# Modeling and Analysis of New Hybrid Clustering Technique for Vehicular Ad Hoc Network
## 3 Clustering Model
### Covering Rough Set Model
#### 3.1.2 Group Implementation
The formulae of the DB index are [28]:

\[Si=\frac{1}{ni}\sum_{v\in ci}d\left(v_{i},v_{k}\right)^{2} \tag{22}\]

\[d_{ij}=d\left(C_{i},C_{j}\right) \tag{23}\]

\[R_{ij}=\frac{Si+Sj}{d_{ij}} \tag{24}\]

\[R_{i}=max\left(R_{ij}\right),\ i\
eq j,\ 0\leq j<n_{c} \tag{25}\]

\[DB=\frac{1}{n_{c}}\ \sum_{i=1}^{n_{c}}R_{i} \tag{26}\]

where:

\(n_{c}\): Number of total clusters;

\(ni\): Number of vehicles in the \(i^{\text{th}}\) cluster;

\(C_{i}\): Value of the center of the \(i^{\text{th}}\) cluster;

\(d\left(C_{i},C_{j}\right)\): Euclidean distance between two centers;

\(v_{i}\) and \(v_{k}\): Two vehicles in the \(i^{\text{th}}\) cluster. ### Dunn Index

The Dunn index (DI) is an index used to calculate the clustering similarity. The DI value is high if the vehicle clusters are well-separated. If the vehicles are in compact and well-separated clusters, the distance between any two clusters is expected to be high, and the cluster diameter is expected to be small. Clusters are compact and well-separated when the inter-cluster distance is maximized and the intra-cluster distance is minimized. A large DI value indicates compact and well-separated clusters. The formulae of the DI are [26]:

\[d\left(C_{i},C_{j}\right)=\left\{d\left(center_{i},center_{j}\right)\right\} \tag{27}\]where \(center_{i}\) and \(center_{j}\) are the centroids of two clusters;

\[diam(C_{i})=max\{d(v_{i},v_{k})\} \tag{28}\]

where \(v_{i}\) and \(v_{k}\) are two vehicles in the same cluster;

\[DI=\frac{min\big{(}d\big{(}C_{i},C_{j}\big{)}\big{)}}{diam(C_{i})} \tag{29}\]

### Cluster Speed Stability

The cluster speed is most important for explaining the cluster lifetime. When the intra-cluster vehicle speed converges, the cluster lifetime is fixed. The effectiveness of using a CRS in the proposed clustering model lies in its distribution of the vehicles based on speed variations. **Definition 5**. : _let \(v.\)speed be the vehicle speed in the \(\hat{n}^{th}\) cluster and \(v.speed=\left[v.speed_{1}\,,\,v.speed_{2}\,,\,\ldots\,,v.speed_{n}\,\,\right]\) be the set of vehicle speeds with \(n\) vehicles._

The average intra-cluster speed \(\upmu\) is defined by

\[\mu=\frac{1}{n}\,\sum_{i=1}^{n}v.speed_{i} \tag{30}\]

The standard deviation of intra-cluster speed \(\sigma\) is defined as

\[\sigma=\sqrt{\frac{1}{n}\,\sum_{i=1}^{n}\left(v.speed_{i}-\mu\right)^{2}} \tag{31}\]

## 5 Simulation and Result

### Vehicle Dataset Initialization

Simulation of Urban Mobility (SUMO) [29; 30] is one of the most popular open-source road vehicle simulators. This simulator allows users to model traffic systems that include public transportation and vehicle mobility. SUMO also includes multi-support tools that can perform tasks such as route searching, the generation of highway networks, and the importing of networks from Open Street Map. SUMO has been used extensively to tackle a variety of research projects. Moreover, datasets can be imported from SUMO and used in other simulations. The imported datasets have different timestamps according to the movement of the vehicles. Figure 8 shows the highway scenario used in this work. A snapshot of the XML dataset file is shown in \[d_{ij}=d\left(C_{i},C_{j}\right) \tag{23}\]0 The XML dataset file described the vehicle id, x-location, y-location, steering angle, vehicle speed, vehicle position, lane number, and vehicle slope. The datasets for each vehicle density were divided into four files based on the timestamp; we used the low-density dataset with 60 vehicles and the

\[d_{ij}=d\left(C_{i},C_{j}\right) \tag{23}\]1 SUMO highway scenario. high-density dataset with 120 vehicles. Table 4 shows the different scenarios used in this work. ### Simulation Environment and Parameter Setting

The simulation environment was run on a Dell PC with the following features: Intel core i7-8550u, dominant frequency 2 \(\times\) 1.99 GHz with 16 GB RAM, Windows 10 Pro, and algorithms coded in MATLAB 2018b.