## 3 System Model Description and Problem Statement
### Resource-Aware Slice Decision Problem and DQN Validation of Resource Allocation  
The primary challenge addressed by the model deployed in the xApp dedicated to slice admission control is the selection of the most suitable slice to accommodate a specific resource demand. This selection process can be optimized using a slice-type decision algorithm, which is modeled as an ILP. The ILP determines the optimal slice type for each request by evaluating the demand alongside the available resources. These resources include MEC resources such as Central Processing Unit (CPU) cores, Random Access Memory (RAM), storage, and bandwidth, as well as RAN resources, where the associated bandwidth is evaluated based on the request's transmission rate and latency requirements. Let us consider an MEC system with CPU cores, RAM, storage, and Bandwidth (BW). Therefore, \(MEC=[MEC^{CPU},MEC^{RAM},MEC^{BW},MEC^{ST}]\) represents the state of the MEC resources. Let us consider an O-RAN instance characterized by the BW and the latency (\(L\)) guaranteed to each connected UE. In this way, \(RAN=[RAN^{BW}]\) represents the RAN resources corresponding to the available transmission rate status. Furthermore, a user request is defined as \([D^{CPU},D^{RAM},D^{ST},D^{BW},R,L]\) requesting \(D^{cpu}\) CPU cores, \(D^{ RAM}\) RAM memory, \(D^{ ST}\) storage, \(D^{ DW}\) BW, \(R\) transfer rate, and \(L\) latency, respectively. It is assumed that, as a strategy for reserving resources for latency-critical services, the required latency solely determines the Bandwidth Part (BWP) to be selected for accommodating the request. The Resource-aware Slice Decision Problem (One-RSDP) can be formulated to ensure that each request is assigned to the most suitable slice type. The objective is to minimize the unused capacity of the relevant set of available resources while ensuring that sufficient resources are allocated to meet each request's requirements. To formulate the One-RSDP problem, we define a binary variable \(x_{s}\) to indicate whether one user resource request is to be assigned to the defined slice \(s,s\in\mathcal{S}\):  
\[x_{s}=\left\{ \begin{array}{l}1\,\text{ if slice type $s$ is decided for the current request}\\ 0\,\text{ otherwise.}\end{array} \right. \tag{1}\]  
In addition, a binary variable \(z\) represents whether the new slice request is allocated in the system. Taking these considerations into account, the ILP formulation for the One-RSDP is as follows:  
\[\text{min}\ \sum_{\forall s\in\mathcal{S}} [w_{1}\cdot(\text{MEC}^{ CPU}-x_{s}\cdot D^{ CPU})+w_{2}\cdot(\text{MEC}^{ RAM}-x_{s}\cdot D^{ RAM})+\] \[w_{3}\cdot(\text{MEC}^{ ST}-x_{s}\cdot D^{ ST})+w_{4}\cdot(\text{MEC}^{ DW}-x_{s}\cdot D^{ DW})]-z \tag{2}\]such that \[x_{s}\cdot D^{\textsc{CPU}}\leq\mathrm{MEC}^{\textsc{CPU}} \forall s\in S\] (3) \[x_{s}\cdot D^{\textsc{RAM}}\leq\mathrm{MEC}^{\textsc{RAM}} \forall s\in S\] (4) \[x_{s}\cdot D^{\textsc{Sf}}\leq\mathrm{MEC}^{\textsc{Sf}} \forall s\in S\] (5) \[x_{s}\cdot D^{\textsc{mv}}\leq\mathrm{MEC}^{\textsc{mv}} \forall s\in S\] (6) \[\sum_{\forall s\in S}x_{s}\leq 1\] (7) if not enough RAN capacity for current request \[\Rightarrow\sum_{\forall s\in S}x_{s}\leq 0\] (8) if \[x_{s}=1\Rightarrow z=1\] \[\forall s\in S\] (9) if \[w_{3}\cdot(\text{MEC}^{ ST}-x_{s}\cdot D^{ ST})+w_{4}\cdot(\text{MEC}^{ DW}-x_{s}\cdot D^{ DW})]-z \tag{2}\]0 (10) \[w_{3}\cdot(\text{MEC}^{ ST}-x_{s}\cdot D^{ ST})+w_{4}\cdot(\text{MEC}^{ DW}-x_{s}\cdot D^{ DW})]-z \tag{2}\]1 (11)  
The objective (2) is to minimize the unused capacity of the relevant set of available resources while maximizing the probability of assigning the current slice request. The formalization of the One-RSDP problem led to a multi-objective model incorporating several objectives. To simplify it into a single-objective model, the weighted sum method was applied. The relative importance of each component in the objective function was determined by assigning weight factors \(w_{i},\ i=\{1,\ldots,4\}\), where \(\sum_{i=1}^{4}w_{i}=1\).