## 7 Case Study
### Comparison of Uniform and Non-Uniform Deployment  
Tables 13 and 13 also show that the optimal sensor deployment schemes are the same in uniform and non-uniform distributions, since the optimal sensor distance is the one that can consume the least energy with one sensor located in the center of each grid. In addition, under the optimal sensor deployment, \(d=58.97\) m and \(L=9\), the total number of sensors required to survive for 365 days in non-uniform deployment is 3006, whereas the number is 3510 in uniform deployment. Moreover, when the numbers of sensors deployed are the same (\(N=3510\)), the lifetime of WSN is 365 days if \begin{table}
\begin{tabular}{c c c} \hline \hline
**Optimal Solutions** & **With Retransmission** & **Without Retransmission** \\ \hline \(d\) & 58.97 m & 50.00 m \\ \(L\) & 9 & 11 \\ \(S\) & 0.99991 & 0.61510 \\ \(\tau\) & 835.06 h & 822.32 h \\ \(N\) & 3006 & 4392 \\ \hline \hline \end{tabular} Note: the optimal lifetime is obtained by the optimization model in Equation (13) under the constraint of \(N^{*}=500\), and the minimal \(N\) is obtained using Equation (13).  
\end{table} 
Table 4: Optimization results of non-uniform deployment. The details of the symbols can be seen in the Appendix. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Optimal Solutions** & **With Retransmission** & **Without Retransmission** \\ \hline \(d\) & 58.97 m & 50.00 m \\ \(L\) & 9 & 11 \\ \(S\) & 0.99991 & 0.61510 \\ \(\tau\) & 715.80 h & 725.44 h \\ \(N\) & 3510 & 5148 \\ \hline \hline \end{tabular} Note: the optimal lifetime is obtained by deploying only one sensor in each regular hexagonal grid using Equation (13), and the minimal \(N\) is obtained using Equation (13).  
\end{table} 
Table 3: Optimization results of uniform deployment. The details of the symbols can be seen in the Appendix. sensors are uniformly deployed, whereas the maximum lifetime can extend to 418 days under the optimal non-uniform deployment via Equation (3.1). All of the above results imply that the non-uniform sensor deployment scheme provides longer lifetime than that of the uniform one. Particularly, the constraints of the maximum number of sensors, the required lifetime and the radius of the circular target area will affect the optimization results. When the constraint \(N^{*}\) (the maximum number of sensors) varies, the optimal lifetime of WSN is illustrated in Figure 3. One can see that the non-uniform deployment is useful in prolonging the lifetime of WSN. As the numbers of packets transmitted and received at some locations are the same (see Figure 3.1), the energy consumptions for sensors in these locations are the same. Thus, one can see from Figure 3.1 that the optimal lifetime for a certain number of sensors does not change continuously. To satisfy a certain required lifetime (\(\tau^{*}\)) varying from 100 to 500 days, the optimal number of sensors deployed in the WSN is shown in Figure 3.1, and the associated energy left when the life ends is illustrated in Figure 3.1. One can see that more sensors are needed to satisfy the long-lifetime requirement. Compared with the uniform deployment, the saving in sensors by the non-uniform WSN is increasingly evident at longer required lifetime. In Figure 3.1, the residual energy for the non-uniform deployment is much less than the uniform one. As the optimal lifetime for a certain number of sensors does not change continuously (see Figure 3.1), the residual energy is not always monotonically increasing in the required lifetime. Figure 5: Optimal lifetime under different \(N^{*}\). Similarly, when the radius of the circular target area \(R_{a}\) varies, the optimal number of sensors deployed (compare the uniform case with the non-uniform case) in the WSN is shown in Figure 10 (for a lifetime of 365 days).