# Performance Model for Video Service in 5G Networks
## 2 Problem Description
Define \(p_{b}^{u,c,m}(t)\) as the power transmitted by the cell \(c\) using the beam \(m\) on the sub-band \(b\) to the user \(u\) at time \(t\). Define \(p(t)=\left\{p_{b}^{u,c,m}(t):u\in\text{U},\text{c}\in\text{C},\text{m}\in \text{M},\text{ b}\in\text{B}\right\}\). This research assumes an SBS system using basic and switched directional antennas. Each user chooses one beam to transmit, referred as the serving beam, and we define the strongest beam from cell \(c\) to user \(u\) as \(\widetilde{m}^{u,c}\). We also assume a constant transmit power \(p_{0}\) per sub-band for each cell. If multiple beams are transmitted, the power will be distributed evenly among the simultaneously transmitted beams. The following Figure 1, Figure 2, and Equations (6)-(8) describe the assumptions. Figure 1 shows the radiation pattern of basic and switched directional antennas with 15, 30, and 60 of half power beam width (HPBW), denoted as \(\theta_{B}\). Equation (6) shows the radiation pattern in decibels [30], where \(G_{0,dB}\) is the maximum gain and \(G_{\text{N},dB}\) is the averaged gain values at the sidelobe. \[G_{l}(\varnothing)=\left\{ \begin{array}{ll}G_{0,\ dB}-12\left(\frac{ \partial}{\partial\theta_{B}}\right)^{2}&|\varnothing|\leq 1.3\ast\theta_{B}\\ G_{\text{N},dB}&|\varnothing|>1.3\ast\theta_{B}\end{array} \right. \tag{5}\]  
\[p_{b}^{u,c,m}=\frac{p_{0}}{\sum_{u\in\text{I}\in U_{c},\text{ m}\in\text{M}}l^{u1,c,m1}}\ast\ G_{l}(\varnothing^{u,c,m}) \tag{6}\]  
where \(\varnothing^{u,c,m}\) is calculated by:  
\[\varnothing^{u,c,m}=\cos^{-1}\frac{\left(\overset{\rightarrow}{u}-\overset{ \rightarrow}{c}\right)\cdot\overset{\rightarrow}{m}}{\left|\overset{ \rightarrow}{u}-\overset{\rightarrow}{c}\right|\overset{\rightarrow}{m}} \tag{7}\]  
\(\With the above assumptions, the optimization problem is written as a function of \(I\), \(\stackrel{{\rightarrow}}{{u}}\), \(\stackrel{{\rightarrow}}{{c}}\), and \(\stackrel{{\rightarrow}}{{m}}\). Problem IC-P1:  
\[\varnothing^{u,c,m}=\cos^{-1}\left(\frac{\stackrel{{\rightarrow}}{{u}} -\stackrel{{\rightarrow}}{{c}})\cdot\stackrel{{ \rightarrow}}{{m}}}{\left|\stackrel{{\rightarrow}}{{u}}- \stackrel{{\rightarrow}}{{c}}\right|\left|\stackrel{{ \rightarrow}}{{m}}\right|}\right. \tag{8}\]  
\[\textit{Maximize}\sum_{t\in T}\sum_{b\in B}\sum_{c\in C}\sum_{u\in U_{c}}w_{u} \sum_{m\in M}r_{b}^{u,c,m}(I) \tag{9}\]  
Over: \(I\). Subject to:  
\[I_{b}^{u,c,m}(t)=\left\{ \begin{array}{cc}1&m=\stackrel{{ \rightarrow}}{{m}}^{u,c}\\ 0&\textit{Else}\end{array} \right. \tag{10}\]  
Figure 1: Gains of a basic antenna. Figure 2: Power received at the UE [29]. \[\sum_{u\in C}l_{b}^{u,c,m}(t)\ \leq M_{max}\quad\forall c\in C,\ t\in T,b\in B \tag{11}\]  
where  
\[r_{b}^{u,c,m}(I)=W\log_{2}\!\left[1+SINR_{b}^{u,c,m}(I)\right] \tag{12}\]  
where  
\[SINR_{b}^{u,c,m}(I)=\frac{g^{u,c}\ p_{b}^{u,c,m}}{\sum_{l\in C,k\in M,\ (l\
eq ck \
eq m)}l_{b}^{u,l,k}*g^{u,l}*p_{b}^{u,l,k}+\sigma_{b}^{u}}. \tag{13}\]  
\(p_{b}^{u,c,m}\) is defined in Equations (6)-(8). \(w_{u}\) is the weight of user \(u\). For video traffic, \(w_{u}\) is calculated using the head of line delay of the video traffic. \(W\) is the bandwidth. \(g^{u,c}\) is the channel gain, which depends on the distance between the cell \(c\) and the user \(u\). \(M_{max}\) is the maximum number of beams per cell. \(\sigma_{b}^{u}\) is the thermal noise. Problem P1 defines UIC approach. GIC approaches allocate time and frequency resources to UEs so that all UEs' interference level are lower than a threshold. The problem is formulated as:  
Problem IC-P2:  
\[Maximize\sum_{c\in C}\sum_{b\in B}\sum_{u\in U_{c}}l_{b}^{u,c}(t)\quad\forall t\in T \tag{14}\]  
Over: \(I\). Subject to:  
\[SINR_{b}^{u,c}(I)>SINRThreshold\qquad\forall c\in C,\ u\in U \tag{15}\]  
where \(SINRThreshold\) is the pre-defined threshold. The IC approaches defined in IC-P1 and IC-P2 are used to improve the performance model. After selecting the best interference coordination approach, we constructed the performance models for each service to map SLA requirements into network attributes.