# Policy-Based Composition and Embedding of Extended Virtual Networks and SFCs for IIoT
## 5 Results
### Evaluation of the Chain Embedding Using a Random Topology  
Our chain embedding algorithm is a form of greedy heuristics, since it adapts to the remaining path length and next path resources when making the placement decision. When the path is long, the algorithm tries to spread the VNFs over the path. When the path is short, the algorithm tries to place the VNFs on the least possible number of servers. We compare it to an existing similar solution to preliminarily show its feasibility. Our VNF consolidation method, the random topology, and the distribution of resources are chosen to represent hierarchical edge computing where the resources might be limited and paths are short. Comparison to the results of other works might be challenging, since there are differences in the objectives and topologies. Figure 6: Generated EVN for AR11. Figure 7: Embedding AR11 in the substrate network. The evaluation structure in Alevin enables the comparison of different algorithms while using random topologies and several metrics. This structure also enables the definition of the number of runs and the assignment of random resource/demand values from specified ranges. The created SN and VNs, their demands and resources, mapping results, and resulting metrics' values are exported to an XML file that includes the results for all runs. In the SFC embedding domain, two graph models are used for generating random networks. The Waxman model [22] used in [5], creates highly randomized networks by placing in each step two nodes on a two-dimensional plane and connecting them with a certain probability. This probability is calculated from their distance and two model parameters \(\alpha\) and \(\beta\), where the large \(\beta\) increases the edge density and small \(\alpha\) increases the probability of shorter edges. The Barabasi Albert model [23] used in [13] creates random scale-free networks by taking preferential attachment into account. With preferential attachment, the degree distribution follows a power law and the probability of connecting two nodes is based on individual nodes degrees. Initially, a certain number of nodes (\(m_{0}\)), time steps, and edges to be added per time step are defined. Based on the defined number of time steps, a new node with \(m\leq m_{0}\) edges is added to the network in each step. A new node is connected to an already existing node i that has a degree \(k_{i}\) with a probability:  
\[P(i)=\frac{k_{i}}{\sum_{|k_{j}}} \tag{11}\]  
Figure 8 is an exemplary generated topology with the parameters:  
* Initial number of nodes: \(m_{0}=1\)
* Number of time steps: 20
* Number of new edges per time step: 1  
Alevin uses directed graphs and the generator adds the reversed edge for each created edge. The Barabasi Albert model is better for representing industrial network levels (as in our use case). We use it for generating the SN for evaluating the chain embedding algorithm. Regarding VNs, each VN has two VNos and a VLi that has a VNF demand, in which we vary the chain length. We adapted our EVN embedding algorithm for this evaluation by just performing node mapping for VNos based on ID demand and using the k-shortest path algorithm to select the path. Subsequently, the chain embedding heuristic is used to map the FG without the locations, domains, and types of nodes. Creating random SN that includes such properties to evaluate the whole system and choosing algorithms to compare with is a challenging future work. Our system runs in an offline mode, where the SN and EVNs are defined in advance. Each test run is repeated 30 times and the mean values with confidence intervals (0.95%) are graphically depicted. We use two typical metrics from VNE; the runtime and acceptance ratio. The runtime is the time needed by the algorithm to embed a number of EVNs on a SN. The acceptance ratio is the ratio of  
Figure 8: Exemplary random topology generated by Barabasi Albert network generator. successfully embedded EVNs to the total number of EVNs in the scenario.