# Policy-Based Composition and Embedding of Extended Virtual Networks and SFCs for IIoT

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Problem Description

#### 3.1. Use Case

#### 3.2. System Model

- A physical topology $SN({N}_{SN},{E}_{SN})$ is defined by the service provider.
- The application requirements $AR({N}_{AR},{E}_{AR})$ are defined by the user.
- A set of policies (defined by the service provider), where each policy is composed of:
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- A pattern P: conditions on the $SN$ and $EVN$ for checking and comparison of properties, demands, and resources.
- –
- Transformation T: if P match is found, then perform $T(EV{N}_{i-1},SN)=EV{N}_{i}$, where $EV{N}_{0}=AR$, and T is a set of rules.
- –
- A rule R adds or copies virtual links or nodes, adds or changes the properties or demands, or adds VNFs to the VNF demand of a VLi.

- The policies are applied with the predefined order on the input AR and SN.
- EVN mapping:
- –
- For each VLi in the final EVN, if the VLi has a VNF demand, perform topological sorting on the VNF demand $TS(vnfDemand,vnfDependencies)=List\phantom{\rule{0.277778em}{0ex}}of\phantom{\rule{0.277778em}{0ex}}FG$, where the output is a set of FGs.
- –
- Map EVN nodes on the SN based on their properties and demands.
- –
- Map EVN simple VLis based on their properties and demands.
- –
- For each VLi with a VNF demand, map a selected FG while using the chain embedding proactive heuristic.

Algorithm 1: System Model |

#### 3.3. General Dependency Graph

## 4. Methodology

#### 4.1. Transformations

#### 4.1.1. Redundancy

#### 4.1.2. Nodes Types

#### 4.1.3. Security VNFs

#### 4.1.4. Low Latency VNF

#### 4.1.5. Bandwidth

#### 4.1.6. CPU and Load Balancing

#### 4.2. Creating FGs Using Topological Sorting

Algorithm 2: Topological sorting |

Algorithm 3: Finding all topological sortings |

#### 4.3. EVN Embedding

#### 4.4. Chain Embedding Heuristic

Algorithm 4: Proactive chain embedding algorithm |

#### 4.5. Complexity of the Proposed Methods

- Neglecting the operations applied by the rule on only a small subset of the EVN entities.
- Considering that the maximum size of the final EVN (nodes M, links L) is known and used to estimate the complexity even for intermediate EVNs.
- (D, D) is the size of the dependency graph. We consider that, in NFV, a function typically has one dependency, so the number of edges in the dependency graph can be approximated to the number of VNFs. The size of the dependency graph is considered the size of each VNF demand.
- We consider that finding one topological sorting of the VNF demand (one FG) is enough, since we do not yet classify the resulting FGs or trying to select the best FG based on a specific criteria.
- P is the diameter of the SN graph (the number of edges in the shortest path between the most distant vertices). We use this worst-case value as the length of the shortest path on which the chain will be embedded.
- We are calculating the complexities in the worst cases without considering locations and domains that might reduce the search space for node and link mapping, in particular, in specific topologies, like in our use case.

- Transformations:
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- Redundancy—O(L): checking the redundancy demand of all EVN links and copying them when needed.
- –
- Node types—O(N.M): comparing certain properties of SNos and VNos and adding the required entities to the EVN with their demands.
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- Security—O(L): comparing the source and destination locations/domains of the EVN links and adding the required VNFs.
- –
- Low latency—O(L): checking the latency demand of the EVN links and adding the TAS VNF where needed.
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- Bandwidth—O(L): checking the bandwidth demand of each VLi and adjusting it based on the bandwidth demands of the incoming edges to its source VNo.
- –
- CPU—O(M): adjusting the CPU demand of all EVN nodes based on the incoming bandwidth, cloning certain nodes, and then adding load balancing function.
- –
- Total complexity of the transformations: O(4L+M (1+N)) = O(L + M.N) when considering that $N\gg 1$.

- Topological sorting—O(L.D) = O(L): the complexity of finding one topological sorting is O(D+D) = O(D). We assume here that D is small and fixed and that we need to calculate a FG for each VLi.
- Node mapping—O (N.M): finding the SNo with the matching ID, properties, or resources of each VNo.
- Chain embedding—O (2P)=O(P): when mapping a VNF is tried on a SNo, the demands are compared to the resources. A VNF can be mapped on the latest SNo used from the path or any other following node. In the worst case, all path SNos are checked twice for each placed VNF and next VNF that cannot be placed on the same SNo, so a following one is checked. The P value varies with the SN size.
- Link mapping: the complexity of the k-shortest path algorithm is O(E+N.log N)) [21]. This algorithm is used for mapping all VLis. We assume for the worst case that all VLis have VNF demands for which chain embedding will be performed. The total complexity of link mapping is O(L.(E+N.log N+P)).

## 5. Results

#### 5.1. Implementation

#### 5.2. Applying the Methods to the Use Case

#### 5.3. Evaluation of the Chain Embedding Using a Random Topology

- Initial number of nodes: ${m}_{0}$ = 1
- Number of time steps: 20
- Number of new edges per time step: 1

#### 5.3.1. Runtime

#### Runtime for Increasing SN Size

#### Runtime for Increased Number of EVNs

#### 5.3.2. Comparison to LightChain

#### Acceptance Ratio with Increasing FG Length

#### Average Path Utilization with Increasing FG Length

#### Acceptance Ratio for Increasing CPU Capacity

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Mandarawi, W.; Rottmeier, J.; Rezaeighale, M.; de Meer, H.
Policy-Based Composition and Embedding of Extended Virtual Networks and SFCs for IIoT. *Algorithms* **2020**, *13*, 240.
https://doi.org/10.3390/a13090240

**AMA Style**

Mandarawi W, Rottmeier J, Rezaeighale M, de Meer H.
Policy-Based Composition and Embedding of Extended Virtual Networks and SFCs for IIoT. *Algorithms*. 2020; 13(9):240.
https://doi.org/10.3390/a13090240

**Chicago/Turabian Style**

Mandarawi, Waseem, Jürgen Rottmeier, Milad Rezaeighale, and Hermann de Meer.
2020. "Policy-Based Composition and Embedding of Extended Virtual Networks and SFCs for IIoT" *Algorithms* 13, no. 9: 240.
https://doi.org/10.3390/a13090240