## 3 System Model
### Pair-Decision Resource Mapping Relations  
In the resource allocation scheme, each substrate node can accommodate multiple VNUs. We assume that each VNU carrying certain VNFs in a slice can only be mapped into one substrate node, which cannot be instantiated to different servers [39]. The VNU is not shared by communication service instances both in the same or different slices for the slice-isolation purpose.  
When mapping a communication service instance into substrate network nodes and links, we introduce a binary variable \(m^{(s,k)}_{u}\) to indicate the mapping relationship between the virtual and substrate nodes. In slice \(s\), we use \(m^{(s,k)}_{u}=1\) to stand for that VNU \(k\) is mapped into the substrate node \(u\in N^{p}\), and \(m^{(s,k)}_{u}=0\) otherwise. The substrate link \((u,v)\in L^{p}\) between the substrate nodes \(u\) and \(v\in N^{p}\) is a segment link of the path \((r,d)\), whose resources are allocated to support adjacent VNUs for processing and forwarding, called processing link. For example, Figure 2 illustrates the process for mapping VNUs and virtual links into substrate nodes and links. The communication service chain \(I\) possesses two VNUs \(k\) and \(l\) and a virtual link \((k,l)\). The path \((r,v)\) is the selected substrate path mapped by the communication service chain \(I\) and the substrate link \(ri,ij\) and so on are segment links of the path \((r,v)\) to transmit the traffic from the source node \(r\) to the terminated node \(v\). In particular, \(u\) and \(v\) are processing substrate nodes that support corresponding VNUs \(k\) and \(l\) respectively, as well as \((u,v)\) is the processing link for supporting virtual link \((k,l)\).  
In order to describe the aforementioned mapping relationship, we also introduce a binary variable \(x_{(u,v)}^{(s,k),(s,l)}\). Let \(x_{(u,v)}^{(s,k),(s,l)}=1\) stand for the virtual link \((k,l)\) in slice \(s\) mapped into substrate link \((u,v)\) if and only if VNN\(k\)\(k\) and \(l\) are mapped to the substrate nodes \(u\) and \(v\) respectively at the same time, and \(x_{(u,v)}^{(s,k),(s,l)}=0\) otherwise. According to the assumption that one VNU can be only mapped onto one substrate node, \(m_{u}^{(s,k)}\) and \(m_{v}^{(s,l)}\) are independent variables. Since both of them are binary variables, \(x_{(u,v)}^{(s,k),(s,l)}\) can be interpreted as a pair-decision variable by (3).  
\[x_{(u,v)}^{(s,k),(s,l)}=m_{u}^{(s,k)}m_{v}^{(s,l)},\forall s\in S,u,v\in N^{p}, k,l\in N^{v}. \tag{3}\]  
To avoid the quadratic constraint, such relationship in (3) can be represented by a triangle inequality [45] as:  
\[m_{u}^{(s,k)}+m_{v}^{(s,l)}-x_{(u,v)}^{(s,k),(s,l)}\leq 1,\forall s\in S,u,v\in N ^{p},k,l\in N^{v}. \tag{4}\]  
We can obtain the value of \(m_{u}^{(s,k)}\) by summing up all the possible values [39] of \(m_{v}^{(s,l)}\) in \(x_{(u,v)}^{(s,l),(s,l)}\), as well as \(m_{v}^{(s,l)}\). Since both of them are independent variables, the relationship in (4) can be expressed as:  
\[\sum_{l\in N^{v}}\sum_{v\in N^{p}}x_{(u,v)}^{(s,k),(s,l)}=m_{u}^{( s,k)},\forall s\in S,u\in N^{p},k\in N^{v}, \tag{5}\] \[\sum_{k\in N^{v}}\sum_{u\in N^{p}}x_{(u,v)}^{(s,k),(s,l)}=m_{v}^{ (s,l)},\forall s\in S,v\in N^{p},l\in N^{v}. \tag{6}\]