# Instantaneous Networking Service Availability for Disaster Recovery
## 5 Restoration of Networking Service Availability for Users in Multiple Areas
### Conventional Restoration Process without LI-Cloud  
Let \(PC_{A\sim A}(t)\) denote the probability of successful communication connection between X and Y, which are in A. Since any user needs to connect with another user through the global network, we then have  
\[PC_{A\sim A}(t)=P_{A}(t)\cdot P_{AG}(t). \tag{4}\]  
In the same way, we can express  
\[PC_{B\sim B}(t)=P_{B}(t)\cdot P_{BG}(t), \tag{5}\]  
and  
\[PC_{C\sim C}(t)=P_{C}(t)\cdot P_{CG}(t). \tag{6}\]  
Let \(PC_{A\sim B}(t)\) denote the probability of successful communication between X and Y in case either X is in A and Y in B, or X is in B and Y in A. We have  
\[PC_{A\sim B}(t)=P_{A}(t)\cdot P_{AG}(t)\cdot P_{B}(t)P_{BG}(t). \tag{7}\]  
To summarize the previous statement and equation, a user in A is able to communicate with B only when local servers are launched in both areas A and B, and both areas A and B are connected with the global network.  
In case either X is in A and Y is in C, or X is in C and Y is in A, we have the probability of successful communication between [PERSON] and Y as  
\[PC_{A\sim C}(t)=P_{A}(t)\cdot P_{AG}(t)\cdot P_{C}(t)P_{CG}(t). \tag{8}\]  
In case X is in A and Y is a global destination, we have the probability of successful communication between [PERSON] and Y as  
\[PC_{A\sim G}(t)=P_{A}(t)\cdot P_{AG}(t). \tag{9}\]  
In case either [PERSON] is in B and [PERSON] is in C, or X is in C and Y is in C, we have the probability of successful communication between [PERSON] and Y as  
\[PC_{B\sim C}(t)=P_{B}(t)\cdot P_{BG}(t)\cdot P_{C}(t)\cdot P_{CG}(t). \tag{10}\]  
In case [PERSON] is in B and Y is a global destination, we have the probability of successful communication between [PERSON] and [PERSON] as  
\[PC_{B\sim G}(t)=P_{B}(t)\cdot P_{BG}(t). \tag{11}\]  
In case [PERSON] is in C and [PERSON] is in C or is a global destination, we have the probability of successful communication between [PERSON] and [PERSON] as  
\[PC_{C\sim G}(t)=P_{C}(t)\cdot P_{CG}(t). \tag{12}\]  
Let CaseAA denote that one user in A attempts to communicate with one user in A, and the communication succeeds. Let \(PC_{CaseAA}\) denote the probability for (CaseAA). We have  
\[PC_{CaseAA}=F_{AA}\cdot P_{A\sim A}(t). \tag{13}\]In the same way, we can express \(PC_{CaseAB}=F_{AB}\cdot P_{A\sim B}(t)\) for Case AB, \(PC_{CaseAC}=F_{AC}\cdot P_{A\sim C}(t)\) for CaseAC, \(PC_{CaseAG}=F_{AG}\cdot P_{A\sim G}(t)\) for CaseAG.  
Let CaseBB denote that one user in B attempts to communicate with one user in B, and the communication succeeds. Let the probability of CaseBB be \(PC_{CaseBB}\). We have  
\[PC_{CaseBB}=F_{BB}\cdot P_{B\sim B}(t). \tag{14}\]  
In the same way, we can express that \(PC_{CaseBC}=F_{BC}\cdot P_{B\sim C}(t)\) for CaseBC, \(C_{CaseBG}=F_{BG}\cdot P_{B\sim G}(t)\) for CaseBG.  
Let CaseCC denote that one user in C attempts to communicate with one user in C, and the communication succeeds. Let Probability of (CaseCC) be \(P_{CaseCC}\). We have  
\[PC_{CaseCC}=F_{CC}\cdot P_{C\sim C}(t). \tag{15}\]  
In the same way, we can express that \(PC_{CaseCG}=F_{CG}\cdot P_{C\sim G}(t)\).  
Consequently, the availability probability of \(X\sim Y\) is then expressed as  
\[PC(X\sim Y) =PC_{CaseAA}+PC_{CaseAB}+PC_{CaseAC}+PC_{CaseAG}\] \[+PC_{CaseBB}+PC_{CaseBC}+PC_{CaseBG}+PC_{CaseCC}+PC_{CaseCG}.\]