# A Lifetime Maximization Relay Selection Scheme in Wireless Body Area Networks
## 6 Performance Evaluation
### Result Analysis  
Firstly, before the lifetime performance comparison, the time complexity performance of the heuristic solution and its convergency loop condition are analyzed. From a theoretical perspective, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Body Part** & **Number of Nodes** & **Range** \\ \hline Head part & 2 & \(\begin{array}{c}x\in[-10,10]\\ y\in[46,70]\end{array}\) \\ \hline Main body part & 6 & \(\begin{array}{c}x\in[-20,20]\\ y\in[-25,45]\end{array}\) \\ \hline Left arm part & 2 & \(\begin{array}{c}x<-20,y<42\\ 0.91x+y+38.2>0\end{array}\) \\ \hline Right arm part & 2 & \(\begin{array}{c}x>20,y<42\\ 0.91x-y-38.2<0\end{array}\) \\ \hline Left leg part & 2 & \(\begin{array}{c}x\in[-20,-5]\\ y\in[-105,-24]\end{array}\) \\ \hline Right leg part & 2 & \(\begin{array}{c}x\in[5,20]\\ y\in[-105,-24]\end{array}\) \\ \hline \hline \end{tabular} The coordinator is at the origin\((0,0)\) of the coordinate graph where right and up are positive directions of \(x\)-axis and \(y\)-axis, respectively. The unit in the graph is centimeters.  
\end{table} 
Table 1: Sensor node distribution. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline
**Network Model** & \\ \hline Number of sensor nodes \(l\) & 16 \\ Number of relay nodes \(m\) & 6 \\ Packet size \(k\) & 1200 bit \\ Time slot duration & 10 ms \\ Superframes duration & 700 ms \\ \hline
**Energy Consumption Model** & \\ \hline \(E_{TXelec}\) & 16.7 nl/bit \\ \(E_{RXelec}\) & 36.1 nl/bit \\ \(E_{amp}\) & 1.97 nl/bit \\ \(\mu\) & 2 J \\ \(\Delta\) & 1 J \\ \(n(LOS)\) & 3.38 \\ \(n(NLOS)\) & 5.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values for simulation. when we use the enumeration method to solve the optimization, the time complexity is \(O(2^{n\times n})\). However, when it comes to the proposed heuristic iterative solution, the time complexity is only \(O(n\times n)\). On the other hand, from the experimental perspective in Table 3, which illustrates the average time cost to solve the optimization in the simulation with the varying number of nodes in the network, the enumeration costs much more than the proposed rapid solution. In more detail, the time taken by enumeration grows very fast with the increasing number of nodes, even more than an hour when the number of nodes is 20 in our simulation. While the proposed solution grows much more smoothly within a hundred microseconds. In a word, the simulation results in Table 3 match the theoretical analysis on time complexity of enumeration and the proposed solution. Figure 2 shows the convergency loop performance of the proposed rapid solution when the iteration ended in the simulation. It can be seen from the figure that the proposed solution always converges to a loop no matter how many nodes are involved (if the iteration can not converge to a loop, then the loop size will be represented by \(\infty\)), which means that the iteration of the proposed solution can always converge to an end in the simulation. The average loop size is close to 2 which indicates that the majority of the loop size in the simulation is 2. Furthermore, it can be seen that when the number of nodes grows, the maximum and average loop size increase due to the fact that the dimensions of \(\mathbf{X}\) become larger, which brings more binary value combinations for the relay selection matrix. Next, we focus our attention on the performance comparison of network lifetime between our proposed scheme with some existing relay selection schemes: benchmark strategy, where each node in the network only uses direct transmission, the algorithm in [13] (sum-rate algorithm), which is a popular relay selection scheme in WSNs aiming to minimize the total energy consumption, and the WBAN-specific relay selection algorithm in [24] (maxi-rate algorithm), which formulates an optimization problem to minimize the maximum energy consumption rate among sensor nodes.