# Reliable Design for a Network of Networks with Inspiration from Brain Functional Networks
## 3 Network of Networks in Virtualized Networks
### Model of Network of Networks with Network Slicing
#### 3.2.2 Availability States of Interfaces  
We define \(v^{V_{k}}_{r,s}(t)\) to express the availability state of \(i^{V_{k}}_{r,s}\) considering the dependence among the PN and VNs. If \(i^{P}_{r,s}\) has no malfunction and \(i^{V_{k}}_{r,s}\) has no packet overflow, then \(i^{V_{k}}_{r,s}\) is regarded as available and \(\sigma^{V_{k}}_{r,s}(t)=1\) holds, otherwise \(\sigma^{V_{k}}_{r,s}(t)=0\). However, the threshold for judging the presence of packet overflow differs among the three types of PV-NoN model as explained above, and details of their definitions are described below. Type-SD  
Because type-SD allocates the divided physical resources to each VN statically, there is no interdependence among the VNs. To derive the variable \(\sigma^{V_{k}}_{r,s}(t)\) for \(i^{V_{k}}_{r,s}\), we begin by defining a variable \(R^{V_{k}}_{r,s}(t)\) that represents the presence of available capacity on \(i^{V_{k}}_{r,s}\). \(\sigma^{V_{k}}_{r,s}(t)\) for type-SD is then defined as follows using \(R^{V_{k}}_{r,s}(t)\):  
\[R^{V_{k}}_{r,s}(t) = \begin{cases}1&\text{if }\mathcal{B}^{P}_{r,s}/N_{r,s}>n^{V_{k}}_{r,s}(t),\\ 0&\text{otherwise},\end{cases}  \tag{8}\] \[\sigma^{V_{k}}_{r,s}(t) = R^{V_{k}}_{r,s}(t)\big{(}1-n^{P}_{r,s}(t)\big{)}, \tag{9}\]  
where \(N_{r,s}\) is the number of VNs that share \(i^{P}_{r,s}\). \(R^{V_{k}}_{r,s}(t)\) is defined so that it is 1 if the number of packets arriving at \(i^{V_{k}}_{r,s}\) at time \(t\) does not exceed \(\mathcal{B}^{P}_{r,s}/N_{r,s}\). Equation (9) defines the availability state of \(i^{V_{k}}_{r,s}\) considering just VN \(k\), which is in contrast to the other two types of PV-NoN model. By multiplying \(R^{V_{k}}_{r,s}(t)\) and \(\big{(}1-n^{P}_{r,s}(t)\big{)}\), \(\sigma^{V_{k}}_{r,s}(t)\) is configured to be 1 only if the interface is available on both the virtual and physical levels. Type-UD  
The resources of the P-interface are undivided in the case of type-UD, and thus there is the interdependence that a traffic increase on a VN can limit the performance of the other VNs. That is, type-UD has logical AND-like interdependence whereby the performance can be guaranteed only when _all_ VNs that are interdependent with each other are not congested, similar to the characteristics of the C-NoN model explained in Section 2. Here, in contrast to \(R^{V}_{r,s}(t)\), we define \(R^{V}_{r,s}(t)\) to express the presence of available capacity considering all VNs. Using \(R^{V}\), the availability \(\sigma^{V_{k}}_{r,s}(t)\) for type-UD is expressed as  
\[R^{V}_{r,s}(t) = \begin{cases}1&\text{if }\mathcal{B}^{P}_{r,s}>\sum_{l\in \mathcal{V}}n^{V_{l}}_{r,s}(t),\\ 0&\text{otherwise},\end{cases}  \tag{10}\] \[\sigma^{V_{k}}_{r,s}(t) = R^{V}_{r,s}(t)\big{(}1-n^{P}_{r,s}(t)\big{)}, \tag{11}\]  
where \(\mathcal{V}\) is the set of all VNs. \(R^{V}_{r,s}(t)\) is 1 if the number of packets arriving at \(i^{P}_{r,s}\) at time \(t\) does not exceed \(\mathcal{B}^{P}_{r,s}(t)\). This reflects the characteristic of type-UD that each V-interface considers the states of all interdependent V-interfaces, in contrast to type-SD. Because traffic fluctuations in one VN can influence all the other VNs, Equation (11) defines \(\sigma^{V_{k}}_{r,s}(t)\) so that \(i^{V_{k}}_{r,s}\) can be regarded as being available only if _all_ interdependent V-interfaces have room for packets. Type-DD  
With type-DD, the resources of P-interfaces are reallocated among the VNs in a dynamic and complementary way depending on the traffic condition. This property allows type-DD to possess _logical OR_-like interdependence, which is seen in the B-NoN model described in Section 2: VN \(k\) can guarantee the performance if there is room in _either_ VN \(k\) or the external VNs.