## 3 System Model and Problem Formulations
### Capacity and Coverage Dimensioning
#### 3.2.1 Coverage Dimensioning  
The path loss model to carry out the minimum radius of macro-RRU is based on 3 GPP TR-38.901 specifications [47], where the path loss model for the Urban Macro (UMa) cell is given as Equation (4).  
\[PL^{UMa}[\text{dB}]=max(PL_{(UMa-LOS)},PL^{\prime}_{UMa-NLOS})\text{ for }10\text{ m}\leq d_{2D}\leq 5\text{ km} \tag{4}\]  
where  
\[PL^{\prime}_{(UMa-NLOS)}=13.54+39.08 log_{10}(d_{3D})+20 log_{10}(f_{c})-0.6(h_{UE }-1.5) \tag{5}\]  
and  
\[PL_{(UMa-LOS)}=\begin{cases}28+22 log_{10}(d_{3D})+20 log_{10}(f_{c}^{UMa})& \text{for }10\text{ m}\leq d_{2D}\leq d^{\prime}_{BP}\\ 28+40 log_{10}(d_{3D})+20 log_{10}(f_{c}^{UMa})-&\\ 9 log_{10}((d^{\prime}_{BP})^{2}+(h_{BS}-h_{UE})^{2})&\text{for }d^{\prime}_{BP} \leq d_{2D}\leq 5\text{ km}\end{cases} \tag{6}\]  
The minimum radius for UMa-LOS is calculated using,  
\[Radius_{(UMa-LOS)}=10^{(MAPL_{UMa-LOS}-28-20 log_{10}(f_{c}^{UMa}))/22} \tag{7}\]  
where \(d_{3D}\) is the 3D distance between transmitter, UE, \(d_{2D}\) is the 2D distance between the transmitter and UE, \(f_{c}\) is the operating center frequency in GHZ, and \(d^{\prime}_{BP}\) is the breakpoint distance [57] and is given as \(d^{\prime}_{BP}=4h_{BS}{}^{\prime}\times h_{UE}{}^{\prime}\times f_{c}\times 1 0^{9/c}\). The 3D distance between the transmitter and UE is calculated using the relation,  
\[d_{3D}=\sqrt{d_{2D}^{2}+(h_{BS}-h_{UE})^{2}} \tag{8}\]  
We carry out link budget analysis using Equation (3) and find MAPL for macro-RRU for uplink and downlink as \(MAPL_{uplink}=101.05\) dB and \(MAPL_{downlink}=98.05\) dBn respectively. The 3D cell radius calculated for uplink and downlink using Equation (7) is 669.829 m and 489.326 m, respectively. Then, the 2D cell radius is calculated for uplink and downlink using Equation (8) and is 669.828 m and 488.853 m, respectively. Similarly, the path loss for Urban Micro (UMi) cell based on 3 GPP TR-38.901 specifications [47] is given as,  
\[PL^{UMi}[\text{dB}]=max(PL_{(UMi-LOS)},PL^{\prime}_{UMi-NLOS})\text{ for }10\text{ m}\leq d_{2D}\leq 5\text{ km} \tag{9}\]  
where  
\[PL_{UMi-LOS}[\text{dB}]=\begin{cases}32.4+21 log_{10}(d_{3D})+20 log_{10}(f_{c}^ {UMi})&\text{for }10\text{ m}\leq d_{2D}\leq d^{\prime}_{BP}\\ 32.4+40 log_{10}(d_{3D})+20 log_{10}(f_{c}^{UMi})-&\\ 9.5 log_{10}((d^{\prime}_{BP})^{2}+(h_{BS}-h_{UE})^{2}))&\text{for }d^{\prime}_{ BP}\leq d_{2D}\leq 5\text{ km}\end{cases} \tag{10}\]  
and\[PL^{\prime}{}_{UMi-NLOS}[\text{dB}]=22.4+35.3 log_{10}(d_{3D})+21.3 log_{10}(f^{UMi}_ {c})-0.3((h_{UE}-1.5) \tag{11}\]  
The minimum radius for UMi-LOS is calculated using,  
\[Radius_{UMi-LOS}=10^{(MAPL_{UMi-LOS}-32.4-20 log_{10}(f^{UMi}_{c}))/21} \tag{12}\]  
After calculating \(MAPL_{RRU_{m}}\) using Equation (3), we calculated minimum 3D radius using Equation (12) and 2D radius using Equation (8) and the 2D cell radius for UMi for micro-RRU is 146.90 m. The calculated values for MAPL, 2D cell radius, 3D radius, and minimum required number of both micro and macro-RRUs in urban and suburban areas for both uplink and downlink are included in Table 4 for both micro-RRU and macro-RRU.  
The surface area for each cell i, \(A^{i}_{Mcell}\) is expressed using the formula, \(A^{i}_{Mcell}=\pi(RRU^{i}_{m})^{2}\).  
After calculating minimum cell radius and surface area for each cell, the total number of \(\text{RRU}_{\text{m}}\) and \(\text{RRU}_{\text{M}}\) is calculated as,  
\[N_{Mcov}=\frac{A_{total}}{A_{Mcell}} \tag{13}\]  
and  
\[N_{mcov}=\frac{A_{total}}{A_{mcell}} \tag{14}\]  
where \(A_{total}\) is the total case-studied area, and \(A_{mcell}\) and \(A_{Mcell}\) are areas covered by each \(\text{RRU}_{\text{m}}\) and \(\text{RRU}_{\text{M}}\) per sector, respectively.