# An Optimal Resource Allocation Scheme with Carrier Aggregation in 5G Network under Unlicensed Band
## 3 System Model
### Reserved Unlicensed Spectrum Minimization Problem with QoS Satisfying Rate-Sensitive UE  
Due to the CSMA/CA feature of UE in the Wi-Fi system, small-cell UE in 5G-U can take over an unlicensed spectrum and cause degradation regarding the performance of the Wi-Fi system, which can be overcome by reserving a partial unlicensed spectrum for the Wi-Fi system [6]. To guarantee the performance of rate-sensitive UE associated with Wi-Fi APs, a portion of unlicensed spectrum needs to be saved, which also guarantees the PCC for rate-sensitive UE. The mean ergodic achievable rate \(C_{w}\) for UE \(u\) in the downlink communication from any Wi-Fi AP \(w\in W\) on condition that the tagged AP is at a distance \(r\) from the UE is shown as follows [28]:  
\[C_{w}(b_{w},p_{w}^{t})=\int_{r>0}e^{-\phi r\pi^{2}}\omega 2\pi\phi rdr, \tag{13}\]  
where  
\[\omega=\int_{T>0}e^{-\eta{p_{w}^{t}}^{-1}r^{\alpha}\sigma^{2}(2^{\frac{1}{Ru^{ 2}}}-1)}L_{\mathbf{I}\mathbf{I}\mathbf{I}_{w}}(\eta{p_{w}^{t}}^{-1}r^{\alpha}( 2^{\frac{1}{Ru^{2}}}-1))dt \tag{14}\]  
with \(b_{w}\) and \(p_{w}^{t}\) as virtual carrier bandwidth and transmit power for Wi-Fi AP to UE and  
\[L_{\mathbf{I}\mathbf{I}_{w}}(\eta{p_{w}^{t}}^{-1}r^{\alpha}(2^{\frac{1}{Ru^{2} }}-1))=exp(-2\pi\phi\int_{r}^{\infty}(1-\frac{1}{1+{p_{w}^{t}}^{-1}(\frac{r}{ \tau})^{\alpha}(2^{\frac{1}{Ru^{2}}}-1)})vdv). \tag{15}\]  
In the Poisson-Voronoi (PV) system, mean average length of connections from Wi-Fi AP to UE, which is also the radius of Wi-Fi cell coverage, is obtained according to Voronoi properties as follows [29]:  
\[E[l_{w}]=\frac{E[L_{w}^{w}]}{E[N_{w}^{w}]}=\frac{\frac{\delta_{w}}{2\delta_{w }}}{\frac{\delta_{w}}{\delta_{w}}}=\frac{1}{2\delta_{w}}^{\frac{1}{2}}, \tag{16}\]  
where \(E[N_{w}^{w}]\) and \(E[L_{w}^{w}]\) are the mean number of UE in one Wi-Fi cell coverage and the mean total length of connections from Wi-Fi AP to UE, respectively. The mean number of UE in one Wi-Fi cell coverage is given as [32]  
\[E[N_{u}^{w}]=\delta_{u}2\pi\int_{0}^{\infty}l_{w}e^{-\delta_{w} \pi l_{w}^{2}}dl_{w}=\frac{\delta_{u}}{\delta_{w}}, \tag{17}\]  
where \(l_{w}\) represents the radius of Wi-Fi cell coverage. By utilizing the same method, the mean number of UE in one small cell coverage is \(E[N_{u}^{s}]=\frac{\delta_{u}}{\delta_{s}}\). With slight modifications in (17), the mean total length of connections from Wi-Fi AP to UE can be shown as follows:  
\[E[L_{u}^{w}]=\delta_{u}2\pi\int_{0}^{\infty}l_{w}^{2}e^{-\delta_{w}\pi l_{w} ^{2}}dl_{w}=\frac{\delta_{u}}{2\delta_{w}}^{\frac{3}{2}}, \tag{18}\]  
and mean total length of connections from small cell AP to UE \(E[L_{u}^{s}]=\frac{\delta_{u}}{2\delta_{s}}\). With the Voronoi properties mentioned above, the mean average length of connections from Wi-Fi AP to UE, which can also be interpreted as the radius of Wi-Fi cell coverage, is obtained as  
\[E[l_{w}]=\frac{E[L_{u}^{w}]}{E[N_{u}^{w}]}=\frac{\frac{\delta_{u}}{2\delta_{ w}}^{\frac{3}{2}}}{\frac{\delta_{u}}{\delta_{w}}}=\frac{1}{2\delta_{w}}^{ \frac{1}{2}}. \tag{19}\]  
As mentioned above, if we uniformly allocate aggregated RBs to Wi-Fi APs and each virtual carrier in downlink communication behavior, then \(b_{w}=\frac{(1-\gamma)B}{Ar\delta_{w}}\), where \(\gamma\) is the ratio of unlicensed spectrum allocated to small-cell APs in area \(Ar\). By applying the same method, \(p_{w}^{t}=\frac{\delta P}{Ar\delta_{w}}\), in which \(\xi\) is the ratio of power allocated to Wi-Fi APs. \(B\) and \(P\) are the total unlicensed spectrum and power, respectively. To protect the performance of rate-sensitive UE associated with Wi-Fi APs, we set a Wi-Fi system performance guard threshold to guarantee that the mean achievable rate for rate-sensitive UE associated with Wi-Fi APs is no less than \(C_{t}\). Then, the pre-reserved unlicensed spectrum minimization problem **P1** for rate-sensitive UE is proposed to achieve the minimum unlicensed spectrum to protect the network performance of Wi-Fi users, as shown in (20). \[