OP and SOP performance of the considered schemes. Next, as shown in figures below, the $\mathrm{SIM}$ and $\mathrm{ANA}$ results align closely, verifying the accuracy of our derivations.

Figure 2 illustrates the probability of the unsuccessful transmission of the packet $p_i$ as a function of $\mathsf{\Delta }\left(\mathsf{\Delta }=P_{\mathrm{SB}}/{\sigma }_0^2\right)$ in dB with $x_{\mathrm{SB}}=0.7$ and $\alpha =0.35$. As seen, both ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ decrease as $\mathsf{\Delta }$ increases. This is due to the fact that increasing $\mathsf{\Delta }$ also increases the transmit power of ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$. Additionally, ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ with $K=5$ are lower than those with $K=3$, which is due to the improved quality of the ${\mathrm{SS}}_i\to {\mathrm{SS}}_j$ links at higher values of $K$. We also observe from Figure 2 that ${\mathsf{\Phi }}_2$ is lower than ${\mathsf{\Phi }}_1$. This is because ${\mathrm{SS}}_2$ is closer to $\mathrm{SB}$ than ${\mathrm{SS}}_1$, resulting in a higher average transmit power for ${\mathrm{SS}}_2$ as compared to ${\mathrm{SS}}_1$.

Figure 3 presents ${\mathsf{\Phi }}_i$ as a function of the fraction of time $\left(\alpha \right)$ allocated for the $\mathrm{EH}$ operation. The system parameters in this figure are set to $\mathsf{\Delta }=15\left(\mathrm{dB}\right)$ and $K=4$. We can see that both ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ change significantly with variations in $\alpha$. It is straightforward that with very low values of $\alpha$, the transmit power of ${\mathrm{SS}}_i$ is also low, resulting in the low channel capacity and high ${\mathsf{\Phi }}_i$. However, when $\alpha$ is very high, the time allocated for the data transmission phase is reduced, which also leads to low channel capacity and high ${\mathsf{\Phi }}_i$. Therefore, ${\mathsf{\Phi }}_i$ reaches its lowest value at a medium value of $\alpha$. For example, with $x_{\mathrm{SB}}=0.3$, ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ obtain their minimum values at $\alpha =0.5$. In addition, the position of $\mathrm{SB}$ significantly impacts on ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$. Indeed, as shown in Figure 3, the value of ${\mathsf{\Phi }}_1\left({\mathsf{\Phi }}_2\right)$ as $x_{\mathrm{SB}}=0.3$ is lowest (highest) because ${\mathrm{SS}}_1\left({\mathrm{SS}}_2\right)$ is nearest (farthest) to $\mathrm{SB}$. When $x_{\mathrm{SB}}=0.65$, ${\mathsf{\Phi }}_2$ is lower than ${\mathsf{\Phi }}_1$ because ${\mathrm{SS}}_2$ is closer to $\mathrm{SB}$ than ${\mathrm{SS}}_1$.

In Figure 4, we present $\mathrm{OP}$ at two sources in the $\mathrm{Cov-Scm}$ scheme as a function of $\mathsf{\Delta }$ (dB) when $x_{\mathrm{SB}}=0.35$, $\alpha =0.5$ and $N_{\max }=6$. With $x_{\mathrm{SB}}=0.35$, the distance from ${\mathrm{SS}}_1$ to $\mathrm{SB}$ is shorter than that from ${\mathrm{SS}}_2$ to $\mathrm{SB}$, resulting in ${\mathsf{\Phi }}_2<{\mathsf{\Phi }}_1$, and $\mathrm{OP}$ at ${\mathrm{SS}}_2$ is lower than that at ${\mathrm{SS}}_1$. Hence, we observe from Figure 4 that the $\mathrm{OP}$ performance of ${\mathrm{SS}}_2$ is better than that of ${\mathrm{SS}}_1$ for all values of $\mathsf{\Delta }$ and $K$. It is also shown that $\mathrm{OP}$ of both sources decreases when the values of $\mathsf{\Delta }$ and $K$ increase. Furthermore, the $\mathrm{OP}$ gap between the two sources also increases as $\mathsf{\Delta }$ increases. It is worth noting that the results obtained in this figure can be used to design/optimize the OP performance. For example, with $K=4$, the OP of both sources in the Cov-Scm is lower than 0.01 when the value of $\mathsf{\Delta }$ is from 11 dB to 20 dB. In other words, the SB station can use a minimum transmit power of 11 dB to ensure that OP of both sources remains below 0.01.

