# User-BS Selection Strategy Optimization with RSSI-Based Reliability in 5G Wireless Networks
## 3 System Model and Problem Formulation
### Optimization Model with RSSI-Based Reliability  
The higher the received signal strength is, the more reliable the connection is. Combined with failure probability \(R_{BS}(i)\) and \(\mathbb{P}[\pi]\) in (3), the reliability of the user at location \(x\), \(R(s,x,R_{BS})\), can be expressed by selecting strategies \(s\), \(x\), and \(R_{BS}\) :  
\[R(s,x,R_{BS})=\sum_{i=1}^{N}s_{i}(x)RSSI(d_{i}(x))(1-R_{BS}(i)). \tag{8}\]Combining Equation (8) with user distribution, the overall reliability of wireless side \(R_{w}\), can be calculated as:  
\[\begin{split}& R_{w}=\iint R(s,x,R_{BS})f(x)dx\\ &=\iint\sum_{i=1}^{N}s_{i}(x)(1-R_{BS}(i))RSSI(d_{i}(x))f(x)dx\\ &=\sum_{i=1}^{N}\iint s_{i}(x)(1-R_{BS}(i))RSSI(d_{i}(x))f(x)dx. \end{split} \tag{9}\]  
The overall optimization problem is:  
\[\max_{s\in\mathcal{S}}\sum_{i=1}^{N}\iint s_{i}(x)(1-R_{BS}(i))RSSI(d_{i}(x))f( x)dx. \tag{10}\]  
\(BS\) capacity refers to the number of channels that should be configured for a base station or a cell. In large cities and megacities, due to the rapid growth of users, each \(BS\) should be equipped with as many available channels as possible. Therefore, \(BS\) capacity becomes user capacity calculated by the number of channels. When there are too many users connected to the same \(BS\), this leads to a decrease in communication quality and reduced reliability. Therefore, the expected number of users connected to each BS \(i\) cannot exceed the limit of the number of connections of \(BS\) capacity \(N_{BS}(i)\). For each \(BS\)\(i\), the probability of a user connecting to it can be expressed as \(\iint s_{i}(x)f(x)dx\), so the capacity condition for \(M\) users can be expressed as:  
\[M\iint s_{i}(x)f(x)dx\leqslant N_{BS}(i), \tag{11}\]  
for \(i=1,2,\ldots,N\).  
Receiving sensitivity refers to the minimal received signal strength with which the receiver can correctly take out the useful signal, which means that the RSSI must be greater than receiving sensitivity \(SEN\):  
\[RSSI>SEN. \tag{12}\]  
\(SEN\)[34] can be expressed as:  
\[SEN=10\log_{10}(KT0)+10\log_{10}(BW)+NF+SNR_{min}. \tag{13}\]  
where, \(10\log_{10}(KT0)\) represents that the noise floor at a room temperature of 25 \({}^{\circ}\)C is \(-174\) dBm, \(BW\) refers to bandwidth, NF is the noise figure of the system that generally refers to the noise figure of the first low noise amplifier, and \(SNR_{min}\) is the minimal signal-to-noise ratio (SNR) requirement of the receiver. Similar to RSSI, the above approach is not restricted by this specific formula and can be straightforwardly extended to any sensitivity forms.  
Taking a gNodeB BS of 5G NR with 20 MHz bandwidth as an example, the \(SEN\) of gNodeB is:  
\[\begin{split} SEN&=-174+10\log_{10}(19.08\times 10^{6} )+6+(-1)\\ &=-96.2,\end{split} \tag{14}\]  
where \(BW=20\) MHz, and the actual bandwidth occupied by the business is 19 MHz; \(NF\ =\ 6\) dB for the system. NF here is the insertion loss before the first-level LNA and the noise coefficient NF of the LNA itself, \(SNR_{min}=-1\) dB.  
According to the RSSI formula [35]:  
\[d=d_{0}10^{\frac{R_{n}+R_{n}-PL(d_{0})}{10n}}10^{-\frac{1}{10n}RSSI(d)}, \tag{15}\]  
so \(RSSI(d_{i}(x))>\) Sensitivity can be transformed into  
\[d_{i}(x)<D=d_{0}10^{\frac{R_{n}-PL(d_{0})}{10n}}10^{-\frac{1}{10n}SEN}, \tag{16}\]  
which means that signal effective range \(C_{i}\) is given to any \(BS\)\(i\), which satisfies that \(\forall x\in C_{i}\), \(d_{i}(x)<D\), and \(D\) represents the maximal signal connection distance. Therefore, the optimization problem under constraints can be expressed as:  
\[s_{i}(x)=0,\quad\text{for }x\
otin C_{i}. \tag{17}\]  
On the basis of Equation (10), considering BS capacity (11) and \(SEN\) (17) constraints at the same time, the optimization problem can be expressed as follows:  
\[\max_{s\in\mathcal{S}}\ R_{w}, \tag{18}\]  
\[\text{s.t.}\begin{cases}M\iint s_{i}(x)f(x)dx\leq N_{BS}(i),\\ s_{i}(x)=0,\end{cases} \tag{19}\]  
for \(i=1,2,\dots,N\) and \(x\
otin C_{i}\).