# Performance Analysis of Downlink 5G Networks in Realistic Environments
## 4 Capacity Bounds Analysis
### Single-Service Capacity Bounds
#### 4.1.2 Capacity Bounds Due to Limited Uplink Transmission Power  
The maximum transmitted power \(P_{\max}\) of the user equipment (UE) depends on the distance d between the UE and Node B. Therefore, the UE may not be able to achieve the desired received power, \(P_{R}\), at Node B due to path loss. Because of this fact, the following is very well known [19]:  
\[P_{R}=P_{\max}\cdot PL_{E}(d_{\max}) \tag{9}\]  
**Notations. **\(P_{R}\) is the power received at Node B, \(P_{\max}\) is the maximum power sent by the UE, \(PL_{E}(d_{\max})\) is the path that depends on the distance between the UE and Node B, and parameter E is the code to be used for the type of environment (free space, urban, dense-urban, suburban, and rural). However, in a real environment, the power budget may not be perfect. Therefore, the power control error \(P_{\text{err}}\) should be considered in the path loss as a substitute for the difference between the real and estimated path losses due to fading. Consequently, (9) can be written as follows:  
\[P_{R}=P_{\max}\cdot PL_{E}(d_{\max})\cdot P_{\text{err}} \tag{10}\]  
Then,  
\[PL_{E}(d_{\max})=\frac{P_{R}}{P_{\max}\cdot P_{\text{err}}} \tag{11}\]  
By substituting the value of \(P_{R}\) in (7) into (11), we obtain the following:  
\[PL_{E}(d_{\max})=\frac{N_{0}}{\bigg{(}S-(n-1)\cdot\frac{V\cdot\varepsilon}{F} \bigg{)}\cdot P_{\max}\cdot P_{\text{err}}} \tag{12}\]  
Recalling [19; 20], the path loss of free space propagation is expressed as follows:  
\[\frac{P_{R}}{P_{\text{Smax}}}=\bigg{(}\frac{\lambda}{4\pi\cdot d_{\max}} \bigg{)}^{2}\cdot g_{b}\cdot g_{m}=PL_{E}(d_{\max}) \tag{13}\]Substituting the \(PL_{E}(d_{max})\) value from (12) into (13) gives \(P_{R}\) being dependent on the number of active users in a single-cell case; thus, the antenna gains \(g_{b}\), \(g_{m}\), and \(P_{\text{err}}\) are all assumed to be 1. We obtain  
\[\left(\frac{\lambda}{4\pi\cdot d_{\text{max}}}\right)^{2}=\frac{N_{0}}{\left(S- \left(n-1\right)\cdot\frac{V\cdot\varepsilon}{F}\right)\cdot P_{\text{Smax}}} \tag{14}\]  
**Notations. **\(\lambda\) is the wavelength \(\lambda=c/f\) (\(c\) is the speed of light and \(f\) is the frequency), and \(d_{\text{max}}\) is the maximum distance between the UE and Node B. For a single-service example, the number of active users in the cell, \(n\), determines the received power, \(PR\). When the load on the cell increases, the cell's effective coverage decreases because an additional user causes more interference for existing users, according to the cell breathing phenomena [17; 21] shown in Figure 2. As a result, people who are near cell borders have the best chance of being in adjacent cells. From (14) and based on the derivation in [12], the new formula for \(d_{\text{max}}\) as a function of \(n\) can be expressed as follows:  
\[d_{\text{max}}(n)=\frac{\lambda}{4\pi}\cdot\sqrt{\frac{P_{\text{Smax}}\cdot PL _{E}(d)}{N_{0}}\cdot\left(S-\left(n-1\right)\cdot\frac{V\cdot\varepsilon}{F}\right)} \tag{15}\]  
Mobile operators should consider the coverage of a particular cell because the maximum number of active users in a cell varies dynamically. As a result, it is crucial to consider cell planning for users who are actively depending on cell coverage and the existing radio link service. Figure 2: Cell breathing. Therefore, from (14), the maximum number of active users, \(n_{\max}\), can be reformulated as follows, where \(n\) is replaced with \(n_{\max}\):  
\[\left(S-\frac{\left(4\pi\cdot d_{\max}\right)^{2}\cdot N_{0}}{P_{\max}\cdot \lambda^{2}}\right)\cdot\frac{F}{V\cdot\varepsilon}=n_{\max}-1 \tag{16}\]  
Afterward, move \(n_{\max}\) to the left to obtain  
\[P_{R}=P_{\max}\cdot PL_{E}(d_{\max})\cdot P_{\text{err}} \tag{10}\]0  
The maximum number of active users in the cell can also be represented by rewriting (17) as follows, presuming that the cell has an ideal circle with a radius, \(r\). \[P_{R}=P_{\max}\cdot PL_{E}(d_{\max})\cdot P_{\text{err}} \tag{10}\]1