Figure 5 shows $\mathrm{OP}$ at two sources in the $\mathrm{Mod-Scm}$ scheme as a function of $\mathsf{\Delta }$ (dB) when $x_{\mathrm{SB}}=0.35$, $\alpha =0.5,$ $K=3$, and $N_{\max }=6$. Furthermore, we consider two scenarios: (i) ${\mathrm{SS}}_1$ transmits first (named $\mathrm{Mod-Scm-1}$); (ii) ${\mathrm{SS}}_2$ transmits first (named $\mathrm{Mod-Scm-2}$). We observe that in $\mathrm{Mod-Scm-1}$, $\mathrm{OP}$ of ${\mathrm{SS}}_2$ is lower than that of ${\mathrm{SS}}_1$ at low and medium $\mathsf{\Delta }$ values, and at high $\mathsf{\Delta }$ values, $\mathrm{OP}$ of ${\mathrm{SS}}_1$ is higher. In $\mathrm{Mod-Scm-2}$, $\mathrm{OP}$ of ${\mathrm{SS}}_2$ is always better than that of ${\mathrm{SS}}_1$ for all values of $\mathsf{\Delta }$. Moreover, the $\mathrm{OP}$ gap between the two sources in $\mathrm{Mod-Scm-2}$ is much higher than that in $\mathrm{Mod-Scm-1}$, and $\mathrm{OP}$ at ${\mathrm{SS}}_2$ and ${\mathrm{SS}}_1$ in $\mathrm{Mod-Scm-2}$ are lowest and highest, respectively. Therefore, Figure 5 shows that $\mathrm{Mod-Scm-1}$ achieves greater performance fairness for two sources.

To determine whether $\mathrm{Mod-Scm-1}$ or $\mathrm{Mod-Scm-2}$ performs better, Figure 6 compares $\mathrm{SOP}$ of all the considered schemes. In this figure, the parameters are set to $x_{\mathrm{SB}}=0.35$, $\alpha =0.5,$ $K=5$, and $N_{\max }=6$. As shown, $\mathrm{Mod-Scm-1}$ obtains the best $\mathrm{SOP}$ performance, while that of $\mathrm{Cov-Scm}$ is worst. We also observe that $\mathrm{SOP}$ of $\mathrm{Mod-Scm-1}$ is much lower than those of $\mathrm{Mod-Scm-2}$ and $\mathrm{Cov-Scm}$. Therefore, in this simulation, the source ${\mathrm{SS}}_1$ in $\mathrm{Mod-Scm}$ should be prioritized to transmit its data first.

Figure 7 presents both $\mathrm{OP}$ and $\mathrm{SOP}$ of the considered schemes as a function of $\alpha$ when $\mathsf{\Delta }=$8.5 dB, $x_{\mathrm{SB}}=0.35,$ $K=3$, and $N_{\max }=7$. It is noted that because SB is located at $\left(0.65,0.5\right)$, we have ${\mathsf{\Phi }}_2<{\mathsf{\Phi }}_1$; hence, in $\mathrm{Cov-Scm}$, $\mathrm{OP}$ of ${\mathrm{SS}}_1$ is lower than that of ${\mathrm{SS}}_2$. As emphasized in Section 3, we can confirm from Figure 4 that $\mathrm{OP}$ of ${\mathrm{SS}}_1$ in $\mathrm{Cov-Scm}$ is equal to that of ${\mathrm{SS}}_1$ in $\mathrm{Mod-Scm-2}$, and $\mathrm{OP}$ of ${\mathrm{SS}}_2$ in $\mathrm{Cov-Scm}$ is equal to that of ${\mathrm{SS}}_2$ in $\mathrm{Mod-Scm-1}$. This figure also shows that there are optimal values of $\alpha$ at which $\mathrm{OP}$ of each user is lowest. For example, in $\mathrm{Mod-Scm-2}$, the $\mathrm{OP}$ performance at ${\mathrm{SS}}_1$ and ${\mathrm{SS}}_2$ is best when $\alpha =0.35$ and $\alpha =0.4$, respectively. For the $\mathrm{SOP}$ performance, we see that $\mathrm{Mod-Scm-2}$ obtains the best performance, while $\mathrm{Mod-Scm-1}$ still outperforms $\mathrm{Cov-Scm}$. Similarly, there exists optimal values of $\alpha$ which provides the best $\mathrm{SOP}$ performance for the considered schemes. Based on the results obtained from Figure 6 and Figure 7, we can conclude that in the $\mathrm{Mod-Scm}$ scheme, if ${\mathsf{\Phi }}_j<{\mathsf{\Phi }}_i$ (then $\mathrm{SOP}$ of $\mathrm{Mod-Scm-}j$ is lower than that of $\mathrm{Mod-Scm-}i$), hence, the source ${\mathrm{SS}}_j$ should be prioritized to transmit data first. Now, let us consider examples of designing the proposed schemes. If the required quality of service (QoS) dictates that the SOP performance must be below 0.01, then, as shown in Figure 7, only the $\mathrm{Mod-Scm-2}$ scheme can satisfy this requirement. Moreover, the value of $\alpha$ must be designed within the range of 0.325 to 0.4. For another example, to determine the optimal value of $\alpha$ in the $\mathrm{Mod-Scm-2}$ scheme, we follow the following steps:

-

Step 1: As shown in Figure 7, the simulation and theoretical results of the SOP performance over a wide range of $\alpha$ were used to confirm the existence of an optimal value of $\alpha$.

-

Step 2: Identifying the interval that contains the optimal value of $\alpha$. For example, in Figure 7, the interval of $\alpha$ is (0.325, 0.4).

-

Step 3: Using the derived expression of SOP (i.e., Equation (27)) to search the optimal value of $\alpha$ within the interval determined in Step 2.

