# An Optimal Resource Allocation Scheme with Carrier Aggregation in 5G Network under Unlicensed Band
## 4 Problem Formulation  
_System Satisfaction Maximization Problem_  
In the Poisson-Voronoi (PV) system, the expected number of UE in one small-cell coverage is given as [29]  
\[E[N_{u}^{s}]=\delta_{u}2\pi\int_{0}^{\infty}l_{s}e^{-\delta_{u}\pi l_{s}^{2}}dl_ {s}=\frac{\delta_{u}}{\delta_{s}}, \tag{31}\]  
where \(l_{s}\) represents the radius of the small-cell coverage. With the same procedure, the expected number of UE in one Wi-Fi cell coverage \(E[N_{u}^{w}]=\frac{\delta_{u}}{\delta_{w}}\). With slight modifications in (31), the expected numbers of small-cell APs and Wi-Fi APs in one MBS cell coverage are \(\frac{\delta_{u}}{\delta_{w}}\) and \(\frac{\delta_{w}}{\delta_{w}}\), respectively. Figure 6: The flowchart of resource allocation process for 5G network under unlicensed band with CA feature. The Jain fairness index (JFI) has been used significantly in the fairness criterion literature for resource allocation [17]. Jain's fairness index is  
\[f(x)=\frac{(\sum_{i=1}^{N}x_{i})^{2}}{N\sum_{i=1}^{N}x_{i}^{2}}, \tag{32}\]  
where \(N\) denotes the total number of users and \(x_{i}\) denotes the received allocation for the \(i_{th}\) user. Based on Jain's fairness index, the fairness index when considering UE's QoS requirements is defined so as to maximize the UE satisfaction valuation. Then, in this section, we consider system utility function based on the satisfaction degree of each UE to the QoS provided by the system, which is defined as the ratio of QoS to system performance threshold. To balance the desperate desire of network operators to provide better network service for delay-sensitive UE and promised network performance for rate-sensitive UE, we consider the weight factor \(\eta\) between system satisfaction of delay-sensitive UE and rate-sensitive UE. The objective of the system is to maximize the overall UE satisfaction, shown as  
\[ \begin{array}{ll}\textbf{P3:}\underset{\gamma\
ot\in\mathcal{A}}{\text{max}}& \eta\sum_{w=1}^{W}\frac{C_{w}}{C_{t}}-(1-\eta)\sum_{s=1}^{S}\frac{D_{s}}{D_{t} }\\ \text{s.t.}&b_{w}\geq b_{w}^{*},w\in\textbf{W}\\ &E[D_{s}](\gamma,\xi,\eta)\leq D_{t}^{*},s\in\textbf{S}\\ &P_{s}^{\epsilon}\geq P_{thr,s}^{\epsilon},s\in\textbf{S}\\ &0\leq\gamma,\xi,\eta\leq 1,\\ &\rho_{s}\leq 1,s\in\textbf{S}\end{array}  \tag{33}\]  
where \(b_{w}^{*}\) and \(D_{t}^{*}\) are the solutions achieved in the optimization problem **P1** and **P2**, which are considered as the performance threshold to guarantee the mean achievable rate and mean packet delay for rate-sensitive and delay-sensitive UE. The minus sign for delay-sensitive UE is due to the fact that delay is preferred to be minimized in the maximization problem. The procedure of system satisfaction maximization via optimal resource allocation by utilizing the Gradient descent method with the Bisection method [33] for 5G under unlicensed band with CA is presented in Algorithm 2. ```
Algorithminput:\(\textbf{G}((\textbf{M},\textbf{S},\textbf{W}),\textbf{U}),\hat{A}_{w},\hat{A}_{w}, \hat{A}_{w},\hat{A}_{w},\hat{A}_{w}\), Step size \(\theta\), tolerance \(\epsilon>0\). Initialize \(\gamma\), \(\xi\) with \(\rho_{s}(\gamma,\xi)\leq 1\) and \(0\leq\gamma,\xi\leq 1\) begin  Obtain \(b_{w}\) from (29). if\(\sum b_{w}=B\)then  Stop; else \(b_{w}^{1}=\sum b_{w}\);  Achieve maximum system satisfaction from (33) and the corresponding \(b_{w}\) by utilizing Gradient descent method with step size \(\theta\); \(b_{w}^{2}=\sum b_{w}\); \(b_{w}^{1}=max(b_{w},\ldots,b_{w}^{L})\). \(\gamma^{*}=1-\frac{\gamma_{w}}{\Gamma}\); if\(||\
abla\sum_{i=1}^{L}||B||(\xi||\tau^{\prime})||_{2}\leq\epsilon\), then \(\xi^{*}=\xi\);  Break;  Update \(\zeta\) by using Bisection method and return to Step 3;  end if endif
```  
**Algorithm 2**System Satisfaction Maximization Algorithm  
```
Algorithminput:\(\textbf{G}((\textbf{M},\textbf{S},\textbf{W}),\textbf{U}),\hat{A}_{w},\hat{A}_{w}, \hat{A}_{w},\hat{A}_{w}\), Step size \(\theta\), tolerance \(\epsilon>0\).