# On Optimal Cell Flashing for Reducing Delay and Saving Energy in Wireless Networks
## 3 Optimal Flashing Ratio
Then, in the interference limited regime, \(f(\beta)\) is a monotone decreasing function, i.e.,  
\[\psi(\beta):=\sum_{i\in\mathcal{B}}\left\{\left(1-\frac{2}{3}\beta\right)( \rho_{i}(\beta)P_{i}+P_{o})+\frac{2}{3}\beta P_{s}\right\}. \tag{7}\]0  
Proof. : Since all base stations are symmetric, we focus on BS \(i\) in the analysis hereafter. Let the path loss factor be \(\alpha\), \(y_{i}\in\mathcal{L}\) be the location of BS \(i\) and \(\mathcal{B}_{f}\) be the set of turned-on base stations when cells flash. Then, for any \(x\in\mathcal{L}_{i}\), \(c_{i}^{0}(x)\) and \(c_{i}^{1}(x)\) are given by:  
\[\psi(\beta):=\sum_{i\in\mathcal{B}}\left\{\left(1-\frac{2}{3}\beta\right)( \rho_{i}(\beta)P_{i}+P_{o})+\frac{2}{3}\beta P_{s}\right\}. \tag{7}\]1  
\[\psi(\beta):=\sum_{i\in\mathcal{B}}\left\{\left(1-\frac{2}{3}\beta\right)( \rho_{i}(\beta)P_{i}+P_{o})+\frac{2}{3}\beta P_{s}\right\}. \tag{7}\]2  
Note that in the interference limited regime, transmit power of the transmitter and the interferers are canceled out. Then, we obtain \(\rho_{i}(\beta)\) as follows:  
\[\psi(\beta):=\sum_{i\in\mathcal{B}}\left\{\left(1-\frac{2}{3}\beta\right)( \rho_{i}(\beta)P_{i}+P_{o})+\frac{2}{3}\beta P_{s}\right\}. \tag{7}\]3  
From Equation (18), we can obtain the derivative function of \(\rho_{i}(\beta)\) as follows:  
\[\psi(\beta):=\sum_{i\in\mathcal{B}}\left\{\left(1-\frac{2}{3}\beta\right)( \rho_{i}(\beta)P_{i}+P_{o})+\frac{2}{3}\beta P_{s}\right\}. \tag{7}\]4  
Figure 2 shows \(\rho_{i}^{\prime}(\beta)\) is negative for all \(\beta\) where the path loss factor is \(2\leq\alpha\leq 7\)[29]. We are only interested in whether \(\rho_{i}^{\prime}(\beta)\) is negative or not, so we set \(W=1\) and \(\gamma=1\). Thus, \(\phi^{\prime}(\beta)\) is also negative for all \(\beta\) and we can know that \(\phi(\beta)\) is a monotone decreasing function in \(\beta\). It should be noted that we numerically check that \(\rho_{i}^{\prime}(\beta)<0\) even when \(\alpha\) is unrealistically high, e.g., \(20\). Since we verify all practically meaningful \(\alpha\), we do not attempt to prove that \(\rho_{i}^{\prime}(\beta)<0\) holds for all \(\alpha\). **Remark 3.1**. : _Note that by cell flashing, we can minimize average delay and average energy consumption simultaneously in the interference limited regime. Interference limited regime is usually the case with small cells. Furthermore, in 5G communication systems, cell coverage will be smaller than now for high data rates and low energy consumption, and thus cell flashing achieves both of them as well as improves the edge user performances._  
**Remark 3.2**. : _When the effect of the noise is high enough, the problem becomes non-trivial because \(\phi(\beta)\) can be an increasing function. In Section 4, we consider the impact of noise throughout all simulations._  
**Remark 3.3**. : _In general, \(f(\beta)\) is not a convex function. However, if \(P_{t}\) is negligible in Equation (7), which means \(P_{o}\) is much larger than \(\rho_{i}(\beta)P_{t}\), \(\psi(\beta)\) can be considered as a decreasing linear function [30]. Thus, \(f(\beta)\) becomes a convex function, and we can easily find the optimal \(\beta^{*}\). It is a reasonable assumption because \(P_{o}\) is much larger than \(P_{t}\) in small cells (see Table 2), and \(\rho_{i}(\beta)\) lies between \(0\) and \(1\). However, when \(P_{t}\) is not negligible, the optimal \(\beta^{*}\) can be found by exhaustive search; this may not be technically challenging because there is only one optimization variable._