Figure 8 studies the impact of the positions of the power station $\left(\mathrm{SB}\right)$ on the $\mathrm{OP}$ and $\mathrm{SOP}$ performance when $\mathsf{\Delta }=$11 dB, $\alpha =0.375$, $K=4$, and $N_{\max }=7$. Due to the symmetry, we can see that in $\mathrm{Cov-Scm}$, $\mathrm{OP}$ of ${\mathrm{SS}}_1$ at $x_{\mathrm{SB}}=a$ equals to that of ${\mathrm{SS}}_2$ at $x_{\mathrm{SB}}=1-a,$ where $0<a<1$. Similarly, $\mathrm{OP}$ of ${\mathrm{SS}}_1$ in $\mathrm{Mod-Scm-1}$ at $x_{\mathrm{SB}}=a$ equals to that of ${\mathrm{SS}}_2$ in $\mathrm{Mod-Scm-2}$ at $x_{\mathrm{SB}}=1-a$. For the $\mathrm{SOP}$ performance, we see that $\mathrm{SOP}$ of $\mathrm{Cov-Scm}$ is symmetrical about $x_{\mathrm{SB}}=0.5$, while $\mathrm{SOP}$ of $\mathrm{Mod-Scm-1}$ at $x_{\mathrm{SB}}=a$ equals to that of $\mathrm{Mod-Scm-2}$ at $x_{\mathrm{SB}}=1-a$. Therefore, as $x_{\mathrm{SB}}<0.5\left(x_{\mathrm{SB}}>0.5\right)$, the source ${\mathrm{SS}}_1\left({\mathrm{SS}}_2\right)$ should be selected to transmit data first. Finally, it is worth noting from Figure 8 that the $\mathrm{SOP}$ of $\mathrm{Cov-Scm}$, $\mathrm{Mod-Scm-1}$, and $\mathrm{Mod-Scm-2}$ is lowest when $x_{\mathrm{SB}}=0.5$, $x_{\mathrm{SB}}=0.3$, and $x_{\mathrm{SB}}=0.7,$ respectively.

In Figure 9, we present the average number of transmissions of $\mathrm{FC}$ packets for the successful data exchange between two sources in the $\mathrm{Mod-Scm}$ scheme as a function of $\mathsf{\Delta }$ (dB) when $x_{\mathrm{SB}}=0.4$, $\alpha =0.2$, and $N_{\max }=7$. It is worth noting that the number of transmissions in $\mathrm{Cov-Scm}$ is always $2N_{\max }$. Figure 9 presents that the average number of transmissions in $\mathrm{Mod-Scm-1}$ and $\mathrm{Mod-Scm-2}$ decreases with the increasing of $\mathsf{\Delta }.$ When the $\mathsf{\Delta }$ values are high enough, the average number of transmissions in $\mathrm{Mod-Scm-1}$ and $\mathrm{Mod-Scm-2}$ will reach $2H_{\min }$. Therefore, the proposed $\mathrm{Mod-Scm}$ scheme not only obtains a better $\mathrm{OP}$ and $\mathrm{SOP}$ performance, but also achieves a lower average number of transmissions. As shown in Figure 9, the average number of transmissions of $\mathrm{Mod-Scm-2}$ is lower than that of $\mathrm{Mod-Scm-1}$. However, at high $\mathsf{\Delta }$ regimes, the performance of $\mathrm{Mod-Scm-1}$ and $\mathrm{Mod-Scm-2}$ is almost the same. Finally, as observed, increasing the number of reflective elements in the $\mathrm{RIS}$ also reduces the average number of transmissions significantly.

Figure 10 presents the impact of $\alpha$ on the average number of transmissions of $\mathrm{FC}$ packets for the successful data exchange between two sources in the $\mathrm{Mod-Scm}$ scheme when $\mathsf{\Delta }=8.5$ dB, $K=5$, and $N_{\max }=7$. As seen in Figure 10, the average number of transmissions in $\mathrm{Mod-Scm-1}$ and $\mathrm{Mod-Scm-2}$ achieves the minimum value as $\alpha =0.5$. Moreover, the positions of the $\mathrm{SB}$ station also impacts the average number of transmissions significantly. In this figure, the average number of transmissions in $\mathrm{Mod-Scm-1}$ and $\mathrm{Mod-Scm-2}$ with $x_{\mathrm{SB}}=0.5$ is the same, and is lowest as compared with $x_{\mathrm{SB}}=0.1$ and $x_{\mathrm{SB}}=0.2$.

5. Conclusions

This paper evaluated the performance of two FC-based TWRCR schemes through both simulation and analysis, focusing on OP, SOP, and the average number of transmissions for the successful data exchange. The results showed that the Mod-Scm scheme achieves better OP and SOP performance compared to the Cov-Scm scheme. Additionally, the Mod-Scm scheme can reduce the average number of FC packet transmissions, which in turn lowers delay and power consumption. In Cov-Scm, the order in which the source transmits data also significantly affects the performance. Finally, key parameters such as $K,$ $N_{\max }$, and $\alpha$ should be appropriately designed to enhance performance for both the Cov-Scm and Mod-Scm schemes.