## 5 Evaluation
### Power Efficiency  
The sensor networks should function for as long as possible since it may be inconvenient or impossible to recharge node batteries. Therefore, all aspects of the node, from the hardware to the protocols, must be designed to be extremely energy efficient. In this paper, power efficiency is defined as sensor nodes have different number of \"one-hop\" sink (sink to node is one-hop) when total them with controlled transmission power over maximum transmission power.  
Equation (2) gives the relationship of transmission and reception power for wireless devices. Thus, the transmission power of sink node to the sub nodes in a cluster is given by:  
\[P_{t}=P_{r}\left/\left[\left(\frac{\lambda}{4\pi d}\right)^{n}g_{t}g_{r}\right.\right] \tag{14}\]  
Assume has \(S\) sink in the network, the total transmission power is given by:  
\[P_{t}=\sum_{i=1}^{s}P_{t,i}=\sum_{t=1}^{s}P_{t,i}\left/\left[\left(\frac{ \lambda}{4\pi d}\right)^{n}g_{t}g_{r}\right]=\frac{P_{r,i}\left(4\pi\right)^{n} }{g_{t}g_{r}\lambda^{n}}\sum_{t=1}^{s}d^{n}\right. \tag{15}\]  
Figure 16 shows the comparisons of the total transmission power between MSCWSNs-PC and maximum transmission power per round. As shown in Figure 16, MSCWSNs-PC achieves an improvement of several orders of magnitude compared to maximum power transmissions and with the increase of the number of sink the gap between the two lines gradually widens. For the two modes of axis ordering orientation, more total transmission power is needed in axis mode, but the gap between the two modes is very small.  
In the well-known energy model of nodes (First Order Radio Model) in [5], the model took into account nodes' transmitting and receiving energy consumptions, compared to the actual work that node energy model more suitable for constant rate and peer-to-peer links in the mobile node.The literature assumes a simple model for the radio hardware energy dissipation where the transmitter dissipates  
Figure 16: Total transmission power at different sink numbers (n=2).  
energy to run the radio electronics and the power amplifier, and the receiver dissipates energy to run the radio electronics. For the experiments described here, both the free space (power loss) and the multipath fading (power loss) channel models were used, depending on the distance between the transmitter and receiver [28]. Power control can be used to invert this loss by appropriately setting the power amplifier - if the distance is less than a threshold, the free space (fs) model is used; otherwise, the multipath (mp) model is used. Thus, to transmit a \(k\)-bit message a distance, the radio expends:  
\[E_{Tx}(k,d)=E_{Tx-elec}(k)+E_{Tx-amp}(k,d)=\begin{cases}kE_{elec}+k\xi_{gs}d^{2},&d<d_{0}\\ kE_{elec}+k\xi_{mp}d^{4},&d\geq d_{0}\end{cases} \tag{16}\]  
and to receive this message, the radio expends:  
\[E_{Rx}(k)=E_{Rx-elec}(k)=kE_{elec} \tag{17}\]  
The electronic energy, \(E_{elec}\), depends on factors such as the digital coding, modulation, filtering, and spread of the signal, whereas the amplifier energy, \(\xi_{gs}d^{2}\) or \(\xi_{mp}d^{4}\), depends on the distance to the receiver and the acceptable bit-error rate. For the experiments described in this paper, the communication energy parameters are set as: \(E_{elec}\) = 50 nJ/bit, \(\xi_{gs}\) = 10 pJ/bit/m\({}^{2}\), and \(\xi_{mp}\) = 0.0013 pJ/ bit/m\({}^{4}\). Using the experimental results in [29], the energy for data aggregation is set as \(E_{{}_{\mathit{DL}}}\) = 5 nJ/ bit/signal.  
The first order radio model shows that we should reduce the communication of each sensor node as much as possible. In MSCWSNs-PC, without considering the following process of clustering routing, not only all sinks save power by constraining their radio range, but also all sensor nodes receive a smaller number of sink advertising messages, which greatly saves sensor node battery energy.  
Figure 17: Mean connectivity at variety node number and sink number (100 rounds).