# Intelligent Dynamic Power Control with Cell Range Expansion for Small-Cells in 5G HetNets
## 2 System Model  
### System Topology and Channel Assignment  
We consider the DL of 5G HetNets based on orthogonal frequency division multiple access and FDD (OFDMA-FDD). Figure 1 shows the system architecture in which SBSs are overlapped on the coverage of MBSs. We assume that there are \(M\) hexagonal macrocells (\(M=7\)) where the target macrocell is surrounded by six macrocells and a set of \(M\) MBSs, \(M=|\mathbf{M}|\), \(\mathbf{M}=\{1,2,\ldots,M\}\), is located at the center of each macrocell. Let \(d_{IS}\) denote the inter-site distance between MBSs. In addition, sets of \(N\) MUEs, \(\mathbf{N}=\{1,2,\ldots,N\}\), and \(S\) SBSs, \(\mathbf{S}=\{1,2,\ldots,S\}\) are uniformly distributed in the coverage of each site of MBSs. SBSs are installed in outdoor environments and have a set of \(T\) SUEs, \(\mathbf{T}=\{1,2,\ldots,T\}\) in the coverage of each SBS (SBSs have no overlapping coverages with each other). Each MBS has a three-sectored directional antenna and thus the coverage of MBSs is divided into three sites, i.e., sites 1, 2 and 3, while SBSs have an omnidirectional antenna. Figure 2 shows the channel assignments of MBSs and SBSs (including CUEs) in 5G HetNets. Each MBS uses a set of \(K\) subchannels, \(\mathbf{K}=\{1,2,\ldots,K\}\), and \(K\) subchannels are divided into three subchannel groups (SGs) (each SG has one-third subchannels), i.e., A, B and C, assigned to site 1, 2, and 3 of MBSs, respectively. On the other hand, all SBSs use \(K\) subchannels to serve their SUEs and CUEs without the transmission power control and thus MBSs and SBSs have strong interference with each other.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Descriptions** \\ \hline
**M** & Set of macro base stations (MBSs) \\
**S** & Set of small-cell base stations (SBSs) \\
**N** & Set of macro user equipments (MUEs) \\
**T** & Set of small-cell user equipments (SUEs) \\
**K** & Set of total subchannels \\ \(\gamma_{mn}^{k}\) & SINR of MUE \(n\) serviced by MBS \(m\) at subchannel \(k\) \\ \(\gamma_{mst}^{k}\) & SINR of SUE \(t\) serviced by SBS \(s\) in the coverage of MBS \(m\) at subchannel \(k\) \\ \(h_{mn}\) & Channel gain between MBS \(m\) and MUE \(n\) \\ \(h_{mst}\) & Channel gain between SBS \(s\) and SUE \(t\) in the coverage of MBS \(m\) \\ \(p_{MBS}\) & Transmission power of MBSs \\ \(p_{SBS}^{SBS}\) & Transmission power of SBSs \\ \(p_{mns}^{SBS}\) & Maximum transmission power of SBSs \\ \(\xi_{mBS}^{MBS}\) & RSRP between MBS \(m\) and MUE \(n\) \\ \(\xi_{mns}^{SBS}\) & RSRP between SBS \(s\) and SUE \(t\) in the coverage of MBS \(m\) \\ \(L_{mn}\) & Path loss between MBS \(m\) and MUE \(n\) \\ \(L_{mst}\) & Path loss between SBS \(s\) and SUE \(t\) in the coverage of MBS \(m\) \\ \(\psi_{m}^{k}\) & Indicator variable to assign subchannel \(k\) from MBS \(m\) to MUE \(n\) \\ \(\psi_{mst}^{k}\) & Indicator variable to assign subchannel \(k\) from SBS \(s\) of MBS \(m\) to SUE \(t\) \\ \(C_{mn}\) & Capacity of MUE \(n\) served by MBS \(m\) \\ \(C_{mst}\) & Capacity of SUE \(t\) served by SBS \(s\) in the coverage of MBS \(m\) \\ \(p_{MIE}^{MIE}\) & Outage probability of MUEs \\ \(\psi_{mBL}^{MIE}\) & Outage probability of SUEs (including CUEs) \\ \(\Gamma_{th}\) & Given target SINR threshold for outage probability \\ \(\Gamma_{sth}\) & Given target SINR threshold for SUEs in the inner and outer zones \\ \(W\) & Bandwidth per subchannel \\ \(\eta\) & Given CRE bias for SBSs \\ \(d_{IS}\) & Inter-site distance between MBSs \\ \(A(\theta)\) & Antenna gain of MBSs \\ \(A_{mg}\) & Maximum antenna gain of MBSs \\ \(A_{ma}\) & Maximum attenuation of MBSs \\ \(\sigma^{2}\) & White noise power \\ \hline \hline \end{tabular}
\end{table}
Table 2: List of main symbols and notations.