# Cooperative Game-Based Energy Efficiency Management over Ultra-Dense Wireless Cellular Networks
## 4 Bargaining Problem Solutions for Ultradense Wireless Cellular Networks
### Simplified Algorithm for Sub-Optimal Solution  
A simplified algorithm based on Algorithm 1 is proposed to obtain a sub-optimal power-coordination solution. As the power of the MeNBs is far larger than that of the SeNBs, we assume that only the inter-tier interference of two-tier macro-small HetNets exists. As denoted in Equation (1), \(h_{m,i}\) is the function of the inter-tier interference, the intra-tier interference, and the noise. In Equation (2), with the increase of small cells, we assume that \(\sum_{n=1}^{N}p_{n}g_{n,\{m,i\}}\gg\sum_{m^{\prime}\
eq m,m^{\prime}=1}^{M}p_{ m}g_{m^{\prime},\{m,i\}}+\sigma_{m,i}^{2}\); thus, it can be rewritten as:  
\[\gamma_{m,i}=\frac{p_{m}g_{m,i}}{\sum\limits_{n=1}^{N}p_{n}g_{n,\{m,i\}}}=p_{m} h_{m,i} \tag{46}\]  
that is:  
\[\frac{1}{h_{m,i}}=\frac{\sum\limits_{n=1}^{N}p_{n}g_{n,\{m,i\}}}{g_{m,i}} \tag{47}\]  
To simplify the power-updating process, we assume that there is always one maximum SeNB\({}_{n}\) interference for the MUE. This assumption was made in [31]. Therefore, we have:  
\[\frac{1}{h_{m,i}}=\frac{p_{n}g_{n,\{m,i\}}}{g_{m,i}} \tag{48}\]  
randomly distributed in the coverage of MeNBs, and SUEs are randomly distributed in the coverage of their SeNBs. For comparison with the simulation in [31], we set similar simulation parameters. The MeNBs use omnidirectional antennae, which are set in the center of their coverage in a 50 m \(\times\) 50 m rectangular-area model. In the macro-cell, SeNBs are overlaid in a dense deployment, in a 5 \(\times\) 5 grid model. The maximum transmit power of the MeNBs is set as 46 dBm, and the number of users per macro-cell is \(I\) = 100. The coverage radius of the macro-cell is 500 m, and that of a small-cell is 25 m. The number of users per small-cell is \(J\) = 10 and the maximal transmit power of the SeNB is 20 dBm. We use the femto base station as the SeNB in our simulations. Femtocells in the 5 \(\times\) 5 grid model of 3 GPP-TR 36.814 are used for pass-loss models and shadowing models, and the detailed parameters are presented in Table A.2.1.1.2-3 [36]. The path-loss model from the SeNB to SUEs is \(L\) = 127 + 30 log\({}_{10}R\) in length, and the other links are \(L\) = 128.1 + 37.6 log\({}_{10}R\) in length, where \(R\) is the distance between the base station and the user in kilometers. The Gaussian noise power spectral density is \(N_{0}\) = \(-\)174 dBm/Hz, where the additive white Gaussian noise power is \(\sigma^{2}=B/N_{0}\). The coefficients \(\alpha_{m}\), \(\beta_{m}\), \(\alpha_{n}\), and \(\beta_{n}\) are related to the distribution of users for each base station. The actual effects on the SE and EE were studied in [35]. In the present work, we set these as 1/3 to guarantee the fairness among the SE, EE, and DE. The other parameters are listed in Table 2. In Figure 2, the convergence property of Algorithm 1 is evaluated with the power iteration difference constraint \(\theta=0.0001\) and the power initializations of 46 dBm for the MeNB and 20 dBm for the other two SeNBs. This algorithm uses a fixed-point iteration process for power coordination, which is proven to converge in [24]. As shown in Figure 2, we choose the similar scenario and set the same simulation parameters as [31]. Finally, the proposed power-coordination algorithm converges in ultra-dense wireless cellular networks. As illustrated in Figure 2, \(p_{m}\), \(p_{\text{1}}\) and \(p_{\text{2}}\) exhibit fast convergence after five or six iterations. Compared with the convergence performance in [31], our algorithm ensures faster convergence.