# AI-Based Resource Allocation in E2E Network Slicing with Both Public and Non-Public Slices
## 2 System Model
### Core Network Domain  
We consider the physical network as a directed graph \(G=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}=\{1,2,\ldots,V\}\) represents the set of server nodes, and \(\mathcal{E}=\{(1,1),(1,2),\ldots,E\}\) represents the set of links between server nodes. The resources of each server usually include CPU, memory, disk, etc. In this paper, in order to simplify the model, various resources on nodes are unified as computing resources. The available computing resources of all server nodes are denoted as \(\mathcal{O}=\{o_{1},o_{2},\ldots,o_{V}\}\). The number of virtual machines (VMs) that the server nodes host is represented as \(\mathcal{M}=\{m_{1},m_{2},\ldots,m_{V}\}\). The set of link bandwidths between the server nodes is represented as \(\mathcal{L}=\{l_{1,1},l_{1,2},\ldots,l_{v,\sigma^{\prime}},\ldots,l_{V,V}\}\), and the set of link delays between server nodes is represented as \(\mathcal{D}_{E}=\{d_{1,1},d_{1,2},\ldots,d_{v,\sigma^{\prime}},\ldots,d_{V,V}\}\). Note that if there is no directed edge from Server node \(a\) to Server node \(b\), then the link bandwidth \(l_{a,b}=0\) and the link delay \(d_{a,b}=\infty\). The link bandwidth from a node to itself is specified to be infinity, i.e., \(l_{a,a}=\infty\), while the link delay is zero, i.e., \(d_{a,a}=0\). VNF is a type of network function implemented in the software and it operates on physical servers. Different VNFs can be arranged in a specific sequence to create a service function chain (SFC), enabling the provision of complex network services. The VNFs we are considering primarily include a firewall (FW), network address translator (NAT), traffic monitor (TM), video optimization controller (VOC), an intrusion detection and prevention system (IDPS), and so on [19]. Different VNFs types are needed, which is represented as \(\mathcal{F}=\{1,2,\ldots,F\}\). For each slice, its SFC consists of several VNFs in a certain order. Thus, the SFC of slice \(s\in\mathcal{S}\) is denoted by \(\mathcal{F}^{s}=\{f^{s}_{1},f^{s}_{2},\ldots,f^{s}_{t_{s}}\}\), where \(f^{s}_{j}\in\mathcal{F}\), \(j\in\{1,\cdots,t_{s}\}\), and \(t_{s}\) is the length of the SFC of Slice \(s\). Each VM on any server node can host at most one VNF, and multiple VNFs in the same SFC can be deployed in the same server node. The processing delay of each VNF is determined by its type. To simplify the model, we assume that the delay of each VNF type is known in advance and it can be represented as \(\mathcal{D}_{V}=\{d_{1},d_{2},\ldots,d_{F}\}\). When the data flow passes through the SFC of a slice, after the processing of each VNF, the data processing rate in the SFC will change. In order to easily depict the variation in the data flow in the SFC, we assume that the data processing rate through different VNFs follow linear changes, and this change is quantified by using the data rate expansion ratio [12]. Note that the data rate expansion ratio depends on the type of the VNF; hence, we collect the data rate expansion ratio of all VNF types and denote it by \(\beta=\{\beta_{1},\beta_{2},\ldots,\beta_{F}\}\). Thus, the data processing rate through the \(j\)-th VNF in the SFC of Slice \(s\) can be denoted by [12]:  
\[R^{s}_{j+1}=R^{s}_{j}\beta_{f^{s}_{j}} \tag{6}\]  
where \(R^{s}_{j}\) is the input data processing rate of the \(j\)-th VNF and correspondingly, \(R^{s}_{j+1}\) is the output data processing rate of the \(j\)-th VNF, which is also the input data processing rate of the \(j+1\)-th VNF. Note that the data processing rate of the first VNF in the SFC of Slice \(s\) is the total rate of slice, \(R^{s}_{1}=r^{s}\), where \(r^{s}\) is given in (4). Figure 2 illustrates an example of the data processing rate in the SFC of Slice \(s\):  
A binary variable \(\eta^{s}_{j,v}\) is defined as an indicator to show the mapping of the VNFs in the SFC to server nodes:  
Figure 2: An example of the date processing rate in the SFC. \[\eta^{s}_{j,\
u}= \begin{cases}1,&\text{if the $j$-th VNF in the SFC of Slice $s$ is deployed on Server node $v$}\\ 0,&\text{Otherwise}\end{cases}