## 6 Discussion  
The reconfiguration of RRP is achieved by dynamically exchanging message among the BS, the DN, ANs and MNs. This section discusses the impact of such a message exchange on the energy consumption during data transmission. This impact is shown by comparing the energy consumption of message exchange with that of data transmission. Figure 15: Number of living nodes _vs._ scanning angle. Suppose the average hop from a node to its nearest CH or UCH is _hopup_, the size of the data packet is _Sizedatapacket_, then transmitting a data packet to the BS requires energy as follows:  
\[E_{data\_transmission}=(E_{{}_{AX}}+E_{{}_{RX}})\times hop_{up}\times\textit{ Size}_{data\_packet}+(E_{{}_{RX}}\times\textit{Size}_{data\_packet}+\alpha) \tag{15}\]  
where \(\alpha\) is a compensation factor. If the last node in the uplink path to the BS is an UCH, then the \(\alpha\) is a positive number, and the (\(E_{{}_{RX}}\times\textit{Size}_{data\_packet}+\alpha\)) equals to the energy required for the UCH transmitting the data packet to the BS with the active transmitter. However, \(\alpha\) equals to 0 if the last node is a real CH which communicates with the BS using passive transmitter. During data transmission, when a node's remaining energy reaches a threshold, it sends an SOS message to the BS. Let _SizeSOSmessage_ be the size of SOS message, then the energy consumption for the SOS message to get to the BS, denoted by _ESOS_, is as follows:  
\[E_{{}_{SOS}}=(E_{{}_{AX}}+E_{{}_{RX}})\times hop_{up}\times\textit{Size}_{ SOS\_message}+(E_{{}_{RX}}\times\textit{Size}_{SOS\_message}+\alpha) \tag{16}\]  
After receiving the SOS message, the BS sends out reconfiguration messages directly to MNs, ANs, and the DN, taking advantage of their location information. Let \(num_{{}_{MN}}\) be the number of MNs, \(num_{{}_{MN}}\) be the number of ANs, and \(\textit{Size}_{{}_{MN}}\), \(\textit{Size}_{{}_{DN}}\) be the sizes of reconfiguration messages for ANs, MNs, and the DN respectively, then the energy consumption for nodes receiving the these messages, denoted by \(E_{{}_{Reconf}}\), is:  
\[E_{{}_{Reconf}}=E_{{}_{RX}}\times\textit{Size}_{{}_{AN}}\times num_{{}_{AN}}+E _{{}_{RX}}\times\textit{Size}_{{}_{MN}}\times num_{{}_{MN}}+E_{{}_{RX}}\times \textit{Size}_{{}_{DN}}\times 1 \tag{17}\]  
The total energy needed for message exchange with respect to one reconfiguration is:  
\[E_{{}_{message\_exchange}}=E_{{}_{SOS}}+E_{{}_{Reconf}} \tag{18}\]  
Actually, the size of SOS message and reconfiguration messages are relatively small compared with the data packet. In our simulation, the SOS message is only 20 bits, including 4 bits for message type, and 16 bits for the DN's ID. The reconfiguration messages for DN, ANs, MNs are 20 bits, 132 bits and 36 bits respectively. The largest \(num_{{}_{AN}}\) and \(num_{{}_{MN}}\) in our simulation sets are 5 and 12 respectively. So if substituting these numbers for corresponding parameters in Equations (16-18), after simplification, we have:  
\[E_{{}_{message\_exchange}}=20\times E_{{}_{AX}}\times hop_{up}+(20\times hop_{up }+1112)\times E_{{}_{RX}}+20\times E_{{}_{RX}}+\alpha \tag{19}\]  
Since the data packet is 2,048 bits, the energy consumption for delivering a data packet is:  
\[E_{{}_{data\_transmission}}=2048\times E_{{}_{AX}}\times hop_{up}+2048\times hop_{up }\times E_{{}_{RX}}+2048\times E_{{}_{RX}}+\alpha \tag{20}\]  
According to Equations (19) and (20), it is intuitive to observe that \(E_{{}_{message\_exchange}}\) is much smaller than \(E_{{}_{data\_transmission}}\) as long as \(hop_{up}\) is a positive number. And we know that this is always the case because the average number of hops never becomes negative. Besides, since each node is able to initiate a reconfiguration only once, the total number of reconfiguration initiation at most equals to the number of nodes in the network. Comparing to the tens of thousands of data packet transmission, the energy spent on message exchange for reconfiguration actually falls into acceptable range.