# An Optimal Resource Allocation Scheme with Carrier Aggregation in 5G Network under Unlicensed Band
## 3 System Model
### Primary and Secondary Cell Association  
Cross-carrier scheduling maximizes the transmission quality by aggregating the control signals of multiple carriers into a single control channel, which makes individual UE  
Figure 1: An illustration of 5G network under unlicensed band with CA feature. experience less interference from neighbor cells [3]. The CC with the highest quality signal among all the aggregated CCs serves as PCC for transmissions of control signals and data, while the remaining aggregated CCs serve as SCCs only for data transmission. Then, the SCCs can be dynamically activated without unnecessary control channel interference in densely deployed cellular networks by arranging the control signals to PCC [3]. In this paper, we assume that, within the coverage area of each CC, RBs for users with same type of APs are uniformly allocated, which turns the RB resource allocation problem into a UE aggregation problem for each CC. The CCs aggregated and utilized by AP \(i\) to communicate with serving UE, which are aggregated in Voronoi cell \(V_{i}\), are shown as [28]  
\[\mathbf{U}_{i}\stackrel{{\text{def}}}{{=}}\cup\{u_{i}:u\in \mathbf{U}\cap V_{i}\}, \tag{2}\]  
where the \(\mathbf{U}_{i}\) is the set of UE associated with the AP \(i\). To analyze the characteristics of UE aggregation for each AP, we introduce a non-negative function \(f(u)\) defined on \(\mathbf{R}^{2}\), which is shown as [29]  
\[f(u_{i})=\sum_{u\in\mathbf{U}}f(u)\mathbf{1}\{u\in V_{i}\}. \tag{3}\]  
where \(\mathbf{1}\) is the identify function between AP \(i\) and UE \(u\) and shown as  
\[\mathbf{1}=\left\{ \begin{array}{ll}1,&\quad\text{if }u\in V_{i},\\ 0,&\quad\text{otherwise}.\end{array} \right. \tag{4}\]  
Then,  
\[E\{f(u_{i})\}=\lambda_{u}E\{\int f(u)\mathbf{1}\{u\in V_{i}\}\}. \tag{5}\]  
Conditions that one CC can be utilized by AP \(i\) for communicating with UE \(u\) are shown as follows [18]:  
\[\{u\in V_{i}\}\stackrel{{\text{def}}}{{=}}\{N^{l}=0,N^{l}=0,d_{ ju}>(\frac{\Psi_{u}p_{iu}^{t}}{p_{ju}^{t}})^{-\alpha-1}d_{iu}\}, \tag{6}\]  
where \(V_{i}\) is the Voronoi cell for AP \(i\). \(N^{l}\) and \(N^{l}\) are the numbers of APs from AP sets \(I\) and \(J\) utilizing the same CC in the coverage centered in \(u\) with radius \(d_{iu}\) and \(d_{ju}\), respectively, i.e., \(I,J\in\{S,W\}\) and \(I\
eq J\). In (6), one CC is aggregated by AP \(i\) to communicate with UE \(u\in\mathbf{U}\) if and only if the following conditions are satisfied:  
1. There are no APs from set \(\mathbf{I}\) utilizing the same CC in the coverage that centered in \(u\) with radius \(d_{iu}\). 2. There are no APs from set \(\mathbf{J}\) utilizing the same CC in the coverage that centered in \(u\) with radius \(d_{ju}\). 3. AP \(j\) shows more disadvantage than AP \(i\) with bias parameter \((\frac{\Psi_{u}p_{iu}^{t}}{p_{ju}^{t}})^{-\alpha-1}\). In the perspective of distance, the conditions shown above are that the serving AP \(i\) is the nearest AP among all the APs in set \(\mathbf{I}\) utilizing the same CC at a distance \(d_{iu}\) to UE \(u\) and shows advantage in distance over the nearest AP \(j\) from set \(\mathbf{J}\) utilizing the same CC at a distance \(d_{ju}\) to UE \(u\) with bias parameter \((\frac{\Psi_{u}p_{iu}^{t}}{p_{ju}^{t}})^{-\alpha-1}\). Then, the CC is aggregated by the AP \(i\) and utilized in the communication with UE \(u\). It is worthwhile to mention that the delay-sensitive UE associated with small-cell APs can utilize 5G and 5G-U, while the rate-sensitive UE associated with Wi-Fi APs can only utilize unlicensed spectra, which means there are no conditions 1 and 2 when it comes to 5G licensed band. Then, the null probability of \(d_{iu}\) is shown as follows [28]:  
\[Pr\{u\in\mathbf{V}_{i}\}=e^{-(\lambda_{I}\pi d_{iu}{}^{2}+\lambda_{J}\pi\{(\frac{ \{\eta w_{iu}l_{iu}^{t}}{r_{iu}^{t}})-\alpha^{-1}d_{iu}\}^{2})}}, \tag{7}\]  
where \(\lambda_{I}\) and \(\lambda_{J}\) are densities of APs aggregating the same CC from \(I\) and \(J\), respectively.