## 2 System Model  
Let us consider the coexistence of multiple slices at one Fifth Generation Base Station (gNB), each providing either non-TCP streaming or TCP elastic service (non-TCP streaming or TCP elastic slice) to customers (Figure 1). Thus, we consider a system model with a <<service-type-based slice>> [3,4,9]. We denote the following notation:  
\(C,0<C<\infty\)--the total system capacity;  
\(\mathcal{K}=\{1,\ldots,K\}\)--the finite set of slices in the system;  
\(\mathcal{K}_{\mathrm{s}}\subseteq\mathcal{K},|\mathcal{K}_{\mathrm{s}}|=K_{ \mathrm{s}}\)--the subset of non-TCP streaming slices;  
\(\mathcal{K}_{\mathrm{e}}\subseteq\mathcal{K},\mathcal{K}_{\mathrm{s}}\cap \mathcal{K}_{\mathrm{e}}=\emptyset,\mathcal{K}_{\mathrm{s}}\cup\mathcal{K}_{ \mathrm{e}}=\mathcal{K},|\mathcal{K}_{\mathrm{e}}|=K_{\mathrm{e}}\)--the subset of TCP elastic slices;  
\(C_{k},0<C_{k}\leq C,\sum_{i=1}^{K}C_{i}\geq C,k\in\mathcal{K}\)--the overall capacity of slice \(k\);  
\(Q_{k},0\leq Q_{k}\leq C_{k},\sum_{i=1}^{K}Q_{i}\leq C\)--the guaranteed capacity of slice \(k\). We note that the model is built at the top level of abstraction, without particular attention paid to the configuration details at the lower layers of the OSI/ISO model. Thus, we consider the bitrate (one bit per second) as the resource unit (r.u.) since the Quality of Service (QoS) requirements for services are primarily specified in terms of bitrate. From this point of view, the slice capacity is measured in r.u. Let the customer flow at a slice \(k\in\mathcal{K}\) be a Poisson flow with intensity \(\lambda_{k}\Big{[}\frac{\mathrm{ customers}}{\mathrm{time\ units\ (t.u.)}}\Big{]}\). We assume that a customer flow \(k_{\mathrm{g}}\in\mathcal{K}_{\mathrm{s}}\) is characterized by the service time with intensity \(\mu_{k_{\mathrm{g}}}\big{[}\frac{\mathrm{ customers}}{\mathrm{t.u. }}\big{]}\) and constant data rate \(b_{k_{\mathrm{g}}}\big{[}\frac{\mathrm{r.u. }}{\mathrm{ customer}}\big{]}\), while a customer flow \(k_{\mathrm{e}}\in\mathcal{K}_{\mathrm{e}}\) denotes an exponentially distributed random size of the transmitted data block with parameter \(\mu_{k_{\mathrm{e}}}^{-1}\) (size units (s.u.)] and minimum data rate \(b_{k_{\mathrm{e}}}\big{[}\frac{\mathrm{r.u. }}{\mathrm{ customer}}\big{]}\). Let us note that r.u. are expressed in s.u. per t.u. (\(\frac{\mathrm{s.u.}}{\mathrm{t.u.}}\)). Table 1 groups the characteristics of the flows in the system. Let us denote the following:  
\(D_{k}=\left\lfloor{Cb_{k}^{-1}}\right\rfloor,0<D_{k}<\infty,k\in\mathcal{K}\)--the maximum number of customers in slice \(k\), given that equality \(C_{k}=C\) holds;  
\(M_{k}=\left\lfloor{C_{k}b_{k}^{-1}}\right\rfloor,0<M_{k}\leq D_{k}\)--the maximum number of customers in slice \(k\);  
\(N_{k}=\left\lfloor{Q_{k}b_{k}^{-1}}\right\rfloor,0<N_{k}\leq M_{k}\)--the guaranteed number of customers in slice \(k\);  
\(n_{k},0\leq n_{k}\leq M_{k}\)--the number of customers in slice \(k\);  
\(\mathbf{n}:=(n_{1},\ldots,n_{K})^{T},\mathbf{n}\in\Omega\)--the gNB state, with the state space  
\[\Omega=\left\{\mathbf{n}\in(\mathbb{N}\cup\{0\})^{K}:\sum_{k=1}^{K}n_{k}b_{k} \leq C\right\},\quad(z)^{+}=\left\{ \begin{array}{ll}z,&z>0;\\ 0,&z\leq 0;\end{array} \right. \tag{1}\]  
\(c_{k}(\mathbf{n}),0\leq c_{k}(\mathbf{n})\leq C_{k}\)--the capacity share of slice \(k\);  
\(x_{k}(\mathbf{n})=n_{k}^{-1}c_{k}(\mathbf{n}),n_{k}>0\)--the data rate allocated to each customer in slice \(k\) (with this, we assume the uniform allocation of the capacity share of slice \(k\)).