## 2 System Model
: _In the Cov-Scm scheme [36], the data transmission of \(\text{SS}_{1}\) and \(\text{SS}_{2}\) operates independently, making the transmission order of \(\text{SS}_{1}\) and \(\text{SS}_{2}\) irrelevant. However, in the Mod-Scm scheme, the system performance depends on whether \(\text{SS}_{1}\) or \(\text{SS}_{2}\) transmits first (this issue will be examined in Section 4). Moreover, the Cov-Scm scheme always uses a total of \(2N_{\text{max}}\) transmission times, while the Mod-Scm scheme uses fewer, because the transmission stops as soon as each source gathers enough desired packets._  
Let \(g_{\text{XY}}\) represent the channel gains between nodes X and Y, and its distribution functions expressed as  
\[F_{g_{\text{XY}}}(x)=1-\exp(-\lambda_{\text{XY}}x),f_{g_{\text{XY}}}(x)= \lambda_{\text{XY}}\exp(-\lambda_{\text{XY}}x), \tag{1}\]  
where \(F_{g_{\text{XY}}}(. )\) and \(f_{g_{\text{XY}}}(. )\) are cumulative distribution function \((\text{CDF})\) and probability density function \((\text{PDF})\) of \(g_{\text{XY}}\), respectively, \(\lambda_{\text{XY}}=\left(d_{\text{XY}}\right)^{\text{PL}}\)[36] with \(\text{PL}\ (2\leq\text{PL}\leq 6)\) being the path-loss factor and \(d_{\text{XY}}\) being the physical distance between X and Y. Let \(d_{\text{X}\text{R}\text{S}_{k}}\) as the distance between the node X and the \(k\) - th reflector component of the RIS, where \(k=1,2,\ldots,K,\text{X}\in\{\text{SS}_{1},\text{SS}_{2}\}\). As assumed in [37], we can assume that all the distances \(d_{\text{X}\text{R}\text{S}_{k}}\) are identical, i.e., \(d_{\text{X}\text{R}\text{S}_{k}}=d_{\text{X}\text{R}\text{S}_{k}}\left(\lambda_{ \text{X}\text{R}\text{S}_{k}}=\lambda_{\text{X}\text{R}\text{S}_{k}}\right)\forall k\). Considering the transmission of a packet \(p_{i}\) from \(\text{SS}_{i}\) to \(\text{SS}_{j}\) via the RIS. Assume that the total delay for each transmission of \(p_{i}\) is normalized to 01 (time unit). Duringthe interval \(\alpha(0<\alpha<1)\), \(\text{SS}_{i}\) harvests energy from SB, and its harvested energy can be calculated as  
\[Q_{i}=\eta\alpha P_{\text{SB}}g_{\text{SSBS}}, \tag{2}\]  
where \(\eta\) represents a conversion efficiency, and \(P_{\text{SB}}\) is the transmit power of SB. The remaining interval \((1-\alpha)\) is allocated for the transmission of \(p_{i}\). Therefore, the average transmit power of \(\text{SS}_{i}\) can be formulated as  
\[P_{\text{SS}_{i}}^{(1)}=\frac{Q_{i}}{1-\alpha}=\chi P_{\text{SB}}g_{\text{SSBS }_{i}}, \tag{3}\]  
where \(\chi=\eta\alpha/(1-\alpha)\). Moreover, the transmit power of \(\text{SS}_{i}\) must satisfy the interference threshold set by PU, i.e.,  
\[P_{\text{SS}_{i}}^{(2)}=\frac{I_{\text{PU}}}{g_{\text{SS}_{i}\text{PU}}}, \tag{4}\]  
where \(I_{\text{PU}}\) is the interference threshold. From Equations (3) and (4), the transmit power of \(\text{SS}_{i}\) can be formulated by  
\[P_{\text{SS}_{i}}=\min\Bigl{(}P_{\text{SS}_{i}}^{(1)},P_{\text{SS}_{i}}^{(2)} \Bigr{)}=\min\Bigl{(}\chi P_{\text{SB}}g_{\text{SSBS}_{i}},\frac{I_{\text{PU}} }{g_{\text{SS}_{i}\text{PU}}}\Bigr{)}. \tag{5}\]  
Next, \(\text{SS}_{i}\) transmits \(p_{i}\) to \(\text{SS}_{i}\), and the received signal at \(\text{SS}_{i}\) is given as (see [38; 39; 40]):  
\[y_{\text{SS}_{i}}=\sqrt{P_{\text{SS}_{i}}}\sum\limits_{k=1}^{K}h_{\text{SS}_{ i}\text{RS}_{k}}r_{\text{RS}_{k}}h_{\text{RIS}_{k}\text{SS}_{i}}P_{i}^{\text{modu}}+n_{ \text{SS}_{i}}, \tag{6}\]  
where \(h_{\text{SS}_{i}\text{RIS}_{k}}\) and \(h_{\text{RIS}_{k}\text{SS}_{i}}\) are channel coefficients of the \(\text{SS}_{i}\rightarrow\text{RIS}_{k}\) and \(\text{RIS}_{k}\rightarrow\text{SS}_{j}\) links, respectively, \(r_{\text{RIS}_{k}}\) is response of the \(k\) - th component of the RIS, \(p_{i}^{\text{modu}}\) is the modulated signals of \(p_{i}\), and \(n_{\text{SS}_{i}}\) is zero-mean Gaussian noise at \(\text{SS}_{i}\). We assume that variance of all Gaussian noises is identical and equal to \(\sigma_{0}^{2}\), i.e., \(\text{Var}\Bigl{\{}n_{\text{SS}_{i}}\Bigr{\}}=\sigma_{0}^{2}\).