# AI-Based Resource Allocation in E2E Network Slicing with Both Public and Non-Public Slices
## 2 System Model
### Performance Metrics  
In order to evaluate the performance of the resource allocation, we introduce the performance metrics of user satisfaction and the network operator's profit. In order to reflect the difference of users' demand in the slice, we propose the personalized requirements of users (PRU). For the eMBB slices, users have individual rate requirements; for the URLLC slices, users have individual delay requirements; and for the NPN slices, users have both individual rate and delay requirements. It is important to note that the user's personalized requirements simply reflect the rate or delay that the user anticipates, which may not be strictly guaranteed during the resource allocation. The set of PRU for these three types of slices are denoted by \(Q_{e}=\{q_{s,u}^{p},u\text{ is an eMBB user}\}\), \(Q_{u}=\{q_{s,u}^{d},u\text{ is an URLLC user}\}\), and \(Q_{np}=\{(q_{s,u}^{p},q_{s,u}^{d}),u\text{ is a NPN user}\}\), respectively. The set of PRU can be represented as \(\mathcal{Q}=\{Q_{e},Q_{u},Q_{np}\}\). Ideally, we would like the network to satisfy all users' personalized requirements. However, given the limited resources, this is not always possible. In our work, we place a higher service priority on the NPN users, i.e., the resource allocation should be that the individual rate requirement and delay requirement of each NPN user is satisfied strictly. For users in other slices, personalized requirements are addressed to the best extent possible, ensuring that the overall user satisfaction remains above a certain threshold. User satisfaction for users in the eMBB, URLLC, and NPN slices are respectively defined as follows:  
\[\xi_{u}^{s}=\min\left(1,\frac{r_{u}^{s}}{q_{s,u}^{s}}\right),\quad\forall u\in \mathcal{U}_{s},\ \forall s\in\mathcal{S}_{e} \tag{16}\]  
\[\xi_{u}^{s}=\min\left(1,\frac{q_{s,u}^{d}}{d_{s,u}^{\text{total}}}\right), \quad\forall u\in\mathcal{U}_{s},\ \forall s\in\mathcal{S}_{u} \tag{17}\]  
\[\xi_{u}^{s}=\frac{1}{2}\left(\min\left(1,\frac{q_{s,u}^{d}}{d_{s,u}^{\text{ total}}}\right)+\min\left(1,\frac{r_{u}^{s}}{q_{s,u}^{s}}\right)\right),\forall u\in \mathcal{U}_{s},\ \forall s\in\mathcal{S}_{np} \tag{18}\]  
Hence, as can be seen, for an eMBB user \(u\), the satisfaction is 1 as long as the achieved rate \(r_{u}^{s}\) is no smaller than the required rate \(q_{s,u}^{p}\). Otherwise, User \(u\)'s satisfaction depends on the ratio of the achieved rate \(r_{u}^{s}\) and the required rate \(q_{s,u}^{s}\). Similarly, for a URLLC user \(u\), the satisfaction is 1 as long as the achieved delay \(d_{s,u}^{\text{total}}\) is no larger the required rate \(q_{s,u}^{d}\). Otherwise, it is the ratio of the required delay \(q_{s,u}^{d}\) and the achieved delay \(d_{s,u}^{\text{total}}\). Since the NPN users have both rate and delay requirements, we define the user satisfaction as the average of the rate satisfaction and the delay satisfaction. Finally, the user satisfaction of a slice \(s\) is defined as the user satisfaction average of all users in Slice \(s\). Recall that we place a higher service priority on the NPN users, i.e., their PRU need to be satisfied strictly. In other words, we require that \(\xi_{u}^{s}=1\) for users of NPN slices, which further means that the user satisfaction of any NPN slice is equal to 1. The network operator's profit is  
\[\psi=\psi_{R}-\psi_{E} \tag{19}\]  
where \(\psi_{R}\) and \(\psi_{E}\) represent the revenue and expenditure, respectively.