## 3 Mathematical Model
### Model with Pre-Emption  
Consider now the system with the pre-emptive priority service. The behavior of this queuing system is similar to the previous one in the sense that the considered process is defined over the same state space \(\mathcal{X}\) as in (8). We assume that the services have different priorities: the highest priority is for the first type of services, the lowest is for \(K\)-type of services. Priority service is implemented in such a way that in the case of insufficient resources in the system to provide the \(k\)-type service with a minimum requirement, the service of one or more lower priority sessions in the shared pool of resources \(c\) could be pre-empted. Figure 2 illustrates the algorithm for selecting sessions that could be pre-empted when a new session arrives to the system. We assume that \(\mathbf{m}=(m_{1},\ldots,m_{K})\)--the numbers of sessions that could be pre-empted. Given the initial condition \(i=K,i>j,m_{k}=0\), \(k=1,\ldots,K\), consider the steps of the algorithm. **Step 1**. Check if there are enough resources for arriving at \(j\)-type session. If the condition is met, the session is accepted. If the condition is not met, go to Step 2. **Step 2**. Check if the service of the \(i\)-type session in the shared pool \(c\) can be pre-empted. If the condition is met, increase \(m_{i}\) by one and go to Step 1. If the condition is not met, go to Step 3. **Step 3**. Check if there are sessions in the system with a higher priority \(i-1>j\). If the condition is met, go to Step 2 with \(i=i-1\). If the condition is not met, the session is rejected. Similarly to the previous model, we also define accepting, loss, and accepting with the maximum number of PRB sets of system states, as \(\mathcal{S}_{k}^{pre}\), \(\mathcal{B}_{k}^{pre}\), and \(\mathcal{S}_{k}^{\max}\), \(k=1,\ldots,K\), respectively. The first set, \(\mathcal{S}_{k}^{pre}\), is defined as  
\[\mathcal{S}_{k}^{pre} =\{\mathbf{n}\in\mathcal{X}:n_{k}<N_{k^{\prime}}(\sum_{l=1}^{k-1} \max(n_{i}b_{i}(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\sum_{i=k+1}^{K}\max(n_{i}b_ {i}^{\min},g_{i})+\] \[+(n_{k}+1)b_{k}^{\min}\leq C\vee(k<K,\sum_{i=1}^{k-1}\max(n_{i}b_ {i}(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\sum_{i=k+1}^{K}\max(n_{i}b_{i}^{\min},g _{i})+\] \[+(n_{k}+1)b_{k}^{\min}-\sum_{i=k+1}^{K}b_{i}^{\min}(n_{i}-N_{i}^{ g})\cdot I(n_{i}>N_{i}^{g})\leq C))\},k=1,\ldots,K, \tag{18}\]  
the second set, \(\mathcal{B}_{k}^{pre}\), is provided by  
\[\mathcal{B}_{k}^{pre} =\{\mathbf{n}\in\mathcal{X}:n_{k}=N_{k}\vee\sum_{l=1}^{k-1}\max(n _{l}b_{i}(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\sum_{i=k+1}^{K}\max(n_{i}b_{i}^{ \min},g_{i})+\] \[+(n_{k}+1)b_{k}^{\min}-\sum_{i=k+1}^{K}b_{i}^{\min}(n_{i}-N_{i}^{ g})\cdot I(n_{i}>N_{i}^{g})>C\},k=1,\ldots,K, \tag{19}\]  
Figure 2: Algorithm for selecting sessions that could be pre-empted. and the last set, \(\mathcal{S}_{k}^{\max}\), is defined in (12). For pre-emptive service, the system of these sets should also be complemented with the pre-emption set of states, \(\Pi_{k}\), \(k=1,\ldots,K-1\), where a \(k\)-type session is accepted to the system, causing the pre-emption of lower priority sessions. This set is given by  
\[\Pi_{k} =\{\mathbf{n}\in\mathcal{X}:n_{k}<N_{k},\sum_{i=1}^{k-1}\max(n_{i }b_{i}(\mathbf{n}+\mathbf{e}_{k}),g_{i})+\sum_{i=k+1}^{K}\max(n_{i}b_{i}^{\min},g_{i})+\] \[+(n_{k}+1)b_{k}^{\min}>C,\sum_{i=1}^{k-1}\max(n_{i}b_{i}(\mathbf{ n}+\mathbf{e}_{k}),g_{i})+\sum_{i=k+1}^{K}\max(n_{i}b_{i}^{\min},g_{i})+\] \[+(n_{k}+1)b_{k}^{\min}-\sum_{i=k+1}^{K}b_{i}^{\min}(n_{i}-N_{i}^ {S})\cdot I(n_{i}>N_{i}^{S})\leq C\},k=1,\ldots,K-1. \tag{20}\]  
Consider now the process of session admission at the system.