# Energy Reduction Multipath Routing Protocol for MANET Using Recoil Technique
## 4 Mathematical Support of Proposed Model for Results
### Lemma-1  
The term mobility refers to the node's average speed during the simulation. The lemma states that the routing overhead increases (in either cases proactive or reactive protocols) with mobility, which can be expressed using Equations (19) and (20).  
\[M_{1}>M_{2},\ \varnothing_{\rm P}(M_{1})>\varnothing_{\rm P}(M_{2}) \tag{19}\]  
\[M_{1}>M_{2},\ \varnothing_{R}(M_{1})>\varnothing_{R}(M_{2}) \tag{20}\]  
**Proof.** Routing overhead is the overhead bear by a protocol to keep updated information about the network. Hence, the number of routing packets is required for the transmission of single data packet. The wired network requires route discovery once and routing overhead incurred only a single time given no mobility in nodes (fixed topology). However, in ad hoc networks, scenario is entirely different as dynamic topology needs frequently route discovery. Additionally, mobility (more frequently changing the topology) of nodes gives rise to multiplier effect to the route discovery and route maintenance means higher mobility, which requires more route discovery, leading to a higher routing overhead.  
Suppose each node has probability \(p_{r}\) to retransmit the RREQ packet to its destination; let the average destination be \(N_{avg}\) and there are m hops to reach the destination. Hence, the routing overhead for the first hop is \(p_{r}\times N_{avg}\). The overall routing overhead from source to destination is denoted by \(\rm R_{e}\) and is given by Equation (21).  
\[R_{e}=1+\ p_{r}.\ N_{avg}+p_{r}^{2}.\ N_{avg}.\ N_{f}+\ p_{r}^{3}.\ N_{avg}.N_{ f}^{2}+\ \ldots+\ p_{r}^{m}.\ N_{avg}.N_{f}^{m-1}\]  
\[R_{e}=1+\ p_{r}.\ N_{avg}\sum_{i=0}^{m-1}\Bigl{(}p_{r}.\ N_{f}\Bigr{)}^{i} \tag{21}\]  
where \(N_{f}\) represent total neighbours number of any node that receive route request packet RREQ, from that node and rebroadcast the same RREQ packet to the next hop with probability \(p_{r}\). The Equation (21) is the summation of finite geometric progression, and \(\rm R_{e}\) reduces to Equation (22).  
\[R_{e}=\ \left\{\begin{array}{l}1+p_{r}.\ N_{avg}\Bigl{(}\frac{1-\bigl{(}p_{r}. \ N_{f}\bigr{)}^{m}}{1-p_{r}.\ N_{f}}\Bigr{)}\\ 1+m.p_{r}.\ N_{avg}\ if\ p_{r}.\ N_{f}=1\end{array}\right.\quad\text{for any $p_{r}$, and $N_{f}$} \tag{22}\]  
Consider the case for reactive routing (e.g., AODV), where intermediate nodes always perform the rebroadcast, hence \(\rm p_{r}=1\), and we obtain  
\[R_{e}^{ AODV}=\ \left\{\begin{array}{l}1+\ N_{avg}\Bigl{(}\frac{1-\bigl{(}\ N_{f} \bigr{)}^{m}}{1-N_{f}}\Bigr{)}\\ 1+m.N_{avg}\ if\ N_{f}=1\end{array}\right.\quad\text{for any $N_{f}$} \tag{23}\]  
When mobility of nodes increases, the frequency of route discovery also increases and the value of routing overhead in single route discovery is expressed by Equation (23). Hence it is proved routing overhead increases with mobility of nodes.  
In simulation result (Figure 13a), the slope of the tangent line would be the same as the slope of the equation of best fit line on the curve \(y=3.72x+17.2\), whose slope is 3.72, which is greater than 0. Hence, the result shows that the routing overhead function is monotonically increasing with the mobility of the node \(\square\).