# An Efficient and Reliable Algorithm for Wireless Sensor Network
## 6 Data Forwarding in Extended ERASCA Protocol (EERASCA)
### Mathematical Analysis for Maintenance of Overhead  
Let assume that \"\(\mu\)\" represents link failure rate due to mobility. For the maintenance of path between the mesh members the effect of overhead is calculated which are as follows:  
* Receiver is not within the radio range of its current (x) IR or MR; (y) transfer within its neighboring IR or MR.
* The secondary core, i.e., mirror core is out of the radio range of its (x) current IR or MR Core node is not accessible to mesh members. The probability for EERASCA is calculated for the scenarios discussed above. The messaging overhead is evaluated for the maintenance of mesh. In Situation 1 (x), the overhead of messaging is \(2k\times 2\) as shown in Figure 5, where \(k\) represents hop and \(r\) represents the radio range. In this scenario, the receiver informs the current IRMR about the status. In addition, after joining a new group it informs the current IRMR and its neighboring IRMR. The probability of exiting IRMR is \(\frac{(2k-1)}{k^{2}}\times\mathbb{V}\), depends on the following derivation:  
On the edge of the group, periphery receivers are \(dk^{2}-d(k-1)^{2}=d\ (2k-1)\) and average of IR and MR is dk2 up to k-hop. The probability of exiting receiver from IRMR is \(\frac{d(2k-1)}{dk^{2}}\) and that of the periphery receiver to exit from IRMR up to total receivers are \(\frac{d(2k-1)IRMR}{R}=\frac{d(2k-1)}{dk^{2}}\). Figure 5: Density of Clusters with 2 kR Distance [4]. \(\mathbb{Y}=\) represents the probability of uncovered area of the mesh where receiver exit, which is shown as: \(\frac{size\ of\ all\ IRMR}{size\ of\ covered\ area}=\frac{\pi(kr)^{2}\times IRMR}{ area}\)  
\[\mathbb{Y}=\left\{ \begin{array}{cc}(1-\frac{\pi(kr)^{2}\times IRMR}{area}),&if \ \frac{\pi(kr)^{2}\times IRMR}{area}<1\\ 0,&otherwise\end{array} \right. \tag{1}\]  
In 1 (y), the messaging overhead is \(2k\) and it is used to inform the current and previous neighbor. It concludes that the receiver is moving between the neighborhoods of the covered area. Hence, the moving receiver probability is \(\frac{(2k-1)}{k^{2}}\times 1-\mathbb{Y}\). In 2 (x), the messaging overhead is \(2k\times 2\) and the departure probability of mirror core to its current MR and IR is \(\frac{(2k-1)}{k^{2}}\times\mathbb{Y}\). The messaging overhead is \(2k\) because mirror core informs the primary core and IRMR, about its mobility and its probability which is \(\frac{(2k-1)}{k^{2}}\times 1-\mathbb{Y}\). In C, the primary core is heading towards the edge because of its mobility and such movement of the core node is tackled by Scenarios 1 and 2. The probability of orphanage receiver, when the core node leaves the mesh is \(\frac{2}{dk^{2}}\). The messages for the new core are represented as \(\frac{1}{2}dk^{2}-1\). To update the receiver group \(2k\times 4 IRMR\) messages are used. When the receiver leaves the mesh of a group then the average number of messages are:  
\[ \begin{array}{cc}\textbf{Message}_{Receiver}=&2[4k\times\frac{(2k-1)}{k^{2}} \times\mathbb{Y}+2k\times\frac{(2k-1)}{k^{2}}\times 1-\mathbb{Y}]\\ &\text{Taking common }2k\ \frac{(2k-1)}{k^{2}}\\ \textbf{Message}_{Receiver}=2[2k\frac{(2k-1)}{k^{2}}\ (2\mathbb{Y}+1-\mathbb{Y})]\\ &\textbf{Message}_{Receiver}=\frac{4(2k-1)}{k}\ (\mathbb{Y}+1)\end{array}  \tag{2}\]  
Equation (2) shows the overall messages created by the receivers of the group. Similarly, the overhead created for the reconfiguration of the next core, which are as follows:  
\[ \begin{array}{cc}\textbf{Message}_{\text{Core}}=&\frac{2}{dk^{2}}\left[ \left(\frac{1}{2}dk^{2}-1\right)+(2k\times d_{IRMR})+4k\times\mathbb{Y}+2k \times(1-\mathbb{Y})\right]\\ \textbf{Message}_{\text{Core}}=&\frac{2}{dk^{2}}\left[\left(\frac{1}{2}dk^{2 }-1\right)+(2k\times d_{IRMR})+2k\ (2\mathbb{Y}+1-\mathbb{Y})\right]\\ \textbf{Message}_{\text{Core}}=&\frac{2}{dk^{2}}\left[\left(\frac{1}{2}dk^{2 }-1\right)+(2k\times d_{IRMR})+2k\ (\mathbb{Y}+1)\right]\end{array}