# A Novel Cloud Approach for Connected Vehicles
## 4 Analytic Model
### Message Dissemination
#### 4.2.4 Dissemination Model  
For each configuration \(X\), we are looking for \(H_{X}\), which is the maximum number of hops needed to cover the segment, and \(M_{X}\) which is the minimum number of hops. \(HM_{X}\) describes the average number of needed hops. We obtain for each configuration the values \(H\), \(M\), and \(HM\) as detailed in Table 7.  
We want to compute \(N_{1}\),\(N_{2}\),...,\(N_{n}\), with \(N_{1}\) being the number of segments without any RSU which are bound to at least one segment with one RSU. Additionally, \(N_{2}\) is the number of segments without an RSU that are bound to at least one segment \(\in N_{1}\), without being bound to a segment \(\in N_{0}\), etc. To get the number of segments at level \(n\), we separate this calculation depending on the number of bound roads, and at the end we sum them. A segment can be bound from \(x=3\) to 6 segments and \(N_{n_{x}}\) is the probability for a segment that is bound to \(x\) roads to be also bound to at least one segment of level \(n-1\). We deduce the following expression:  
\[N_{n}=\sum_{x=3}^{6}O_{x}N_{n_{x}}=1 \tag{16}\]  
and \(N_{n_{x}}\) is expressed as follow:  
\[N_{n_{x}}=\sum_{y=1}^{x}C_{x}^{y}\left(\frac{A*B}{C}\right) \tag{17}\]  
With: \(A=\left(\prod_{w=0}^{y-1}N_{n-1}-w\right)\),  
\(B=\left(\prod_{m=0}^{x-y-1}N_{S}-N_{RSU}-\sum_{k=0}^{n-1}N_{k}-m\right)\) and  
\(C=\prod_{z=1}^{x}N_{S}-N_{RSU}-z-\sum_{k=0}^{n-2}N_{k}\)  
The variable \(x\) refers to the number of roads bound to a segment, this is given by the global formula (16) and, then, takes a value between 3 and 6, and \(y\) refers to the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(H_{X}\) & \(M_{X}\) & \(HM_{X}\) \\ \hline Config. 1 & \(\left[\frac{L}{\prod_{w=0}^{y-1}}\right]\) & \(\left[\frac{L}{\prod_{w=0}^{y-1}}\right]\) & \(\left(\frac{H_{1}-M_{1}}{2}\right)\) \\ Config. 2 & \(\left[\frac{L-P_{RSU}}{\prod_{w=0}^{y-1}}\right]\) & \(\left[\
umber of these roads which are from level \(n-1\) and takes a value between \(1\) and \(x\). Then, \(N_{n_{k}}\) (17) is computed by summing the probability of having a segment linked to a road of level \(n-1\) with the one of being bound with two roads of level \(n-1\) and three roads of level \(n-1\), etc., until \(x\) roads. The computing results of the probability of having \(y\) roads of level \(n-1\) has to be multiplied by the number of combinations of \(y\) among \(x\), because the road or the road of level \(n-1\) may have different locations. For example, if a segment from \(\mathcal{O}_{5}\), which is bound to five roads, is adjacent to one road of level \(n-1\), this last one could take five different locations. The formula \(\prod_{n=0}^{y-1}N_{n-1}-w\), expresses the probability of being bound to w segment of level \(n-1\), to which we have to subtract, at each stage, the segments already processed, that is why \(w\) is used. Then, we obtain the following result : \((N_{n-1}-0)*N_{n-1}-1)*...*(N_{n-1}-w)\). We multiply the result of this expression with the probability for the other segments of not being at a lower level than n. This is expressed as follows: \(\prod_{n=0}^{x-y-1}N_{S}-N_{RSU}-\sum_{k=0}^{n-1}N_{k}-m\). \(N_{s}-N_{U}BR-\sum_{k=0}^{n-1}N_{k}\), this allows expressing the number of segments of a level higher than \(n\), which means the total number of segments in the network to which we subtract the segments having two RSUs and the segments of a level lower than \(n\). We also have to subtract the segments of a level higher than \(n\) that we already processed in the computing, and they are represented by \(m\). The variable may take a value between \(0\) and \(x-y-1\), and if we would develop, we would obtain an expression with the following shape: \((N_{s}-N_{RSU}\sum_{k=0}^{n-1}N_{k})*(N_{s}-N_{RSU}-\sum_{k=0}^{n-1}N_{k}-m)*...*(N_{s}-N_{RSU}-\sum_{k=0}^{n-1}N_{k}-m)\). Then, we divide it by the number of segments of a level higher than \(n-1\).