## 2 System Model
In addition, we assume the SU-Tx has perfect CSI. Without loss of generality, we consider the first SU to have the strongest channel strength, whereas the \(L\)-th SU has the weakest. As such, the order for channel gain between the SU-Tx and the SUs can be expressed as follows:  
\[\|\mathbf{g}_{L}\|^{2}\leq\|\mathbf{g}_{L-1}\|^{2}\leq...\leq\|\mathbf{g}_{1} \|^{2}. \tag{3}\]  
In NOMA systems, by using the successive interference cancelation procedure, the \(l\)-th SU (SU\({}_{l}\)) should be capable of successively extracting and decoding the signal intended for the weaker SUs \(\text{SU}_{L},...,\text{SU}_{l+1}\) from the received signal. Therefore, the following condition for SIC in MISO-NOMA networks with different channel strengths should be satisfied:  
\[\left|\mathbf{g}_{l}^{H}\mathbf{v}_{L}\right|^{2}\geq\left|\mathbf{g}_{l}^{H} \mathbf{v}_{L-1}\right|^{2}\geq...\geq\left|\mathbf{g}_{l}^{H}\mathbf{v}_{1} \right|^{2},l\in\mathbb{Q}. \tag{4}\]  
Figure 1: Frame scheme of the wireless-powered NOMA-MEC network. The condition expressed in (4) will be referred to as the SIC constraint in the rest of the paper. The condition (4) guarantees that at the \(l\)-th SU, we have a higher SINR from messages of more distant users since these messages need to be decoded prior to decode the \(l\)-th user's own message. Accordingly, the received signal after removing the last \(L-l\) user signals can be given by  
\[y_{\text{SU},l}=\mathbf{g}_{l}^{H}\mathbf{v}_{l}s_{l}^{\text{SU}}+\mathbf{g}_{l }^{H}\sum\limits_{l^{\prime}=1}^{l-1}\mathbf{v}_{l^{\prime}}s_{l^{\prime}}^{ \text{SU}}+\mathbf{g}_{l}^{H}\sum\limits_{m=1}^{M}\mathbf{w}_{m}s_{m}^{\text{EH }}+\mathbf{g}_{l}^{H}\mathbf{f}r+n_{\text{SU},l}, \tag{5}\]  
where the first term in (5) denotes the intended signal for the \(l\)-th SU, whereas the second term represents the interference caused by the first \(l-1\) signals intended for the SUs \(\{\text{SU}_{1},...,\text{SU}_{l-1}\}\). The PS structure at the SUs is based on a PS factor which divides the incoming RF signals into two streams; one to be used for ID and another to harvest energy. It is noteworthy that optimizing the PS factor is required to achieve an optimal trade-off. Then, the achievable SINR for the \(l\)-th SU when decoding the signal intended for the \(i\)-th SU can be expressed as  
\[\text{SINR}_{l}^{i}=\frac{\theta_{l}\big{|}\mathbf{g}_{l}^{H}\mathbf{v}_{l} \big{|}^{2}}{\theta_{l}\left(\sum\limits_{l^{\prime}=1}^{i-1}\big{|}\mathbf{g }_{l}^{H}\mathbf{v}_{l^{\prime}}\big{|}^{2}+\sum\limits_{m=1}^{M}\big{|} \mathbf{g}_{l}^{H}\mathbf{w}_{m}\big{|}^{2}+\big{|}\mathbf{g}_{l}^{H}\mathbf{ f}\big{|}^{2}+\sigma_{l}^{2}\right)+\sigma_{cn,l}^{2}},l\in\mathbb{Q},\forall_{l} \leq i, \tag{6}\]  
where \(\theta_{l}\) is the PS factor to divide the incoming signal into the ID and EH modules, and the additive circuit noise in the ID module of the \(l\)-th SU is defined as \(n_{\text{SU},cn,l}\sim\mathcal{CN}\left(0,\sigma_{cn,l}^{2}\right)\). It is worth highlighting that (6) is true only after performing SIC on the preceding \(L-i\) signals. Moreover, the energy harvested by the EH module in the \(l\)-th SU can be written as follows:  
\[\text{EH}_{\text{SU},l}=\eta_{\text{SU},l}\left(1-\theta_{l}\right)\left(\sum \limits_{l^{\prime}=1}^{L}\left|\mathbf{g}_{l}^{H}\mathbf{v}_{l^{\prime}} \right|^{2}+\sum\limits_{m=1}^{M}\left|\mathbf{g}_{l}^{H}\mathbf{w}_{m}\right| ^{2}+\left|\mathbf{g}_{l}^{H}\mathbf{f}\right|^{2}+\sigma_{l}^{2}\right), \forall_{l}. \tag{7}\]  
In (7), \(\eta_{\text{SU},l}\) is the energy-harvesting efficiency for the \(l\)-th SU. For simplicity, we assume that energy-harvesting efficiency is always equal to 1, \(\eta_{\text{SU},l}=1\).