# Modeling and Analyzing Preemption-Based Service Prioritization in 5G Networks Slicing Framework
## 2 System Model
### Case Example of Three NSIs  
Consider the case example of three NSIs, i.e., \(\mathcal{S}=\{1,2,3\}\), instantiated at a common 5G BS. Differently to the previous subsection, introduce Priority levels [3; 24; 29; 33; 34] to clarify the Pre-emption scheme. Let  
* The highest priority be assigned to all servicing requests at the 1 st NSI;
* The medium priority be assigned to all servicing requests at the 2 nd NSI;
* The lowest priority be assigned to all servicing requests at the 3 rd NSI.  
Thus, consider that with an arriving request bound to the via pre-emption admission path at the \(s^{\text{th}}\) NSI, \(s\in\{1,2,3\}\), the following consecutive events occur at the system. Firstly, a number  
\[u_{\hat{s}}^{(s,\mathbf{n})}=\min\biggl{\{}(\mathbf{n}-\mathbf{N }^{\mathbf{\hat{s}}})\cdot\mathbf{e}_{\hat{s}},\left\lceil\frac{(\mathbf{n}+ \mathbf{e}_{\hat{s}})\cdot\mathbf{b}-C}{\mathbf{e}_{\hat{s}}\cdot\mathbf{b}} \right\rceil\biggr{\}},\\ \hat{s}=\max\{i\in\{1,2,3\}\setminus\{s\}:(\mathbf{n}-\mathbf{N }^{\mathbf{\hat{s}}})\cdot\mathbf{e}_{\hat{i}}>0\}, \tag{4}\]  
of violators servicing at the \(s^{\text{th}}\) NSI are discarded. Secondly and last, a number  
\[u_{\hat{s}}^{(s,\mathbf{n})}=\left\lceil\frac{\left(\mathbf{n}+\mathbf{e}_{ \hat{s}}-u_{\hat{s}}^{(s,\mathbf{n})}\mathbf{e}_{\hat{s}}\right)\cdot\mathbf{b }-C}{\mathbf{e}_{\hat{s}}\cdot\mathbf{b}}\right\rceil,\quad\hat{s}\in\{1,2,3 \}\setminus\{s,\hat{s}\}, \tag{5}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & \multicolumn{1}{c}{**Description**} \\ \hline \(\mathcal{S}\) & The set of NSIs at the 5G BS, \(\mathcal{S}\subset\mathbb{N}\setminus\{0\}\), [units (u.)] \\ \(S\) & The number of NSIs at the 5G BS, \(S=|\mathcal{S}|\), [u.] \\ C & The total network capacity of the 5G BS, [capacity units (c.u.)] \\ \(C_{s}\) & The overall network capacity of the \(s^{\text{th}}\) NSI, \(s\in\mathcal{S}\), \(C_{1}+\ldots+C_{S}\geq C\), [c.u.] \\ \(Q_{s}\) & The guaranteed network capacity of the \(s^{\text{th}}\) NSI, \(Q_{s}\leq C_{s}\), \(Q_{1}+\ldots+Q_{S}\leq C\), [c.u.] \\ \(\lambda_{s}\) & The arrival rate of requests at the \(s^{\text{th}}\) NSI, \(\lambda=(\lambda_{1},\ldots,\lambda_{S})\), [requests per time units (requests/t.u.)] \\ \(\mu_{s}^{-1}\) & The average service time for a request at the \(s^{\text{th}}\) NSI, \(\mu=(\mu_{1},\ldots,\mu_{S})\), [t.u.] \\ \(\rho_{s}=\lambda_{s}/\mu_{s}\) & The offered load at the \(s^{\text{th}}\) NSI \\ \(b_{s}\) & The requirement for starting service of a request at the \(s^{\text{th}}\) NSI, \(b_{s}\leq Q_{s}\), \(\mathbf{b}=(b_{1},\ldots,b_{S})\), [c.u.] \\ \(\lfloor C_{s}/b_{s}\rfloor\) & The maximum number of requests that may be admitted for service with the overall network capacity of the \(s^{\text{th}}\) NSI, \(\mathbf{N}^{\text{max}}=(\lfloor C_{1}/b_{1}\rfloor,\ldots,\lfloor C_{S}/b_{ \hat{s}}\rfloor)\), [u.] \\ \(\lfloor Q_{s}/b_{s}\rfloor\) & The maximum number of requests that may be admitted for service with the guaranteed network capacity of the \(s^{\text{th}}\) NSI, \(\mathbf{N}^{\text{B}}=(\lfloor Q_{1}/b_{1}\rfloor,\ldots,\lfloor Q_{S}/b_{ \hat{s}}\rfloor)\), [u.] \\ \(n_{s}\) & The current number of servicing requests at the \(s^{\text{th}}\) NSI, \(\mathbf{n}=(n_{1},\ldots,n_{S})\), [u.] \\ \(\mathbf{e}_{\hat{s}}\) & The \(s^{\text{th}}\) row of the size \(S\times S\) identity matrix \\ \(\mathbf{j}\) & The \(S\)-dimensional all-ones vector \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameters.  
of violators servicing at the \(s^{\text{th}}\) NSI are also discarded.  
As an example application, with an arriving request bound to the via pre-emption path at the 2 nd NSI, exactly \(u_{3}^{(2,\mathbf{n})}\), then \(u_{1}^{(2,\mathbf{n})}\) violators servicing, respectively, at the 3 rd and 1 st NSIs are discarded.  
Thus, generalize the Pre-emption scheme for multiple NSIs in the following subsection.