# Combined Sweeping and Jumping Method to Enhance Node Insertion Algorithm for Wi-Fi Sensor Networks
## 3 Numerical Results  
For wireless sensor networks where \(l\) = 10, 20, 30, \(\ldots\), 100, 200, 400, 600, 800, 1000, sensors are randomly located over the circular service area with a radius of \(r\), and an AP is centered; the performances of the node insertion algorithm with and without the enhancement in the combined sweeping and jumping method are compared. The transmission signal from each sensor is assumed to be propagated in the circular form. The transmission range of sensor \(i\) for \(i\) = 1, 2, \(\ldots\), \(l\) is assumed to vary as \(r+0.1\times(i\ mod\ 6)\times r\). To obtain the accurate numerical results, 50 random networks were generated for each case of the node insertion algorithm with and without the proposed combined sweeping and jumping method, and \(l^{\prime}\)s. The mean numbers of serially connected multipolling sequences for \(l\) sensors, and the mean times taken to derive the serially connected multipolling sequences, are compared for the node insertion algorithm with and without the enhancement in the proposed combined sweeping and jumping method in Table 2. Additionally, for the stress test for the node insertion algorithm with and without the proposed combined sweeping and jumping method, \(l\) sensors are let to be concentrated near the boundary of the circular service area so that the sensors are randomly located outside the distance of \(0.8r\) from the AP. By concentrating the sensors on the donut-shaped area near the boundary of the circular service area, the connectivities between the sensors located at the opposite sides are cut off. The circular and donut-shaped service areas are depicted in Figure 3. When the sensors are located over the donut-shaped area, the mean numbers of serially connected multipolling sequences, and the mean times taken to derive the serially connected multipolling sequences, are also presented for the node insertion algorithm with and without the combined sweeping and jumping method in Table 3. The simulations are conducted using a computer of 1.6 GHz CPU and 8 GB RAM. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & & & & & & \\ \cline{2-13}  & & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** & **90** & **100** & **200** & **400** & **600** & **800** & **1000** \\ \hline W/o & Avg. \# of & & & & & & & & & & & & & & & \\ Sween & Time Taken & & 1.5 & 1.5 & 1.5 & 1.2 & 1.4 & 1.3 & 1.3 & 1.2 & 1.2 & 1.3 & 1.2 & 1.1 & 1.1 & 1.1 & 1.1 \\ and Jump & Time Taken & & 0.02 & 0.1 & 0.2 & 0.4 & 0.8 & 1.3 & 1.9 & 2.2 & 2.0 & 1.6 & 7.7 & 46.8 & 153.1 & 362.8 & 709.4 \\ \hline With & Avg. \# of & & & & & & & & & & & & & & & \\ Sween & Time Taken & & 0.006 & 0.01 & 0.04 & 0.07 & 0.1 & 0.09 & 0.09 & 0.1 & 0.1 & 0.3 & 0.5 & 0.8 & 1.2 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean number of serially connected sequences and mean time taken for derivation (circular service area). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & & & & & \\ \cline{2-13}  & & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** & **90** & **100** & **200** & **400** & **600** & **800** & **1000** \\ \hline W/o & Avg. \# of & & & & & & & & & & & & & & & & \\ Sween & Time Taken & & 1.88 & 1.5 & 1.1 & 1.1 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ and Jump & Time Taken & & 0.03 & 0.09 & 0.2 & 0.4 & 0.7 & 1.3 & 1.5 & 2.3 & 2.3 & 2.9 & 6.3 & 54.8 & 190.8 & 441.2 & 708.6 \\ \hline With & Avg. \# of & & & & & & & & & & & & & & & \\ Sween & Time Taken & & 0.01 & 0.03 & 0.07 & 0.1 & 0.1 & 0.1 & 0.09 & 0.2 & 0.2 & 0.4 & 0.7 & 1.0 & 2.1 & 2.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean number of serially connected sequences and mean time taken for derivation (donut-shaped service area).