# Power Control Method for Energy Efficient Buffer-Aided Relay Systems
## 4 Buffer Size Design  
To determine an efficient buffer size, we evaluate the average achievable rate of \(R_{B}(t)\) in Equation (5), which is defined as  
\[R_{B}(P_{r})=\frac{1}{T}\sum_{t=1}^{T}R_{B}(t). \tag{7}\]Figure 3 shows the average achievable rates in Equations (4) and (7) by varying the buffer size \(b\) when \(\gamma_{sr}=20\) dB. Data transmission time is given as \(T=10^{5}\). Obviously, the relay system without a buffer is a special case of the relay system with a buffer whose size is zero, i.e., \(b=0\). By comparing the rates at \(b=0\) and \(b>0\), it is clearly observed that the buffer can improve the achievable rate of the cooperative systems. It is also observed that the achievable rate increases and it is saturated as buffer size \(b\) increases. As observed in the results, the average achievable rate is almost saturated when the buffer size \(b=10\), regardless of \(\gamma_{rd}\). From the results, the size of an effective buffer is determined by ten for the considered relay system in this study.  
Now, to justify that the proposed buffer-aided relay system with \(b=10\) outperforms the conventional relay system without a buffer, the average achievable rates \(R(P_{r})\) and \(R_{B}(P_{r})\) defined in Equations (4) and (7), respectively, are evaluated over the transmit power of RN, i.e., \(P_{r}\).  
In Figure 4, \(R(P_{r})\) in Equation (4) and \(R_{B}(P_{r})\) in Equation (7) are compared across \(P_{r}\). In simulation, it is assumed that the channel attenuation follows the model in works by the authors of [31; 32; 33; 34] as follows  
\[\rho_{A}=G-128+10\log_{10}(d_{A}^{-\alpha})\ \mathrm{dB}, \tag{8}\]  
where \(A\in\{sr,rd\}\), \(G\) includes the transceiver feeder loss and antenna gains, \(d_{A}^{-\alpha}\) is the path loss where \(d_{A}\) is the distance in kilometer between nodes, and \(\alpha\) is a path loss exponent. The transmit power of SN is fixed as \(P_{s}=23\) dBm. In model (8), we set \(G=5\) dB (2 dB and 0 dB feeder losses at the transmitter and receiver, respectively; and 7 dBi and 0 dB gains for the transmit and receive antennas, respectively [31]), \(\alpha=3.76\) (for urban or suburban environment [35]), \(\sigma_{r}^{2}=\sigma_{d}^{2}=-174\) dBm/Hz for AWGN power [31], and \(P_{s}=23\) dBm for small-size BS [34]. On the other hand, the transmit power of RN varies between 17 dBm and 33 dBm (note that 33 dBm, 21 dBm, and 17 dBm for the transmit power of micro, pico, and femto BSs, respectively [34].). The distance between GBS and RN is set as \(d_{sr}=0.6\) km.  
From the results, it is clearly shown that \(R_{B}(P_{r})\geq R(P_{r})\). For given \(P_{s}\), the average data rate increases up to a certain point and is saturated as \(P_{r}\) increases. The saturation point of \(P_{r}\) of the buffer-aided relay is lower than that of the relay without a buffer. From this, it is verified that the buffer-aided relay system efficiently utilizes the second-link channel in virtue of opportunistic forwarding; therefore, the greater average achievable rate is achieved. Thus, from this fact, the transmit power of RN can be reduced sustaining the throughput such that it is identical to the throughput without a buffer, i.e., \(R_{B}(P_{r}^{o})=R(P_{r})\), where \(P_{r}^{o}<P_{r}\). Consequently, it is expected that EE of the network can be improved.  
Figure 3: Average achievable rate \(R_{B}\) in Equation (7) over the buffer size \(b\) bits when \(\gamma_{sd}=20\) dB.