# Cooperative Game-Based Energy Efficiency Management over Ultra-Dense Wireless Cellular Networks
## 4 Bargaining Problem Solutions for Ultradense Wireless Cellular Networks  
### Problem Simplification  
The problem in Equation (16) is too complicated to be analyzed and solved. Therefore, some simplifications are made. On the basis of numerous studies focusing on the SINR-based optimal functions in [20; 23], the CINR functions in Equations (3) and (5) are used instead of the achievable capacity in Equations (6) and (7) in the utility functions of Lemma 2. According to the analysis in [15], \(c_{0}\) is far smaller than \(c_{1}\), which indicates that we can omit the CapEx \(c_{0}\) in the utility functions of Lemma 2 for simplicity:  
**Corollary 1:**_The optimization problem of P1 in Equation (16) is simplified as:_  
\[\textbf{P2}:\max_{0\leq p_{m}\leq p_{m}^{MAX},0\leq p_{n}\leq p_{n}^{MAX}} \psi=\prod\limits_{m=1}^{M}{\frac{p_{m}h_{m}}{(a_{ikim^{m}}*p_{m}+b_{ikim})^{ \alpha_{m}+p_{m  
\[u_{n}=\left[ \begin{array}{c}\frac{\int\limits_{r-1}^{l}\log_{2}(1+p_{n}h_{n,j}) }{a_{sistem}*p_{n}+b_{sistem}}\\ \end{array} \right]^{\alpha_{n}}\cdot\left[ \begin{array}{c}\frac{\int\limits_{r -1}^{l}\log_{2}(1+p_{n}h_{n,j})}{c_{1n}*(a_{sistem}*p_{n}+b_{sistem})+c_{0n}} \\ \end{array} \right]^{\beta_{n}}\cdot\left[ \begin{array}{c}\int\limits_{j=1}^{l} \log_{2}(1+p_{n}h_{n,j})\\ \end{array} \right]^{1-\alpha_{n}-\beta_{n}} \tag{20}\] \[=\frac{\int\limits_{r-1}^{l}\log_{2}(1+p_{n}h_{n,j})}{(a_{sistem}* p_{n}+b_{sistem})^{\alpha_{n}}\cdot(c_{1n}*(a_{sistem}*p_{n}+b_{sistem})+c_{0n})^{ \beta_{n}}}\]  
As the CINR definitions in Equations (3) and (5) are used instead of the achievable capacity from Equations (6) and (7), the sum achievable capacities of \(\rm{MeNB_{m}}\) and \(\rm{SeNB_{n}}\) are:  
\[C_{m}=\sum\limits_{i=1}^{l}p_{m}h_{m,i}=p_{m}h_{m} \tag{21}\]  
\[C_{n}=\sum\limits_{j=1}^{l}p_{n}h_{n,j}=p_{n}h_{n} \tag{22}\]  
where \(\rm{h_{m}}=\sum\
olimits_{i=1}^{l}h_{m,i}\) and \(\rm{h_{n}}=\sum\
olimits_{j=1}^{l}h_{n,j}\) are the aggregate CINRs from all the MUEs and SUEs of the corresponding MeNBs and SeNBs, respectively. Thus, the utility functions in Equations (19) and (20) can be simplified as:  
\[u_{m}=\frac{p_{m}h_{m}}{\left(a_{sistem}*p_{m}+b_{sistem}\right)^{\alpha_{m} }\cdot\left(c_{1m}*(a_{sistem}*p_{m}+b_{sistem})+c_{om}\right)^{\beta_{m}}} \tag{23}\]  
\[u_{n}=\frac{p_{n}h_{n}}{\left(a_{sistem}*p_{n}+b_{sistem}\right)^{\alpha_{n} }\cdot\left(c_{1n}*(a_{sistem}*p_{n}+b_{sistem})+c_{on}\right)^{\beta_{n}}} \tag{24}\]  
with the assumptions of \(EE_{m}^{0}=DE_{m}^{0}=SE_{m}^{0}=0\) and \(EE_{n}^{0}=DE_{n}^{0}=SE_{n}^{0}=0\). In addition, according to the analysis, the CapEx \(\rm{c_{0}}\) is far smaller than the OpEx coefficient \(\rm{c_{1}}\); that is, the CapEx \(\rm{c_{0m}}\) and \(\rm{c_{0n}}\) in the utility functions of Equations (23) and (24) can be omitted compared with \(\rm{c_{1m}}^{*}(\rm{a_{sistem}}^{*}\rm{p_{m}}^{*}\rm{b_{sistem}})\) and \(\rm{c_{1n}}^{*}(\rm{a_{sistem}}^{*}\rm{p_{n}}^{*}\rm{b_{sistem}})\). Thus, the utility functions can be further simplified:  
\[u_{m}=\frac{p_{m}h_{m}}{\left(a_{sistem}*p_{m}+b_{sistem}\right)^{\alpha_{ m}+\beta_{m}}\cdot c_{1m}^{\beta_{m}}} \tag{25}\]  
\[u_{n}=\frac{p_{n}h_{n}}{\left(a_{sistem}*p_{n}+b_{sistem}\right)^{\alpha_{ n}+\beta_{n}}\cdot c_{1n}^{\beta_{n}}} \tag{26}\]  
Therefore, the optimization problem of P2 can be easily obtained with mathematical derivations by Equations (25) and (26) in Equation (16). **Corollary 2:**_The problem P2 in Equation (18) can be equivalently converted into:_  
\[\textbf{P3}:\ \textbf{max}\ \xi \tag{27}\]  
\[s.t.\ C1:0\leq p_{m}\leq p_{m}^{MAX}\]  
\[C2:0\leq p_{n}\leq p_{n}^{MAX} \tag{28}\]  
_where the utility function is \(\xi=\ln(\psi)\)._  
**Proof:** Because the logarithmic function does not change the convexity and the utility function in Equation (18) is proven to be convex according to [33], P3 in Equation (27) is equal to P2 in Equation (18). Then, the problem in Equation (27) can be expanded as:  
\[