# Performance Modeling and Analysis of LTE/Wi-Fi Coexistence
## 3 Modeling
### Mathematical Formulation of the Model
This probability is given as: \[P_{blocking}=\sum_{i=0}^{N}P_{i}*B(P_{i})\] (12) where \(B(P_{i})\) is a constant, which is equal to one, if the average bit rate of the user is not satisfied, which means that the average bit rate for the given user is less than the minimum bit rate assumed for the cell, otherwise the value of \(B(P_{i})\) will be zero. \(P_{i}\) is the probability to be in a state \(i\). \(N\) is the number of users in the cell. 3. Loss probability: the probability of dropping an active user in the system because the average bit rate of the user is dropped down below the threshold value assumed in  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  
1. Mean total number of users in all zones: First, we need to calculate the mean number of users related to each service in each zone and all zones: \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]0 (4) \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]1 (5) \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]2 (6) \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]3 (7) \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]4 (8) \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]5 (9) where \(z=1,\ldots,k\): \(k\) is the total number of zones. \(N_{z}\): is the number of users in each zone related to each service. \(P_{z}\): is the probability to be in this zone for a given service. Then, we calculate the mean number of users in each zone: \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]6 (10) Therefore, the mean total number of users in all zones is given as: \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]7 (11)
2. New call-blocking probability: the probability to block the new user from being admitted to the system. This probability is given as: \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\]8 (12) where \(B(P_{i})\) is a constant, which is equal to one, if the average bit rate of the user is not satisfied, which means that the average bit rate for the given user is less than the minimum bit rate assumed for the cell, otherwise the value of \(B(P_{i})\) will be zero. \(P_{i}\) is the probability to be in a state \(i\). \(N\) is the number of users in the cell. 3. Loss probability: the probability of dropping an active user in the system because the average bit rate of the user is dropped down below the threshold value assumed in \begin{table} \end{table} 
Table 2: Rules for generating the transition states of the matrix. the system. Evidently, this type of probability should be minimized in the system, as blocking a new user is less harmful to the system than dropping the user who is already admitted and active in the system. \[P_{loss}=\sum_{i=0}^{N}P_{i}*L(P_{i}) \tag{13}\]  
where \(L(P_{i})\) is a constant, which is equal to one if the average bit rate of the user is higher than the minimum bit rate assumed for the cell, otherwise it is zero. However, for some reason, the average bit rate of this user is suddenly dropped due to technical reasons, which result in the loss of the call. 4. Mean total bit rate in the whole cell: According to the LTE modulation, The average bit rate of zone 1 is calculated as follows: First, we calculated the bit rate in each zone as follows (assuming that the inner zone is always given the maximum achievable bandwidth because it represents the dense area with more traffic density than the outer zones). Based on the assumed modulation technique (in our case, two modulation levels are assumed as a compensation for the signal strength and quality), the maximum achievable bit rate in each zone is: \(C_{i}=2*C_{i+1}\), \(i=1,\ldots,Z-1\). Therefore, The bit rate in zone 1 (inner zone): \(BR_{1}=Max\_Cell\_Cap\). The bit rate in zone 2 (outer zone): \(BR_{2}=Max\_Cell\_Cap/2\).