## 3 Mathematical Model  
According to the Radio Admission Control (RAC) scheme given in Section 2 and considering the Poisson flows of customers, the system behavior can be described by the \(K\)-dimensional Markov process \(\mathbf{X}(t)=\big{(}X_{1}(t),\ldots,X_{K}(t),t>0\big{)}^{T}\), where \(X_{k}(t),k\in\mathcal{K}\), is the number of customers in slice \(k\) at time \(t\) over the system state space  
\[\mathcal{S}=\Bigg{\{}\mathbf{n}\in\Omega:\bigwedge_{k_{\mathrm{e}}\in\mathcal{ X}_{\mathrm{e}}^{+}(\mathbf{n})}\big{(}x_{k_{\mathrm{e}}}(\mathbf{n})\geq b_{k_{ \mathrm{e}}}\big{)},\sum_{k=1}^{K}(n_{k}-M_{k})^{+}=0,\sum_{k=1}^{K}c_{k}( \mathbf{n})\leq C\Bigg{\}}. \tag{13}\]  
We note that system state space \(\mathcal{S}\) contains several subspaces ([19], pp. 66-100), ([20], pp. 2-14). For a slice \(k\in\mathcal{K}\), we denote the following:  
\(\mathcal{S}_{k}^{(1)}\)--the state subspace for a direct admission (no preemption) of a customer;  
\(\mathcal{S}_{k}^{(2)}\)--the state subspace for an admission via preemption;  
\(\mathcal{\overline{S}}_{k}\)--the state subspace for blocking (or rejection):  
\[\mathcal{S}_{k}^{(1)}=\Bigg{\{}\mathbf{n}\in\mathcal{S}:\bigwedge_{k_{\mathrm{ e}}\in\mathcal{X}_{\mathrm{e}}^{+}(\mathbf{n}+\mathbf{e}_{k})}\big{(}x_{k_{ \mathrm{e}}}(\mathbf{n}+\mathbf{e}_{k})\geq b_{k_{\mathrm{e}}}\big{)},n_{k}< M_{k}\Bigg{\}}, \tag{14}\]  
\[\mathcal{S}_{k}^{(2)}=\Bigg{\{}\mathbf{n}\in\mathcal{S}:\bigvee_{k_{\mathrm{ e}}\in\mathcal{X}_{\mathrm{e}}^{+}(\mathbf{n}+\mathbf{e}_{k})}\big{(}x_{k_{ \mathrm{e}}}(\mathbf{n}+\mathbf{e}_{k})<b_{k_{\mathrm{e}}}\big{)},n_{k}<N_{k} \Bigg{\}}, \tag{15}\]  
\[\mathcal{\overline{S}}_{k}=\Bigg{\{}\mathbf{n}\in\mathcal{S}:\bigvee_{k_{ \mathrm{e}}\in\mathcal{X}_{\mathrm{e}}^{+}(\mathbf{n}+\mathbf{e}_{k})}\big{(}x _{k_{\mathrm{e}}}(\mathbf{n}+\mathbf{e}_{k})<b_{k_{\mathrm{e}}}\big{)},n_{k}+1 >N_{k}\lor n_{k}=M_{k}\Bigg{\}}, \tag{16}\]  
\[\mathcal{S}=\mathcal{S}_{k}^{(1)}\cup\mathcal{S}_{k}^{(2)}\cup\mathcal{ \overline{S}}_{k}. \tag{17}\]  
Given the above-defined state space \(\mathcal{S}\) and its subspaces, the RAC function for the customers flow \(k\in\mathcal{K}\) can be described as \(f_{k}(\mathbf{n})=\Bigg{\{} \begin{array}{ll}1,&\mathbf{n}\in\mathcal{S}_{k} ^{(1)}\cup\mathcal{S}_{k}^{(2)},\\ 0,&\text{otherwise }(\mathbf{n}\in\mathcal{\overline{S}}_{k}),\end{array} , \mathbf{n}\in\mathcal{S}.\)  
Let \(\mathbf{m}_{k}(\mathbf{n}):=\big{(}m_{k}^{1}(\mathbf{n}),\ldots,m_{k}^{K}( \mathbf{n})\big{)}^{T},k\in\mathcal{K},\mathbf{n}\in\mathcal{S}.\)  
A depiction of the system state transition diagram between \(\mathbf{n}\in\mathcal{S}\) and all its neighbors \(\mathbf{n}+\mathbf{e}_{k},\mathbf{n}-\mathbf{e}_{k},\mathbf{n}+\mathbf{e}_{k }-\mathbf{m}_{k}(\mathbf{n}),\mathbf{n}-\mathbf{e}_{k}+\mathbf{m}_{k}( \mathbf{n})\) is provided in Figure 5.  
Figure 5: State transition diagram between \(\mathbf{n}\in\mathcal{S}\) and its neighbors \(\mathbf{n}\pm\mathbf{e}_{k},\mathbf{n}\pm\mathbf{e}_{k}\pm\mathbf{m}_{k}( \mathbf{n})\). Let \(\mathbf{p}^{T}:=\left(p(\mathbf{n})\right)_{\mathbf{n}\in\mathbb{S}}\) denote the stationary probability distribution of the Markov process \(\mathbf{X}(t)\), with \(p(\mathbf{n}),\mathbf{n}\in\mathbb{S}\), denoting the stationary probability that the system is in state \(\mathbf{n}\).