# A New Hybrid Hierarchical Roadside Unit Deployment Scheme Combined with Parking Cars
## 4 Scenario Description and Basic Definition
### Optimize RSUs Deployment with Already-Existing t-RSUs  
In this section, the main problem lies in how to make the RSUs to be deployed maximize the RSU connectivity and the number of covered vehicles with the constraints of limited cost. When the cars equipped with OBUs are running on the roads, maximizing the connectivity for all of them is the most important issue. First, we divide the running cars on the links or roads into many clusters, which is defined by Equation (2):  
\[Cluster_{i}=\left[\frac{L_{i}}{R}\right] \tag{2}\]  
where \(i\) refers to the identification (ID) of different roads. \(L\) is the length of the road. R is the average communication range of all the vehicles equipped with OBUs. The notation \([\cdot]\) denotes rounding the value up to the nearest integer. \(Cluster_{i}\) means the number of clusters for a given road \(i\), as shown in Figure 5.  
Figure 5: Tram and vehicle’s trajectory and their Markov State Transition diagram. Figure 5 gives a schematic diagram of the cluster and its connectivity. By calculating the number of clusters, the internal connectivity of the cluster is further analyzed. After calculating the cluster numbers of a road, the intra- and inter-cluster connectivity are thus to be analyzed. \[Intra\_R_{i,k}=\frac{\sum_{m=n+1}^{N_{i,k}}\sum_{n=0}^{N_{i,k}-1}R_{m,n}}{C_{N_{i, k}}^{N_{i,k}}}=\frac{\sum_{m=n+1}^{N_{i,k}}\sum_{n=0}^{N_{i,k}-1}R_{m,n}}{\frac{N_{i,k} -\left(N_{i,k}-1\right)}{2}}=r-\frac{\sum_{m=n+1}^{N_{i,k}}\sum_{n=0}^{N_{i,k}- 1}d_{m,n}}{\frac{N_{i,k}-\left(N_{i,k}-1\right)}{2}} \tag{3}\]  
where \(Intra\_R_{i,k}\) means the intra-connectivity for cluster \(k\) in road \(i\); \(R_{m,n}\) is the path robustness between vehicle \(m\) and \(n\); \(d_{m,n}\) refers to the Euclidean distance between vehicle \(m\) and vehicle \(n\); \(\min(r_{m},r_{n})\) equals the minimum transmission range between vehicle \(m\) and vehicle \(n\); \(N_{i,k}\) means the total number of vehicles for cluster \(k\) in road \(i\). For Equation (3), since the transmission ranges of all the vehicles are exactly the same which has been assumed before, \(\min(r_{m},r_{n})\) equals to \(r\). It can be easily obtained from the above equation that the robustness of a single cluster ranges from 0 to \(r\). For inter-cluster connectivity robustness, the circumstance becomes more complex due to the potential presence of varying numbers of t-RSU clusters (which could range from none to many) among the clusters. Thanks to the special conditions described above, it is necessary to design an appropriate strategy for solving this problem. First, we think about the fundamental circumstance. When it comes to the two adjusted clusters, we must check if there are t-RSU clusters existing between them or connecting them thus making them be in a larger and new cluster. In that way, then take them as a whole, i.e., a new cluster. We would apply this strategy until there are no t-RSU clusters existing between them or connecting them. If there are no other t-RSU clusters, then we begin to calculate the inter-cluster connectivity robustness. The inter-cluster robustness is defined as the transmission range minus the maximum Euclidean distance projected over a \(\pi\) or \(-\pi\) degree motion angle between the vehicle from one cluster and the vehicle from the adjacent cluster. In that way, it is easy to conclude that the bigger the new cluster is, the inter-cluster robustness is probably smaller and probably negative. Furthermore, if the two clusters are both initial ones without expansion, the inter-cluster robustness value range is \([-r,r]\). The formula is given in Equation (4):  
\[Inter\_R=\min(r_{m},r_{n})-\max P_{\text{roj}}(Ed(Vel_{m},Vel_{n}))=r-\max P_{ \text{roj}}(Ed(Vel_{m},Vel_{n})) \tag{4}\]  
where \(Inter\_R\) refers to the inter-cluster robustness. \(\min(r_{m},r_{n})\) means the minimum vehicle transmission range in the two adjacent clusters, which equals \(r\). \(m\) and \(n\) denote the vehicle ID in the two adjacent clusters, respectively. \(P_{\text{roj}}\) represents the projection.