## 3 Mathematical Model
### Model without Pre-Emption  
Let us start with a partial sharing strategy with a non-pre-emptive priority. In the state space \(\mathcal{X}\) we can identify three important sets, namely \(\mathcal{S}_{k}\), \(\mathcal{B}_{k}\), and \(\mathcal{S}_{k}^{\max}\), \(k=1,\ldots,K\). The former set, \(\mathcal{S}_{k}\), \(k=1,\ldots,K\), called the \"accepting\" set, contains all the states in which sessions are accepted to the system and is provided by  
\[\mathcal{S}_{k}=\{\mathbf{n}\in\mathcal{X}:n_{k}<N_{k},\sum_{i=1 }^{k-1}\max(n_{i}b_{i}(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\] \[\qquad\quad+(n_{k}+1)b_{k}^{\min}+\sum_{i=k+1}^{K}\max(n_{i}b_{i }^{\min},g_{i})\leq C\},k=1,\ldots,K. \tag{10}\]  
The second set of states \(\mathcal{B}_{k}\), \(k=1,\ldots,K\), called the \"loss\" set, is a set of system states in which arriving sessions are dropped,  
\[\mathcal{B}_{k}=\{\mathbf{n}\in\mathcal{X}:n_{k}=N_{k}\vee\sum_{i=1 }^{k-1}\max(n_{i}b_{i}(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\] \[\qquad\quad+(n_{k}+1)b_{k}^{\min}+\sum_{i=k+1}^{K}\max(n_{i}b_{i }^{\min},g_{i})>C\},k=1,\ldots,K. \tag{11}\]  
Finally, in the accepting set \(\mathcal{S}_{k}\), we can select a subset \(\mathcal{S}_{k}^{\max},k=1,\ldots,K\), a set of system states in which the arriving sessions will be accepted to the system using \(b_{k}^{\max}\) PRBs, i.e.,  
\[\mathcal{S}_{k}^{\max}=\{\mathbf{n}\in\mathcal{X}:n_{k}<\left\lfloor\frac{C- \sum_{i\
eq k}g_{i}}{b_{k}^{\max}}\right\rfloor,\sum_{i=1}^{k-1}\max(n_{i}b_{i }(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\]  
\[\qquad\quad+(n_{k}+1)b_{k}^{\max}+\sum_{i=k+1}^{K}\max(n_{i}b_{i}^{\min},g_{i} )\leq C\},k,i=1,\ldots,K. \tag{12}\]  
Consider now the process of session admission to the system. Particularly, when a new session arrives to the system, the following may happen:* If (i) upon arrival of the new session of type \(k\) there are more than \(b_{k}^{\min},k=1,\ldots,K\), PRBs available, and (ii) the current amount of sessions in service is smaller than \(N_{k},k=1,\ldots,K\), this session is accepted at the system;
* In any other case, the session is rejected.  
Denote by \(\mathbf{p}=\{p(n_{1},\ldots,n_{K}),(n_{1},\ldots,n_{K})\in\mathcal{X}\}\) the steady-state probability distribution of the CTMC \(\mathbf{X}(t)\). Since the considered stochastic process \(\mathbf{X}(t)\) is reversible, by solving the system of local balance equations,  
\[\lambda_{k}p(\mathbf{n})=(n_{k}+1)\mu_{k}p(\mathbf{n}+\mathbf{e }_{k}),\] \[p(\mathbf{n}+\mathbf{e}_{k})=\frac{\lambda_{k}}{(n_{k}+1)\mu_{k} }p(\mathbf{n}), \tag{13}\]  
recursively, we obtain the steady-state probability distribution of the system in the product-form (14), that is,  
\[p(n_{1},\ldots,n_{K})=\frac{\rho_{1}^{n_{1}}}{n_{1}!}\cdot \ldots\frac{\rho_{K}^{n_{K}}}{n_{K!}}p(0,\ldots,0),\,n_{k}=0,1,\ldots,N_{k},\] \[p(0,\ldots,0)=\left(\sum_{n_{1}=0}^{N_{1}}\ldots\sum_{n_{K}=0}^{ N_{K}}\frac{\rho_{1}^{n_{1}}}{n_{1}!}\cdot\ldots\frac{\rho_{K}^{n_{K}}}{n_{K!}}\right)^{-1}, \tag{14}\]  
where \(\rho_{k}=\lambda_{k}/\mu_{k}\), \(k=1,\ldots,K\).  
Once the steady-state distribution vector \(\mathbf{p}\) is found, we may proceed to determine the performance measures of the system:  
* Drop probability of \(k\)-type session, \(p_{B_{k}}\), is given by \[p_{B_{k}}=\sum_{n_{1}=0}^{N_{1}}\ldots\sum_{n_{K}=0}^{N_{K}}p(\mathbf{n})I\{ \mathbf{n}\in\mathcal{B}_{k}\},k=1,\ldots,K;\] (15)
* Average amount of resources occupied by \(i\)-type sessions, \(\overline{k}_{i}\), is given by \[\overline{k}_{i}=\sum_{n_{1}=0}^{N_{1}}\ldots\sum_{n_{K}=0}^{N_{K}}n_{i}b_{i} (\mathbf{n})p(\mathbf{n})I\{\mathbf{n}\in\mathcal{X}\},i=1,\ldots,K;\] (16)
* The fraction of utilized resources, \(U\), is provided as \[U=\sum_{i=1}^{K}\sum_{n_{1}=0}^{N_{1}}\ldots\sum_{n_{K}=0}^{N_{K}}n_{i}b_{i}( \mathbf{n})p(\mathbf{n})I\{\mathbf{n}\in\mathcal{X}\}.\] (17)