## 3 Mathematical Model
### Model with Pre-Emption
Particularly, when a new session arrives to the system, the following may happen:  
* If (i) upon arrival of the new \(k\)-type session there are more than \(b_{k}^{\min},k=1,\ldots,K\), PRBs available, and (ii) the current amount of \(k\)-type sessions in service is smaller than \(N_{k},k=1,\ldots,K\), this session is accepted to the system;
* If (i) the arriving \(k\)-type session, \(k=1,\ldots,K-1\), observes less than \(b_{k}^{\min}\) free PRBs in the system, and (ii) the current amount of \(k\)-type sessions in service is smaller than \(N_{k}\), and (iii) there are more than \(b_{k}^{\min}\) PRBs occupied by lower priority sessions in the shared pool of resources, \(c\), the session is admitted to the system causing the pre-emption of \(\sum_{i=2}^{K}m_{i}\) lower priority sessions;
* In any other case, the session is rejected. By utilizing the rules specified above, we can fully characterize the stochastic process \(\mathbf{X}(t)\) describing the service of URLLC and eMBB sessions with the partial reservation strategy and pre-emptive priority. The arbitrarily chosen \"central\" state of the process is shown in Figure 3.  
\[a(\mathbf{n},\mathbf{n}^{\prime})= \begin{cases}\lambda_{k},&\text{if }\mathbf{n}^{ \prime}=\mathbf{n}+\mathbf{e}_{k},\mathbf{n}\in\mathcal{S}_{k},\,k=1,\ldots,K,\\ &\text{or }\mathbf{n}^{\prime}=\mathbf{n}-\mathbf{m}+\mathbf{e}_{k},\mathbf{n} \
otin\mathcal{S}_{k},\mathbf{n}\in\mathcal{S}_{k}^{pre},n_{i}>m_{i},\,k,i=1, \ldots,K,\\ n_{k}\mu_{k},&\text{if }\mathbf{n}^{\prime}=\mathbf{n}-\mathbf{e}_{k},n_{k}>0,\,k=1, \ldots,K;\\ \star,&\text{if }\mathbf{n}^{\prime}=\mathbf{n},\,k=1,\ldots,K;\\ 0,&\text{otherwise.}\end{cases}  \tag{21}\]  
Figure 3: Transition probabilities of the central state. As opposed to the non-pre-emptive system, the CTMC \(\mathbf{X}(t)\) is non-reversible, implying that no product-form solution is available for this system. In this case, the steady-state distribution can be estimated numerically by solving the following set of linear equations:  
\[\mathbf{p}^{T}\mathbf{A}=\mathbf{0}^{T},\mathbf{p}^{T}\mathbf{1}=1, \tag{22}\]  
where \(\mathbf{A}\) is an infinitesimal generator having elements \(a(\mathbf{n},\mathbf{n}^{\prime})\) defined in (21), with the shorthand notation \(*\) provided by  
\[*=-[\sum_{k=1}^{K}\lambda_{k}I\{\mathbf{n}\in\mathcal{S}_{k}\}+ \lambda_{k}I\{\mathbf{n}\
ot\in\mathcal{S}_{k},\mathbf{n}\in\mathcal{S}_{k}^ {pre},n_{i}>m_{i},i=1,\ldots,K\}+n_{k}n_{k}]. \tag{23}\]  
Once the steady-state distribution vector \(\mathbf{p}\) is obtained, one may proceed with the performance measures of the considered system that can be expressed as follows:  
* Drop probability of \(k\)-type session, \(p_{\mathbb{R}^{pre}_{k}}\), is given by \[p_{\mathbb{R}^{pre}_{k}}=\sum_{n_{1}=0}^{N_{1}}\ldots\sum_{n_{K}=0 }^{N_{K}}p(\mathbf{n})I\{\mathbf{n}\in\mathcal{B}_{k}^{pre}\};\] (24)
* The pre-emption probability, \(p_{pre_{k}}\), that is, the probability that arbitrarily chosen sessions are dropped during ongoing service when \(k\)-type session is accepted to the system \[p_{pre_{k}}=\sum_{n_{1}=0}^{N_{1}}\ldots\sum_{n_{K}=0}^{N_{K}}p( \mathbf{n})I\{\mathbf{n}\in\Pi_{k}\}.\] (25) Note that \(\overline{k}_{i}\), \(U\) are defined in (16) and (17), respectively.