# Performance Modeling and Analysis of LTE/Wi-Fi Coexistence
## 3 Modeling
### Mathematical Formulation of the Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(k\) & The total number of zones & \(MEAN_{VOLTEz}\) & The mean number of users related to the VOLTE service in the respected zone \\ \hline \(NWB\_Zi(t)\) & The number of web-browsing sessions in zone \(i\) at time \(t\) & \(MEAN_{VOLTE\_ALL}\) & The mean number of users related to the VOLTE service in all zones \\ \hline \(NSTR\_Zi(t)\) & The number of video streaming sessions in zone \(i\) at time \(t\) & \(N_{z}\) & The number of users in each zone \\ \hline \(NVOLTE\_Zi(t)\) & The number of voice calls sessions in zone \(i\) at time \(t\) & \(P_{z}\) & The probability of being in the respected zone for a given service \\ \hline \(S(t)\) & A system state at a given point of time & \(MEAN_{z}\) & The mean number of users in each zone with respect to each service \\ \hline \(N_{TOTAL}\) & The total number of users in all zones & \(MEAN_{Total}\) & The mean total number of users in all zones \\ \hline \(\mathcal{Q}\) & The transition matrix & \(P_{blocking}\) & New call-blocking probability \\ \hline \(q_{lm}\) & The transition rate from state \(l\) to state \(m\) & \(P_{loss}\) & Loss probability of the call \\ \hline \(R\) & The total number of possible states & \(Max\_Cell\_Cap\) & Maximum assumed capacity in the cell \\ \hline \(WB\) & Web-browsing service category & \(Zone1_{Cap}\) & Maximum assumed capacity for zone 1 \\ \hline \(STR\) & Video streaming service category & \(Zone2_{Cap}\) & Maximum assumed capacity for zone 2 \\ \hline \(VOLTE\) & Voice over LTE service category & \(WB_{Ratio}\) & The ratio of the WB service out of 100\% \\ \hline \(ABR\_i\) & Average bit rate of zone \(i\) & \(STR_{ratio}\) & The ratio of STR service out of 100\% \\ \hline \(BR\_min\) & Minimum required threshold bit rate & \(VOLTE_{ratio}\) & The ratio of the VOLTE service out of 100\% \\ \hline \(\lambda\) & Arrival rate & \(TMBR_{cell}\) & Mean total bit rate in the cell \\ \hline \(\alpha_{i}\) & The density of the users in zone \(i\) & \(ABR_{cell}\) & Average bit rate in the cell \\ \hline \(BR\_i\) & Bit rate of zone \(i\) & \(MCC\) & Maximum cell capacity \\ \hline \(MEAN_{WB2}\) & The mean number of users related to the WB service in the respected zone & \(AABR_{z}\) & Aggregate average bit rate per zone \\ \hline \(MEAN_{WB_{ALL}}\) & The mean number of users related to the WB service in all zones & \(AABR_{WB_{z}}\) & Aggregate average bit rate per WB service in the respected zone \\ \hline \(MEAN_{STRz}\) & The mean number of users related to the STR service in the respected zone & \(U\) & Utilization \\ \hline \(MEAN_{STR\_ALL}\) & The mean number of users related to the STR service in all zones & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols used in the mathematical formulation of the model. The MOSEL-2 simulation generated the following performance indicators:  
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: \[MEAN_{WBz}=(N_{z-WB}*P_{z-WB})\] (4) \[MEAN_{WB_{ALL}}=\sum_{z=1}^{k}(N_{z-WB}*P_{z-WB})\] (5) \[MEAN_{STEx}=(N_{z-STR}*P_{z-STR})\] (6) \[MEAN_{STR\_ALL}=\sum_{z=1}^{k}(N_{z-STR}*P_{z-STR})\] (7) \[MEAN_{VOLTEz}=(N_{z-VOLTE}*P_{z-VOLTE})\] (8) \[MEAN_{VOLTE\_ALL}=\sum_{z=1}^{k}(\ N_{z-VOLTE}*P_{z-VOLTE})\] (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: \[MEAN_{z}=\sum_{z=1}^{k}(MEAN_{WB_{z}}+MEAN_{STR\_z}+MEAN_{VOLTE\_z})\] (10) Therefore, the mean total number of users in all zones is given as: \[MEAN_{Total}=\sum_{z=1}^{k}(MEAN_{z})\] (11)
2. New call-blocking probability: the probability to block the new user from being admitted to the system.