# Green Small Cell Operation of Ultra-Dense Networks Using Device Assistance +
## 2 System Model
### Problem Formulation  
When UE \(i\) is associated with AP \(j\), the signal-to-interference ratio (SINR) between UE \(i\) and AP \(j\) is given by:  
\[SINR_{i,j}(\mathbf{x})=\frac{g_{i,j}P_{j,Tx}(x_{j})}{N_{0}+\sum_{j^{\prime}\in \mathcal{J}}\left(g_{i,j^{\prime}}\ P_{j^{\prime},Tx}(x_{j^{\prime}})\right)}\]  
where \(g_{i,j}\) is the channel gain between UE \(i\) and AP \(j\), and \(N_{0}\) is the noise power. Then, the maximum spectral efficiency of UE \(i\) served by AP \(j\) is \(\log_{2}(1+SINR_{i,j}(\mathbf{x}))\). We assume that when an AP \(j\) has multiple associated UEs, the associated UEs are scheduled in a temporally fair way or equally when sharing the frequency channel; thus, the data rate of UE \(i\) is given by:  
\[R_{i}(\mathbf{x})=\frac{\log_{2}\left(1+SINR_{i,j}(\mathbf{x})\right)}{S_{j}} \tag{2}\]where \(S_{j}=\sum_{i^{\prime}=1}^{I}s_{i^{\prime}j}\). Energy consumption of APs can be calculated by summing the operational power of turned-on APs. Thus, the total power consumption of the network becomes \(\sum_{j=1}^{J}(P_{j,Op}\cdot 1(S_{j}))\) where an indicator function is \(1(S_{j}):=\ \left\{\begin{array}{ll}1&\text{if }S_{j}>0,\\ 0&\text{if }S_{j}=0.\end{array}\right.\)  
Our objective function is the energy efficiency \(F(\mathbf{x})\), also called the global function given by:  
\[F(\mathbf{x})=\frac{\sum_{i=1}^{I}R_{i}(\mathbf{x})}{\sum_{j=1}^{J}P_{j,Op} \cdot 1(S_{j})}.\]  
Then, we define the local energy efficiency \(f_{i}(\mathbf{x})\), also called the local function, which is given by \(f_{i}(\mathbf{x})=\frac{R_{i}(\mathbf{x})}{\sum_{j=1}^{J}P_{j,Op}\cdot 1(S_{j})}\). From the definition, the local function is a part of the global function that UE \(i\) yields; thus, the global function is given by:  
\[F(\mathbf{x})=\sum_{i=1}^{I}f_{i}(\mathbf{x}).\]  
Our problem is to find on/off state vector \(\mathbf{x}\) that maximizes the global function \(F(\mathbf{x})\). Thus, our optimization problem is:  
\[\max_{\mathbf{x},\mathbf{5}} F(\mathbf{x}) \tag{3}\] \[s.t x_{j}\in\{0,1\},\ \ \forall j\in\mathcal{J}\] (4) \[\sum_{i=1}^{I}\sum_{j\in\{j^{\prime}|x_{j}^{\prime}=0\}}s_{i,j}=0,\] (5) \[\sum_{j=1}^{I}s_{i,j}=1,\ \ \forall i\in\mathcal{I}. \tag{6}\]  
The first constraint of (4) is from determining a set of turned-on APs. The second constraint of (5) means that turned-off APs should not be associated with any UEs. The third constraint of (6) shows that all UEs should be associated with only one APs. To tackle the problem given in (3) to (6), we need to determine the set of turned-on APs, as well as the UE-AP association. While maximizing the energy efficiency in (3), some users may experience low quality of service (QoS). Therefore, although the requirement about QoS is not explicitly expressed in the constraints of the optimization problem, minimum QoS thresholds can be set in our algorithms to ensure a minimum level of QoS, as we discuss in Section 4. Note that one can solve the optimization problem for the given traffic information, but by solving the proposed problem repeatedly, it can be also possible to adaptively manage the on/off states of APs if network traffic is changed over time. Thus, we focus on solving the above problem where the notion of time is excluded. The problem is known as NP-hard, implying that it is hard to find an optimal solution in polynomial time [9]. Thus, to solve the optimization problem, an exhaustive search should be done.  
In the case of small APs, APs are usually randomly deployed, which makes the BS coordination for time/frequency orthogonal transmission very challenging. This is why we focus on the simple ON/OFF operation to improve the energy efficiency of (ultra-)dense wireless networks. Note that our problem is still very challenging because the complexity of the problem is NP-hard, and in ultra-dense networks, the computational complexity increases exponentially in the number of APs. In the next section, we apply BP to our framework and find a solution that is close to the optimal performance.