## 2 System Model
\begin{table}
\begin{tabular}{c c c c c} \hline
**Flow \(k\)** & \(\lambda_{k},\mathbf{0<\lambda_{k}<\infty}\) & \(\mathbf{\mu_{k}^{-1},0<\mu_{k}^{-1}<\infty}\) & \(\mathbf{b_{k},0<b_{k}\leq C_{k}}\) & \(\mathbf{\rho_{k},0<\rho_{k}<\infty}\) \\ \hline \(k\in\mathcal{K}_{\mathrm{s}}\) & Intensity of the customers’ arrival & Mean service time of one customer & Constant data rate of one customer & Offered load \(\lambda_{k}\mu_{k}^{-1}\) \\ (Streaming service) & Intensity of the customers’ arrival & Mean size of one data block & Minimum data rate of one customer & Offered load \(\lambda_{k}\mu_{k}^{-1}\) \\ \hline \(k\in\mathcal{K}_{\mathrm{e}}\) & Intensity of the data blocks’ arrival & Mean size of one data block & Minimum data rate of one customer & Offered load \(\lambda_{k}\mu_{k}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The characteristics of a customer flow \(k\in\mathcal{K}\). According to [15], the state space \(\Omega=\Omega^{\max}\cup\Omega^{\mathrm{opt}}\), where \(\Omega^{\max}\) denotes the feasible states that all customers can be allocated the maximum data rates:  
\[\Omega^{\max}=\Bigg{\{}\mathbf{n}\in\Omega:\sum_{k_{\mathrm{e}}\in\mathbb{X}_{ \mathrm{e}}}n_{k_{\mathrm{e}}}b_{k_{\mathrm{e}}}+\sum_{k_{\mathrm{e}}\in \mathbb{X}_{\mathrm{e}}}n_{k_{\mathrm{e}}}C_{k_{\mathrm{e}}}\leq C\Bigg{\}}; \tag{2}\]  
and \(\Omega^{\mathrm{opt}}=\Omega\setminus\Omega^{\max}\) denotes all other feasible states. We note that two essential slicing features are illustrated in the states of \(\Omega^{\mathrm{opt}}\):  
* The total system capacity is occupied, which ensures that an r.u. is not idle if it can be used to provide services with the highest quality;
* The optimization of r.u. utilization can be performed while ensuring slice isolation for guaranteed QoS. We denote \(\mathcal{K}_{\mathrm{v}}(\mathbf{n})\subseteq\mathcal{K},\mathbf{n}\in\Omega\) as the subset of slices violating the mandatory isolation among the slices (slice violators) in an arbitrary system state \(\mathbf{n}\). Thus, a slice violator is a slice (non-TCP streaming or TCP elastic slice) that uses r.u. guaranteed to another slice (slice owner--also non-TCP streaming or TCP elastic slice) during time intervals when those r.u. are not required/claimed by the other slice due to a lack of customers. We assume that the guaranteed capacity of a slice \(k_{\mathrm{o}}\in\mathcal{K}\) (slice owner) could be utilized by slices violators \(k_{\mathrm{v}}\in\mathcal{K}_{\mathrm{v}}(\mathbf{n})\) in arbitrary system state \(\mathbf{n}\). With this feature, QoS management and isolation among slices can be a problem in the system. Therefore, we propose an analytical solution method--the Dynamic Resource Reallocation (DRR). Through capacity share reduction, DRR can preempt customers' service in slices violators \(k_{\mathrm{v}}\), whenever required for a customer admission to slice owner \(k_{\mathrm{o}}\). We assume that DRR starts its job from slice violator \(k_{\mathrm{v}}\) with the lowest Priority Level (PL) and keeps up until the required capacity is gathered [10; 11]. To gather capacity shares of non-TCP streaming slices violators (\(k_{\mathrm{v}}\in\mathcal{K}_{\mathrm{s}}\)), DRR utilizes service preemption [10], while for TCP elastic slices violators (\(k_{\mathrm{v}}\in\mathcal{K}_{\mathrm{e}}\)), it does so after reducing the data rate allocated to each customer in TCP elastic slices violators \(k_{\mathrm{v}}\) down to the minimum [11; 15]. The DRR method is analytically described in Definition 2. A depiction of the considered \(K\)-dimensional Queueing System (QS) is given by the scheme model in Figure 1.  
Let us assume that in arbitrary system state \(\mathbf{n}\in\Omega\), \(\mathcal{K}^{+}(\mathbf{n})=\{k\in\mathcal{K}:n_{k}>0\}\) and \(\mathcal{K}_{\mathrm{e}}^{+}(\mathbf{n})=\{k_{\mathrm{e}}\in\mathcal{K}_{ \mathrm{e}}:n_{k_{\mathrm{e}}}>0\}\). Note that each non-TCP streaming slice can be seen as 'TCP elastic-friendly' [16], meaning that the minimum data rate allocated to one customer equals the maximum, which makes it constant.