## 2 System Model
A depiction of the RAC scheme for accessing a slice \(k\in\mathcal{K}\) is given in Figure 3. Let us assume that in arbitrary system state \(\mathbf{n}\), \(\mathcal{K}_{\mathrm{v}}(\mathbf{n})=\{k_{\mathrm{v}}\in\mathcal{K}:n_{k_{ \mathrm{v}}}>N_{k_{\mathrm{v}}}\}\). We analytically describe the DRR method governing the preemption for a customer admission to a slice owner \(k_{\mathrm{o}}\in\mathcal{K}\) in arbitrary system state \(\mathbf{n}\in\Omega\). We denote the following: \(u_{k_{\mathrm{o}}}^{k_{\mathrm{v}}}(\mathbf{n}),k_{\mathrm{v}}\in\mathcal{K} _{\mathrm{v}}(\mathbf{n})\)--the capacity share of slice violator \(k_{\mathrm{v}}\) to reallocate for a customer admission to slice owner \(k_{\mathrm{o}}\); \(m_{k_{\mathrm{o}}}^{k_{\mathrm{v}}}(\mathbf{n})\)--the corresponding number of customers to preempt. **Definition 2**. : _The DRR method is based on the following sequence of numbers in arbitrary system state \(\mathbf{n}\in\Omega\):_  
\[u_{k_{\mathrm{o}}}^{K}(\mathbf{n})=\min\Bigl{(}\left(c_{K}( \mathbf{n})-Q_{K}\right)^{+},\left(\delta_{k_{\mathrm{o}}}(\mathbf{n})-C \right)^{+}\Bigr{)},\quad k_{\mathrm{o}}\in\mathcal{K}, \tag{10}\] \[u_{k_{\mathrm{o}}}^{l}(\mathbf{n})=\min\Biggl{(}\left(c_{l}( \mathbf{n})-Q_{l}\right)^{+},\left(\delta_{k_{\mathrm{o}}}(\mathbf{n})-\sum_{ j=l+1}^{K}u_{k_{\mathrm{o}}}^{j}(\mathbf{n})-C\right)^{+}\Biggr{)},\quad l= \overline{K-1,1}, \tag{11}\]  
Figure 2: Explanations for Definition 1 (Dynamic Resource Sharing (DRS)). **(a)** Details about Equation (7); **(b)** Details about Equation (8); **(c)** Details about Equation (9). _with \(\delta_{k_{\rm o}}(\mathbf{n})=\sum_{j=1}^{K}c_{j}(\mathbf{n})+b_{k_{\rm o}}\) means that we require capacity \(b_{k_{\rm o}}\). Therefore, we obtain_  
\[m_{k_{\rm o}}^{k}(\mathbf{n})=\left(n_{k}-\left\lfloor\frac{c_{k}(\mathbf{n})-u _{k_{\rm o}}^{k}(\mathbf{n})}{b_{k}}\right\rfloor\right)^{+},\quad k\in\mathcal{ K}. \tag{12}\]  
Using the DRR method for a customer admission to a slice owner \(k_{\rm o}\in\mathcal{K}\) in an arbitrary system state \(\mathbf{n}\in\Omega\), one may note the following:  
\(0\leq u_{k_{\rm o}}^{k_{\rm v}}(\mathbf{n})\leq C_{k_{\rm v}}-Q_{k_{\rm v}}\), \(0\leq m_{k_{\rm o}}^{k_{\rm v}}(\mathbf{n})\leq M_{k_{\rm v}}-N_{k_{\rm v}}\), \(k_{\rm v}\in\mathcal{K}_{\rm v}(\mathbf{n})\)--the capacity share of slice violator \(k_{\rm v}\) to reallocate cannot exceed \(C_{k_{\rm v}}-Q_{k_{\rm v}}\), which means that the corresponding number of customers to preempt cannot exceed \(M_{k_{\rm v}}-N_{k_{\rm v}}\);  
\(0\leq\sum_{j=1}^{K}u_{k_{\rm o}}^{j}(\mathbf{n})\leq b_{k_{\rm o}}\)--the total capacity to reallocate cannot exceed the data rate \(b_{k_{\rm o}}\). Explanations for Definition 2 are given by the illustrations in Figure 4.  
Given the above-described RAC scheme and the analytical DRS and DRR methods as fundaments of the second version of the Preemption-based service Prioritization (PP2), we can move to the mathematical model construction in the next section. Figure 4: Explanations for Definition 2 (Dynamic Resource Reallocation (DRR)). (**a**) Details about Equations (10) and (11); (**b**) Details about Equation (12). Figure 3: Formalization of the RAC scheme for accessing a slice \(k\in\mathcal{K}\) in the system.