# AI-Based Resource Allocation in E2E Network Slicing with Both Public and Non-Public Slices
## 2 System Model
### E2E Optimization Problem  
The optimization objective of this paper is to maximize the network operator's profit while satisfying the constraint of total resources, the minimum rate, the maximum delay, the user satisfaction requirement of each slice, and the security and privacy of the NPN slices. More specifically, the E2E optimization problem is formulated as follows:  
\[\max_{\{x_{k,\mu}^{k},\eta_{j,\
u}^{s}\}} \psi\] (22a) s.t. \[\sum_{s\in\mathcal{S}}\sum_{u\in\mathcal{U}_{s}}\sum_{k\in \mathcal{K}}x_{s,\mu}^{k}\leq K, \tag{22b}\] \[\sum_{s\in\mathcal{S}}\sum_{j=1}^{t_{s}}\eta_{j,\
u}^{s}\leq m_{ \
u},\quad\forall\
u\in\mathcal{V},\] (22c) \[\sum_{s\in\mathcal{S}}\sum_{j=1}^{t_{s}}\eta_{j,\
u}^{s}c_{j}^{s} \leq o_{\
u},\quad\forall\
u\in\mathcal{V},\] (22d) \[\sum_{s\in\mathcal{S}}\sum_{j=1}^{t_{s}-1}\eta_{j,\
u}^{s}\eta_{j +1,\
u^{\prime}}^{s}R_{j+1}^{s}\leq l_{\
u,\
u^{\prime}},\quad\forall l_{\
u,\
u^{\prime}}\in\mathcal{L},\] (22e) \[r_{u}^{s}\geq r_{\min}^{s},\quad\forall u\in\mathcal{U}_{s}, \forall s\in\mathcal{S},\] (22f) \[d_{s,u}^{\text{total}}\leq d_{\max}^{s},\quad\forall u\in \mathcal{U}_{s},\forall s\in\mathcal{S},\] (22g) \[\frac{1}{|\mathcal{U}_{s}|}\sum_{u\in\mathcal{U}_{s}}\xi_{u}^{s} \geq\xi_{\min}^{s},\quad\forall s\in\mathcal{S},\] (22h) \[\text{and }(\ref{eq:1}),(\ref{eq:2}),(\ref{eq:2}),(\ref{eq:2}),( \ref{eq:2}).\]  
where the objective function is as defined in (19). Note that the constraint in (22b) ensures that the number of RBs allocated to the users do not exceed the total number of RBs in the resource pool; constraint (22c) indicates that the number of VNFs deployed on each server node cannot exceed the number of VMs hosted by the server node; constraint (22d) specifies that the sum of the computing resources required by all VNFs deployed on a server node cannot exceed the available computing resources of the server node; constraint (22e) indicates that the sum of the data traffic on each link does not exceed the bandwidth limit of the link; constraint (22f) ensures that the data rate of each end-user in Slice \(s\), i.e., \(r_{u}^{s}\) as defined in (3), is not lower than the minimum rate threshold \(r_{\min}^{s}\); constraint (22g) ensures that the E2E delay of each end-user in Slice \(s\), i.e., \(d_{s,u}^{\text{total}}\) as defined in (13), cannot exceed the maximum delay threshold \(d_{\max}^{s}\); and constraint (22h) indicates that the user satisfaction of Slice \(s\) should not be lower than the satisfaction threshold \(\xi_{\min}^{s}\), and, according to previous discussions, we have \(\xi_{\min}^{s}\leq 1\) when \(s\) is an eMBB or URLLC slice and \(\xi_{\min}^{s}=1\) when \(s\) is a NPN slice.  
The proposed problem of (22) is a binary nonlinear integer programming problem. The time complexity for solving this problem is \(O(2^{K}\!U2^{V}\!\sum_{s\in\mathcal{S}}t_{s})\); furthermore, it can be reduced to a knapsack problem, which is NP hard. Hence, it is inefficient and infeasible to exhaust all possible allocation options to find the optimal one. However, Artifical Intelligence (AI) has powerful decision making and information extraction capabilities, especially deep reinforcement learning. Furthermore, many researchers have shown that AI-based resource allocation methods are more efficient than other methods [19; 23]. Therefore, in this paper, we resort to machine learning and deep reinforcement learning approaches to solve the proposed problem (22).