# Joint Deployment Optimization of Parallelized SFCs and BVNFs in Multi-Access Edge Computing
## 3 System Model and Problem Statement
### Problem Statement
We record the comparison set that satisfies the delay requirement and the reliability set that consists of all edge servers which deploy parallel \(VNF^{l}\) instances used to enhance reliability. The next step is to find the target edge server to deploy parallel VNF instances;
2. Then, we define the feasible edge server with the highest capacity as the selected edge server. Next, we calculate the end-to-end delay when replacing one edge server in the comparison set one by one with the selected edge server. If the end-to-end delay satisfies the delay requirement, we replace other edge servers in the comparison set one by one with the edge server in the reliability set. Then, if the end-to-end delay  
Figure 4: Explanation of the delay-reduction scheme in our algorithm. If the delay exceeds the delay requirement, this algorithm reduces the delay by deploying parallel VNF instances in lines 16-23. It calculates the delay-reduction priority of all VNFs in SFC and executes the delay-reduction scheme to find critical VNFs and target servers and determine the split and merge of user requests. This procedure continues until the delay requirement is satisfied. However, if the delay requirement is not satisfied when the number of parallel instances for all \(VNF^{n}\) in SFC equals the maximum number of sub-flows or no edge server resources are available to deploy VNF instances, it terminates the loop, drops user request \(r_{i}\), and releases the resource it preempts. Otherwise, it deploys parallel VNFs to satisfy the reliability requirement. Algorithm 1 calculates the reliability-enhancement priority for each VNF in SFC and then executes the reliability-enhancement scheme. This procedure continues until the reliability requirement is satisfied. However, if the reliability requirement is not satisfied when the number of parallel VNF instances for all \(VNF^{n}\) in SFC equals the maximum number of sub-flows or no edge server resources are available to deploy VNF instances, the next step is to deploy BVNFs for user request \(r_{i}\) on lines 27-36. Figure 6: Explanation of the reliability-enhancement scheme in our algorithm. ```
1:Input: \(G,R\);
2:Output: \(A,P\);
3:\(A=0\);
4:FOR all \(V_{i},\ V_{i}\in V\) DO
5: Calculate P[\(V_{i}\)][\(V_{j}\)] by Dijkstra algorithm;
6:END FOR
7:FOR all \(U_{i}\in U\), \(BS_{j}\in BS\) DO
8: Calculate D[\(U_{i}\)][\(BS_{j}\)] and find the nearest base station for \(U_{i}\);
9:END FOR
10:FOR all \(r_{i}\in R\) DO
11:FOR all \(VNF^{l}\in SFC_{i}\) DO
12:Find the edge server \(V_{k}\) with the lowest transmission distance;
13:IF \(C_{k}>C_{f_{VNF^{l}}}\) THEN
14:Deplog one \(VNF^{l}\) instance in \(V_{k}\);
15:Calculate the delay and reliability of \(r_{i}\);
16:END IF
17:END FOR
18:WHILE the delay of \(r_{i}>D_{i}\) DO
19:Execute the delay-reduction scheme;
20:IF all \(\left|VNF^{l}\right|>=n_{i}\) THEN
21:Drop \(r_{i}\);
22:Release all resources of \(r_{i}\);
23:BREAK;
24:END IF
25:END WHILE
26:IF the delay of \(r_{i}<=D_{i}\) DO
27:WHILE the reliability of \(r_{i}<R_{i}\) DO
28:Execute the reliability-enhancement scheme;
29:IF all \(\left|VNF^{l}\right|>=n_{i}\) THEN
30:Find \(VNF^{l}\) with the highest priority;
31:Find server \(V_{k}\) with the lowest communication distance;
32:IF \(C_{k}>C_{f_{VNF^{l}}}\) THEN
33:Deplog one backup instance in \(V_{k}\);
34:Calculate the delay and reliability of \(r_{i}\);
35:ELSE
36:BREAK;
37:END IF
38:END IF
39:IF the delay of \(r_{i}<=D_{i}\) && \(r_{i}>=R_{i}\) DO
40:\(A=A+1\);
41:END IF
42:END FOR
43:Return A, P;
```  
**Algorithm 1** Throughput Maximization Algorithm  
This algorithm also calculates the reliability-enhancement priority for each VNF in SFC and selects the VNF with the highest priority, and selects \(VNF^{l}\) with high priority as the critical VNF. Then, it finds the nearest edge server that deploys an active \(VNF^{l}\) instance and a sub-flow that traverses the edge server.