## 2 System Model
Thus, following [11], we consider the next optimization problem with a utility function of the log type that was proposed by [PERSON] [17] for proportionally fair  
Figure 1: \(K\)-dimensional QS with customer blocking and service preemption. Dynamic Resource Sharing (DRS), which, in our case, coincides with max-min fairness ([18], pp. 111-204):  
\[\sum_{k\in\mathcal{X}^{+}(\mathbf{n})}W_{k}(n_{k})n_{k}\ln(x_{k}) \longrightarrow\max, \tag{3}\] \[\text{subject to}\sum_{k\in\mathcal{X}^{+}(\mathbf{n})}x_{k}n_{k}=C,\] (4) \[\text{over}\ \mathbf{x}\in\mathcal{P}=\left\{\mathbf{x}=(x_{1}, \ldots,x_{K})^{T}:\bigwedge_{k\in\mathcal{X}^{+}(\mathbf{n})}(b_{k}\leq x_{k} \leq C_{k})\right\}, \tag{5}\]  
where \(W_{k}(n_{k})=\left\{ \begin{array}{cc}1,&n_{k}\leq N_{k};\\ n_{k}^{-1}N_{k},&n_{k}>N_{k}.\end{array} ,\right.k\in\mathcal{X}\), denotes the weight functions [15]. We propose an analytical DRS solution method, allocating a data rate to each customer in the considered system. The DRS method is described by the following recursive sequence of numbers:  
\[w_{k}^{(0)}(\mathbf{n})=\left\{ \begin{array}{cc}n_{k}b_{k},\ k\in\mathcal{ X}_{\text{s}},\\ \min(n_{k}b_{k},Q_{k}),\ k\in\mathcal{X}_{\text{e}},\end{array} \right.k= \overline{1,K}, \tag{6}\]  
\[w_{k}^{(l)}(\mathbf{n})=\left\{ \begin{array}{cc}w_{k}^{(l-1)}(\mathbf{n})r _{k}^{(l-1)}(\mathbf{n}),\ k\in\mathcal{X}_{\text{s}},\\ \min\Bigl{(}C_{k},w_{k}^{(l-1)}(\mathbf{n})+\Bigl{(}C-\sum_{i=1}^{K}w_{i}^{(l- 1)}(\mathbf{n})\Bigr{)}^{+}r_{k}^{(l-1)}(\mathbf{n})\Bigr{)},\ k\in\mathcal{X} _{\text{e}},\end{array} \right.l=\overline{1,K_{\text{e}}+1}, \tag{7}\]  
where  
\[r_{k}^{(0)}(\mathbf{n})=\left\{ \begin{array}{cc}1,\ k\in \mathcal{X}_{\text{s}},\\ \left(\frac{n_{k}\mathbf{I}\left(\sum_{k-1}^{K}(n_{i}-N_{i})^{+}=0\right)}{ \sum_{j\in\mathcal{X}_{\text{e}}}n_{j}}+\frac{(n_{k}-N_{k})^{+}\mathbf{I} \left(\sum_{k-1}^{K}(n_{i}-N_{i})^{+}>0\right)}{\sum_{j\in\mathcal{X}_{\text{e }}}(n_{j}-N_{j})^{+}}\right)^{+},\ k\in\mathcal{X}_{\text{e}},\end{array} \right. \tag{8}\] \[r_{k}^{(l)}(\mathbf{n})=\left\{ \begin{array}{cc}1,\ k\in \mathcal{X}_{\text{s}},\\ \left(\frac{n_{k}}{\sum_{j\in\mathcal{X}_{\text{e}}}n_{j}\bigl{(}C_{j}-w_{j}^{ (l)}(\mathbf{n})>0\bigr{)}}\right)^{+},\ k\in\mathcal{X}_{\text{e}},\end{array} \right. \tag{9}\]  
given \(\mathbf{I}(\text{event}\ A)=\left\{ \begin{array}{cc}1,&\text{if $A$ occurs},\\ 0,&\text{otherwise},\end{array} \right.\), which is the indicator function. Let us denote the following:  
\(c_{k}(\mathbf{n}):=w_{k}^{(K_{\text{e}}+1)}(\mathbf{n})\)--the capacity share of slice \(k\) in arbitrary system state \(\mathbf{n}\in\Omega\);  
\(\sum_{k=1}^{K}c_{k}(\mathbf{n})=c_{1}(\mathbf{n})+\ldots+c_{K}(\mathbf{n})\)--the allocated capacity of the system;  
\(C-\sum_{k=1}^{K}c_{k}(\mathbf{n})\)--the unallocated capacity of the system. Explanations for Definition 1 are given by the illustrations in Figure 2.  
Let us denote \(\mathbf{e}_{k},k\in\mathcal{X}\) as the \(k\)-th row of a \(K\times K\) identity matrix. Given the above DRS method, we organize the Radio Admission Control (RAC) scheme in such a way that, upon a customer's arrival at a slice \(k\in\mathcal{X}\) in arbitrary system state \(\mathbf{n}\in\Omega\), three paths are available:  
1. [leftmargin=*,noitemsep,topsep=0 pt]
2. Admission via preemption--when the data rate allocated to each customer in any of the TCP elastic slices will be less than the minimum, and the number of customers in the slice \(k\) is less than guaranteed. 3. Blocking (or rejection) in two cases: 1. [leftmargin=*,noitemsep,topsep=0 pt]
2. When the data rate allocated to each customer in any of the TCP elastic slices will be less than the minimum, and the number of customers in the slice \(k\) will be more than that guaranteed. 3. When the number of customers in the slice \(k\) is equal to the maximum. Direct admission (no preemption)--when the data rate allocated to each customer in each TCP elastic slice will be equal or more than the minimum, and the number of customers in the slice \(k\) is less than the maximum.