# An Energy-Efficient Coverage Algorithm for Macrocell--Small Cell Network Systems
## 3 Problem Description
### Problem Formulation and Goal  
Given the system model under consideration, the aim of this study was to minimize the total amount of power consumed by the transceivers of all the different types of BSs while ensuring that all coverage holes were completely filled. Define \(\Omega_{\mathrm{s}}\) as the set of all combinations of transmission power configurations for the \(K\) SBSs' transceivers. Next, we define \([a_{1}^{(z)},a_{2}^{(z)},\ldots,a_{K}^{(z)}]\) as the transmission power status vector of the \(K\) SBSs' transceivers, where \(z\) is the index for the employed configuration in \(\Omega_{\mathrm{s}}\), \(z=1,2,\ldots,|\Omega_{\mathrm{s}}|\). It follows that \(|\Omega_{\mathrm{s}}|=3^{K}\). Given a configuration \(z\), if SBS \(k\)'s transceiver is in the off state, then \(a_{k}^{(z)}=0\); if SBS \(k\)'s transceiver is in the low power output state, then \(a_{k}^{(z)}=1\); if SBS \(k\)'s transceiver is in the high power output state, then \(a_{k}^{(z)}=2\). Let \(P_{\mathrm{trans},k}^{(z)}\) denote the transmission power from SBS \(k\) in configuration \(z\). \(p_{\mathrm{trans},k}^{(z)}\) can thus be expressed as:  
\[P_{\mathrm{trans},k}^{(z)}= \begin{cases}0,&a_{k}^{(z)}=0\\ P_{\mathrm{s},k}^{(\mathrm{low})},&a_{k}^{(z)}=1\\ P_{\mathrm{s},k}^{(\mathrm{high})},&a_{k}^{(z)}=2\end{cases}  \tag{9}\]  
In turn, define \(\Omega_{\mathrm{m}}\) as the set of all of the transmission power configurations for the MBS transceiver. Next, we define \([b_{\mathrm{m}}^{(h)}]\) as the transmission power status vector of the MBS transceiver, where \(h\) is the index for the employed configuration in \(\Omega_{\mathrm{m}}\), \(h=1,2,\ldots,|\Omega_{\mathrm{m}}|\). It follows that \(|\Omega_{\mathrm{m}}|=N_{\mathrm{m}}+1\). Given a configuration \(h\), if the number of used MBS channels is \(j\), with \(j=0,1,\ldots,N_{\mathrm{m}}\), then \(b_{\mathrm{m}}^{(h)}=j\). Let \(P_{\mathrm{m},\mathrm{trans}}^{(h)}\) denote the transmission power for the MBS in configuration \(h\). \(P_{\mathrm{m},\mathrm{trans}}^{(h)}\) can thus be expressed as:  
\[P_{\mathrm{m},\mathrm{trans}}^{(h)}= \begin{cases}0,&b_{\mathrm{m}}^{(h)}=0\\ P_{\mathrm{m},\mathrm{t},\mathrm{channel}},&b_{\mathrm{m}}^{(h)}=1\\ \vdots&\vdots\\ N_{\mathrm{m}}P_{\mathrm{m},\mathrm{t},\mathrm{channel}},&b_{\mathrm{m}}^{(h)}= N_{\mathrm{m}}\end{cases}  \tag{10}\]  
Moreover, we define three different binary decision indicators \(x_{k},y_{k}^{\ell}\), and \(y_{\mathrm{m}}^{\ell}\). First, the value for \(x_{k}\) indicates whether or not SBS \(k\) is active. If yes, \(x_{k}=1\); otherwise, \(x_{k}=0\). Next, \(y_{k}^{\ell}\) indicates whether or not the selected position \(\ell\) is able to be served by SBS \(k\). If yes, \(y_{k}^{\ell}=1\); otherwise, \(y_{k}^{\ell}=0\). Following that, \(y_{\mathrm{m}}^{\ell}\) indicates whether or not the selected position \(\ell\) is able to be served by the MBS. If yes, \(y_{\mathrm{m}}^{\ell}=1\); otherwise, \(y_{\mathrm{m}}^{\ell}=0\). For the purpose of mathematically modeling the complete coverage of all coverage  
Figure 2: Conceptual depiction of a two-layer cellular network system. holes, we introduce the set, Y, of dense measurement positions. For convenience, define \(\mathbb{N}\) as the set of all the channels of the \(K\) SBSs and the MBS. Let the indicator \(I_{\delta}^{(i)}\) denote whether user \(i\) uses channel \(\delta\). If yes, \(I_{\delta}^{(i)}=1\); otherwise, \(I_{\delta}^{(i)}=0\). Additionally, define \(P_{\text{total}}\) as the total power consumption of all the BSs' transceivers in the considered model, and define \(\aleph^{(z)}\) as the set of active SBSs in configuration \(z\). Given the above conditions, the aim with respect to optimization is to obtain the minimum \(P_{\text{total}}\) by choosing the appropriate values for \(z\) and \(h\) at each scheduling instant, while still providing the required coverage to all users under the constraints of the allowed configuration sets, as well as the assumption that only one channel is used by each user at any given time.