# RIS-Assisted Cooperative Time-Division Multiple Access
## 2 RIS-Assisted Cooperative Time-Division Multiple Access
### Discussion of System Performance  
The objective of the proposed algorithm is that each UE meets the required capacity. In other words, the proposed algorithm ensures QoS, i.e., required SINR or capacity, for all UE as much as possible, not focusing on maximizing the sum capacity for all UEs. Thus, we define the remaining required capacity to measure the performance of the proposed algorithm, indicating the amount of capacity that was not achieved compared with the required capacity.  
When the \(k\)th UE is served only in the allocated non-cooperative time slot (i.e., the dedicated time slot), its remaining required capacity is given as  
\[\mathbf{C}_{short,RIS}^{k}=[\mathbf{C}_{req}-\mathbf{C}_{n}]^{+}, \tag{18}\]  
where \([v]^{+}=\max(0,v)\). When the \(k\)th UE is served in the allocated non-cooperative time slot, and it is also served in the allocated cooperative time slot as the weak-channel-conditioned UE, its remaining required capacity is given as  
\[\mathbf{C}_{short,RIS}^{k}=[\mathbf{C}_{req}-\mathbf{C}_{n}-\mathbf{C}_{c,w}] ^{+}. \tag{19}\]  
when the \(k\)th UE is served in the allocated cooperative time slot as the strong-channel-conditioned UE, its remaining required capacity is given as  
\[\mathbf{C}_{short,RIS}^{k}=[\mathbf{C}_{req}-\mathbf{C}_{c,s}]^{+}=0, \tag{20}\]  
because the proposed algorithm performs power allocation to maximize the weak-channel-conditioned UE's SINR in the cooperative time slot, and it leads to the decrease in the strong-channel-conditioned UE's SINR to become the same value as the required SINR, i.e., \(\mathbf{C}_{req}=\mathbf{C}_{c,s}\). Therefore, the total remaining required capacity can be given as  
\[\mathbf{C}_{short,RIS}=\sum_{k=1}^{K}\mathbf{C}_{short}^{k}. \tag{21}\]  
In the conventional TDMA system, the remaining required capacity for \(k\)th UE can be defined as  
\[\mathbf{C}_{short,BS}^{k}=[\mathbf{C}_{req}-\mathbf{C}_{k}]^{+}, \tag{22}\]where \(\mathbf{C}_{k}\) is \(k\)th UE's capacity in each time slot under the assumption that MRT beamforming is used. Then, the total remaining required capacity in the conventional TDMA system can be given as  
\[\mathbf{C}_{short,BS}=\sum_{k=1}^{K}\mathbf{C}_{short,BS}^{k}. \tag{23}\]  
In the proposed algorithm, RIS is turned on to reflect the incident signals only in cooperative time slots. In the cooperative time slots, transmit beamforming vectors at BS are designed to maximize UEs' SINRs, considering the reflecting effect on the RIS. The on/off mechanism in the proposed algorithm leads to an increase in power efficiency compared with systems that do not use RISs. For example, for the conventional TDMA systems that do not use RISs, additional data should be transmitted using additional time slots, which can provide additional data transmission to users who do not meet the required capacity. Thus, this causes power consumption at the BS. Since the RIS, which delivers signals using passive reflecting elements, is more power-efficient than BS, the proposed algorithm can show better power efficiency than the conventional TDMA. Thus, we can define the effective capacity from the power-efficiency perspective, using the power consumption ratio between BS and RIS. In the conventional TDMA system, the effective capacity can be represented as  
\[\mathbf{C}_{eff,BS}=\frac{\sum_{k=1}^{K}\min(\mathbf{C}_{k},\mathbf{C}_{req}) }{KP_{c}}, \tag{24}\]  
where \(P_{c}\) is the power consumption of BS in each time slot. In the proposed algorithm, the effective capacity can be represented as  
\[\mathbf{C}_{eff,RIS}=\frac{K\mathbf{C}_{req}-\mathbf{C}_{short}}{KP_{c}+\tau P _{c}N_{c}}, \tag{25}\]  
where \(\tau\) is the power consumption ratio between BS and RIS, i.e., \(0<\tau<1\), and \(N_{c}\) is the number of cooperative time slots. In Equations (24) and (25), the effective capacities can be interpreted as measuring satisfaction with the required capacity, i.e., QoS, compared with the power consumption.