# Cooperative Game-Based Energy Efficiency Management over Ultra-Dense Wireless Cellular Networks
## 2 System Model and Network Energy-Efficiency Model  
### System Model  
As previously mentioned, the ultradense deployment of low-power nodes is an important method for increasing the system capacity. Thus, we consider a two-tier heterogeneous network, as illustrated in Figure 1, where multiple co-channel low-power nodes are overlaid on an existing macrocell controlled by a MeNB. We assume that both tiers share the spectrum. As shown in Figure 1, many SeNBs dominate their respective small cells. With the increase of SeNBs, inter- and intra-tier interference seriously affects and restricts the network performance. For example, the macrocell user equipment (MUE) suffers significant performance degradation owing to the nearest SeNB. Therefore, it is crucial to research the interference influence on network EE. Here, we focus on the resource allocation in the downlink of this network. Let M = {1, 2, 3,..., _M_} represent the MeNB set. Every macrocell is covered with \(N\) SeNBs, and N = {1, 2, 3,..., _N_} represents the SeNB set. In each macro-cell, there are \(I\) MUEs and \(J\) small-cell user equipments (SUEs) associated with each SeNB. We set \(g_{m,i}\) and \(g_{n,j}\) as the channel power gains from MeNB and SeNB to their associated users MUE and SUE, respectively, where \(m\)\(\in\){1, 2,..., _M_}, \(i\)\(\in\){1, 2,..., _I_}, \(n\)\(\in\){1, 2,..., _N_}, \(j\)\(\in\){1, 2,..., _J_}; \(p_{m}\) and \(p_{n}\) are the transmission power for the MeNB\({}_{\rm{m}}\) and SeNB\({}_{\rm{n}}\), respectively, in the downlink of the macro-cell and the small-cell. Thus, the received SINR of the _i_th MUE\({}_{\rm{i}}\) at the \(m\)th MeNB\({}_{\rm{m}}\) is given by:  
\[\gamma_{m,i}\ =\ \frac{p_{m}g_{m,i}}{\sum\limits_{m^{\prime}\
eq m,m^{\prime} =1}^{M}p_{m^{\prime}}g_{m^{\prime},\{m,i\}}+\sum\limits_{n=1}^{N}p_{n}g_{n,\{m,i\}}+\sigma_{m,i}^{2}} \tag{1}\]  
Figure 1: Ultradense deployment scenario of a wireless cellular network. where \(p_{m}\) is the transmission power of the other MeNBs besides MeNB\({}_{\rm m}\), \(g_{m^{\prime},\{m,j\}}\) is the channel gain on MUE\({}_{\rm i}\) from other MeNBs, \(g_{n,\{m,i\}}\) is the channel gain on MUE\({}_{\rm i}\) from SeNBs overlaid on the coverage of MeNB\({}_{\rm m}\), and \(\sigma_{m,1}^{2}\) is the additive white Gaussian noise (AWGN) power. In the aforementioned equation, \(\sum_{n=1}^{N}p_{n}g_{n,\{m,i\}}\) represents the inter-tier interference on MUE\({}_{\rm i}\), and \(\sum_{m^{\prime}\
eq m,m^{\prime}=1}^{M}p_{m^{\prime}}g_{m^{\prime},\{m,j\}}\) represents the intra-tier interference on MUE\({}_{\rm i}\). The channel-to-interference-plus-noise ratio (CINR) of MUE\({}_{\rm i}\) is:  
\[h_{m,i}=\frac{g_{m,i}}{\sum\limits_{m^{\prime}\
eq m,m^{\prime}=1}^{M}p_{m^{ \prime}}g_{m^{\prime},\{m,i\}}+\sum\limits_{n=1}^{N}p_{n}g_{n,\{m,j\}}+\sigma_ {m,i}^{2}} \tag{2}\]  
Hence, Equation (1) can be simplified as:  
\[\gamma_{m,i}=p_{m}h_{m,i} \tag{3}\]  
Similarly, the received SINR of the \(j\)th SU\({}_{\rm j}\) at the \(n\)th SeNB\({}_{\rm n}\) is given by:  
\[\gamma_{n,j}=\frac{p_{n}g_{n,j}}{\sum\limits_{n^{\prime}\
eq n,n^{\prime}=1}^{ N}p_{n^{\prime}}g_{n^{\prime},\{n,j\}}+\sum\limits_{m=1}^{M}p_{m}g_{m,\{n,j\}}+ \sigma_{n,j}^{2}} \tag{4}\]  
where \(p_{n^{\prime}}\) is the transmission power of the other SeNBs besides SeNB\({}_{n}\), \(g_{n^{\prime},\{n,j\}}\) is the channel gain on SU\({}_{\rm j}\) from other SeNBs, \(g_{m,\{n,j\}}\) is the channel gain on SU\({}_{\rm j}\) from the MeNBs around SeNB\({}_{\rm n}\), and \(\sigma_{n,j}^{2}\) is the AWGN power. \(\sum_{n=1}^{M}p_{m}g_{m,\{n,j\}}\) represents the inter-tier interference on SU\({}_{\rm j}\), and \(\sum_{n^{\prime}\
eq n,n^{\prime}=1}^{N}p_{n^{\prime}}g_{n^{\prime},\{n,j\}}\) represents the intra-tier interference on SU\({}_{\rm j}\). We assume that \(h_{n,j}\) is the CINR of SU\({}_{\rm j}\).