## 4 The Proposed Method: SLP-E
### Protocol Execution Phase
#### 4.2.2 Scenario-2
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notations** & **Description** & **Notations** & **Description** \\ \hline \(d_{\phi}\) & Threshold distance & \(E_{dec}=50\) nJ/bit & Loss in transmission circuit \\ \hline \(e_{\phi}=\) & Power amplification & \(l=2048\) bits & Packet’s length \\ \(10\) P/bit/m\({}^{2}\) & in free space model & & \\ \hline \(e_{amp}\) + 0.00013 & Power amplification & \(d\) & Distance between transmitter \\ P/bit/m\({}^{4}\) & in fading model & & and receiver \\ \hline \(R\) & Radius of deployment area & \(r_{s}\) & Radio range of nodes \\ \hline \(P_{r}\) & Probability of relaying & \(H_{11}\) & Hops in phase-1 scenario-1 \\ \hline \(H_{12}\) & Hops in phase-2 scenario-1 & \(H_{13}\) & Hops in phase-3 scenario-1 \\ \hline \(H_{14}\) & Hops in phase-4 scenario-1 & \(H_{15}\) & Hops in phase-5 scenario-1 \\ \hline \(H_{1}\) & Max. no. of hops in scenario-1 & \(\theta\) & Hop length in angular walk \\ \hline \(H_{21}\) & Hops in phase-1 scenario-2 & \(H_{22}\) & Hops in phase-2 scenario-2 \\ \hline \(H_{23}\) & Hops in phase-3 scenario-2 & \(H_{24}\) & Hops in phase-4 scenario-2 \\ \hline \(H_{25}\) & Hops in phase-5 scenario-2 & \(H_{22}\) & Max. no. of hops in scenario-2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and descriptions. Figure 4: Maximum delay analytically (in hops). * _Max Delay in Phase-1_: There is a backward random walk between \(R/3\) and \(2R/3\). Let \(d\) be the distance between the source node and the sink. The BRW packets travel a distance of \(\left(\frac{2R}{3}-d\right)\) units. Therefore, the number of hops in _phase-1_ is: \[H_{11}=\frac{\frac{2R}{3}-d}{r_{s}\times p_{r}},\] (2) where \(r_{s}\) is the radio range of the sensors and \(p_{r}\) is the proportion of neighbors at equal and higher depths to all neighbors. * _Max Delay in Phase-2_: The farthest distance that may be covered in _phase-2_ is \(180^{\circ}\); therefore, we have: \[H_{12}=\frac{180^{\circ}}{\theta},\] (3) where \(\theta\) is calculated as follows: from Figure 5 and using the cosine angle rule, we have \[\theta=\cos^{-1}\left(\frac{2{d^{\prime}}^{2}-4{r_{s}}^{2}}{2{d^{\prime}}^{2} }\right),\] (4)
* In this phase, packets move through a distance to reach a sensor that is at \(R/4\) units as measured from the BS. The maximum distance traveled in this phase is \(\frac{2R}{3}-\frac{R}{4}\), i.e., \(\frac{5R}{12}\) units. Therefore, the number of hops in this phase is given by \[H_{13}=\frac{5R}{12*r_{s}\times p_{r}},\] (5)
* _Max Delay in Phase-4_: The greatest distance covered at this phase is \(90^{\circ}\). Consequently, the number of hops in this phase is provided by \[H_{14}=\frac{90^{\circ}}{\theta},\] (6)
* _Max Delay in Phase-5_: The distance traversed by the packets in this phase is \(R/4\) units; therefore: \[H_{15}=\frac{R}{4r_{s}},\] (7) Summing up, we have: \[H_{t_{1}}=\frac{\frac{2R}{3}-d}{r_{s}P_{r}}+\frac{180^{\circ}}{\theta}+\frac{ 2R}{3r_{s}}+\frac{90^{\circ}}{\theta}+\frac{R}{4r_{s}}.\] (8)

#### 6.1.2 Scenario-2

The maximum delay in all five stages of SLP-E is as follows if the distance between the source site and sink (measured in hops) is greater than the permitted threshold:

Figure 5: Theta estimation. * _Max Delay in Phase-1_: Between \(R/2\) and \(R\) is the backward random walk. Then, \(R-d\) units are offered for BRW. Hence, the maximum number of hops in BRW is given by Figure 4:0 (9)
* _Max Delay in Phase-2_: Phase-2's maximum distance traveled is \(180^{\circ}\); hence, the number of hops in this phase is: Figure 4:1 (10)
* _Max Delay in Phase-3_: Here, four hops are the minimum, and \(2R/3\) hops are the maximum. The distance traveled by packets in phase-2 is at most up to \(R\); hence, the maximum delay or hops in this phase is \(2R/3r_{s}\), in the inward direction from the network edge.