## 3 System Model
### Baseline Algorithm for Resource Reallocation  
Let us present a detailed description of a baseline algorithm for resource reallocation that is designed to maximize resource utilization. Resource reallocation may occur when a signal is received from the controller.  
Resource reallocation is triggered under the following conditions:  
* The resources in the 1-slice are fully occupied, while there are still users waiting for service. However, there are available resources in the 2-slice. In this case, some of the idle resources in the 2-slice are reassigned to the 1-slice.
* Similarly, if the resources in the 2-slice are fully occupied and there are users waiting for service, but there are available resources in the 1-slice, some of the idle resources in the 1-slice are reassigned to the 2-slice.  
\begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Description** \\ \hline \multicolumn{3}{c}{Network slice service provider (communication service provider) parameters} \\ \hline \(b\) & Minimum-bit-rate guarantee for transmitting elastic traffic, bps \\ \(N=\lfloor\frac{\mathcal{C}}{b}\rfloor\) & Maximum number of all providers users jointly transmitting elastic traffic \\ \(R_{k}\) & Maximum number of the \(k\)-provider users (\(k\)-users) waiting for delayed \\  & elastic traffic transmission (size of \(k\)-buffer) \\ \hline \multicolumn{3}{c}{Network slice service user (communication service customer) parameters} \\ \hline \(\lambda_{k}\) & Arrival rate of requests for elastic traffic transmission from \(k\)-users, 1/s \\ \(\mu_{k}^{-1}\) & Average volume of elastic traffic transmitted by \(k\)-users, bit \\ \(\varepsilon_{k}\) & Abandonment rate due to impatience of \(k\)-users in \(k\)-buffer, 1/s \\ \hline \multicolumn{3}{c}{Markov decision process (MDP)} \\ \hline \(m\) & Maximum number of 1-users jointly transmitting elastic traffic (size of 1-slice) \\ \(n_{1}\) & Number of 1-users transmitting elastic traffic and waiting in 1-buffer \\ \(n_{2}\) & Number of 2-users transmitting elastic traffic and waiting in 2-buffer \\ \(\mathbf{s}=(m,n_{1},n_{2})\) & State of the system \\ \(\mathcal{S}\) & Set of states \\ \(a=m\) & Action, i.e., size of 1-slice \\ \(\mathcal{A}_{\mathbf{s}}\) & Set of actions available from state \(\mathbf{s}\) \\ \(q(\mathbf{s}^{\prime}|\mathbf{s},a)\) & Transition rate that action \(a\) in state \(\mathbf{s}\) leads to state \(\mathbf{s}^{\prime}\) \\ \(R(\mathbf{s})\) & Reward received while in state \(\mathbf{s}\), i.e., the weighted objective with \(w_{1}\), \(w_{2}\), and \(w_{3}\) \\ \((\mathcal{S},\mathcal{A}_{\mathbf{s}},\mathbf{Q}_{\mathbf{z}},\mathbf{R})\) & Markov decision process (MDP) \\ \hline \end{tabular}
\end{table}
Table 3: _Cont._On the other hand, resource reallocation does not take place if any of the following conditions occur:  
* The system is empty.
* Not all resources are occupied (i.e., there are available resources in both the 1-slice and 2-slice).
* All resources are occupied, but there are no users waiting for service.
* All resources are occupied, and there are users waiting for service from only one slice.
* All resources are occupied, and there are users waiting for service from both the 1-slice and 2-slice.