# AI-Based Resource Allocation in E2E Network Slicing with Both Public and Non-Public Slices
## 2 System Model
### Radio Access Network Domain  
We assume a specific area composed of a set of gNodeBs \(\mathcal{B}=\{1,2,\ldots,B\}\). The three types of slices considered, namely the eMBB slices, the URLLC slices, and NPN slices, are denoted by \(\mathcal{S}_{e}\), \(\mathcal{S}_{u}\), and \(\mathcal{S}_{np}\), respectively. The set of slices is represented as \(\mathcal{S}=\{1,2,\ldots,S\}=\mathcal{S}_{e}\cup\mathcal{S}_{u}\cup\mathcal{ S}_{np}\). Randomly located in the area are the end-users  
Figure 1: Illustration of the E2E network slicing structure. \(\mathcal{U}=\{1,2,\ldots,U\}\), which are eMBB users, URLLC users, and NPN users, i.e., each user belongs to one and only one slice. The set of users belonging to Slice \(s\) can be represented as \(\mathcal{U}_{s}\), \(s\in\mathcal{S}\). We assume that each end-user \(u\in\mathcal{U}\) is served by the nearest gNodeB, denoted as \(b(u)\). Considering the limited coverage of the NPN, we assumes that the NPN slice is served by only one gNodeB. We consider a downlink (DL) transmission scenario based on orthogonal frequency division multiple access (OFDMA). The spectrum resources can be represented as a shared RB pool \(\mathcal{K}=\{1,2,\ldots,K\}\), where the RB are organized as a resource grid and each RB is the minimum scheduling unit [18]. we define a binary variable \(x_{s,u}^{k}\) as an indicator for assigning RB \(k\) to user \(u\) in slice \(s\):  
\[x_{s,u}^{k}= \begin{cases}1,&\text{If $k\in\mathcal{K}$ is assigned to $u\in\mathcal{U}_{s}$}\\ 0,&\text{Otherwise}\end{cases}  \tag{1}\]  
We assume that each RB can be assigned to at most one user, thus reducing interference and ensuring the isolation of slices. This constraint can be expressed as follows:  
\[\sum_{s\in\mathcal{S}}\sum_{u\in\mathcal{U}_{s}}x_{s,u}^{k}\leq 1\quad\forall k \in\mathcal{K} \tag{2}\]  
We further assume that the gNodeBs have a near perfect knowledge of the channel state information (CSI). The achievable data rate of the user \(u\) belonging to slice \(s\) is defined as follows:  
\[r_{u}^{s}=\sum_{k\in\mathcal{K}}x_{s,u}^{k}W\log_{2}\left(1+\frac{P_{d}G_{s,u} }{\sigma^{2}}\right) \tag{3}\]  
where \(W\) represents the bandwidth of an RB; \(P_{d}\) is the downlink transmission power; and \(G_{s,u}\) is the channel gain between the gNodeB \(b(u)\) and the end-user \(u\in\mathcal{U}_{s}\), including Pathloss and Rayleigh fading [19]. \(\sigma^{2}\) is the power of the additive white Gaussian noise (AWGN). We assume that the downlink transmission power is the same for all RBs. We assume that the set of gNodeBs we consider is a gNodeB sharing group, more specifically, all the gNodeBs share a common RB pool [20]. Therefore, there is no frequency reuse in the same gNodeB sharing group. Furthermore, we have made the assumption that each RB can be assigned to at most one user. Thus we do not consider interference in Equation (3). Therefore, the total rate of slice \(s\) can be expressed as:  
\[r^{s}=\sum_{u\in\mathcal{U}_{s}}r_{u}^{s}\quad\forall s\in\mathcal{S} \tag{4}\]  
To calculate the RAN delay of user \(u\) in slice \(s\), we follow [10, 21, 22], where the queue traffic model is formulated as an M/M/1 queuing model. We make the following assumptions: (i) the arrival process of each user packets follows a Poisson distribution, and (ii) the inter-arrival times of the packets are independent and follow an exponential distribution with mean \(1/\lambda_{s,u}\). Thus, the packets arriving and serving process in slice \(s\) can be modeled as \(|\mathcal{U}_{s}|\) independent M/M/1 queuing systems. By applying [PERSON]'s law, the RAN delay of the user \(u\) in slice \(s\) is:  
\[d_{\text{RAN}}^{s,u}=\frac{1}{r_{u}^{s}/\omega_{s,u}-\lambda_{s,u}} \tag{5}\]  
where \(\omega_{s,u}\) and \(\lambda_{s,u}\) denote the packet size and the packet arrival rate of user \(u\) in slice \(s\), respectively. The first and second terms in the denominator represent the processing and arrival rates of user packets, respectively. We assume that users within the slice have the same packet size and packet arrival rate.