## 5 Dynamic Tree (DT) Protocol
### Analysis of DT Protocol  
We analyze the DT protocol under the ideal case and discuss the problems occurring in practical applications in this section. First, we make several assumptions to simplify the analysis:  
1. The collectors in the EMMNet are distributed as a homogeneous spatial Poisson process[] in two-dimensional space. 2. All collectors and concentrators transmit at the same power level and have the same radio range \(r\). Figure 4: Structure of EMMNet with eight collectors. 3. A distance of \(d\) between any collector and its concentrator is equivalent to [d/(0.75 r)] hops. This can be explained as follows: as shown in Figure 5, two hops require at least an interval of \(r+\varepsilon\), and three hops require at least an interval of \(r+2\varepsilon\), where \(\varepsilon\) is an arbitrary small value. This can then be extended to the situation of \(2n\) hops by concatenating \(n\) 2-hop topological graphs. In the same way, \(2n+1\) hops can be extended by concatenating \(n-1\) 2-hop topological graphs and one 3-hop topological graph. We can conclude that 2n-hop or \(2n+1\)-hop requires at least an interval of \(nr\). Thus, with the fact that 2 \(n\) hops require at most an interval of \(2 nr\), it could be deduced that on average there are [d/(0.75 r)] hops between any collector and its concentrator. (4) Taking into account the short distance in one hop, we use the free space channel model for estimating the radio hardware energy dissipation:  
\[E_{TX}=E_{e_{\text{lele}}}k+e_{\beta}d^{2}k \tag{1}\]  
To receive this message, the radio expends:  
\[E_{RX}=E_{e_{\text{lele}}}k \tag{2}\]  
The electronics energy (\(E_{e_{\text{lele}}}\)) depends on such factors as digital coding, modulation, and filtering. The variable \(d\) indicates the distance to the receiver. The amplifier energy, \(\text{e}_{\text{fs}}\), depends on the distance \(d\) and the acceptable bit-error rate [21]. \(k\) is the length of the message. According to the results in Reference [22], the expectation value of the total length of all segments connecting the collectors to the concentrator, \(\text{L}_{\
u}\), is given by:  
\[E[L_{\
u}]\approx\frac{1}{2n^{3/2}} \tag{3}\]  
where \(n\) is the number of collectors. With this formula and the third assumption mentioned above, we can calculate the number of hops, \(J\), in an EMMNet as:  
\[J=\frac{E[L_{\
u}]}{r\times 3/4}=\frac{2}{3n^{3/2}r} \tag{4}\]  
The energy consumption of the communication between the collectors and the concentrator, \(C\), can be evaluated as:  
\[E[C]=J[E_{TX}+E_{RX}]=Jk[(E_{e_{\text{lele}}}+e_{\beta}r^{2})+E_{e_{\text{lele }}}] \tag{5}\]  
\(E[C]\) is minimized by the value of \(r\) that is a solution of:  
Figure 5: (a) Two hops require at least an interval of \(r+\varepsilon\). (b) Three hops require at least an interval of \(r+2\varepsilon\). \[-\frac{2E_{g_{esc}}}{r^{2}}+e_{\beta_{i}}=0 \tag{6}\]  
This equation shows how to maximize the performance of the sensor network. However, a significant gap remains between the assumptions and the application environment. Certain problems may arise in a practical application:  
1. In the conventional literature on sensor networks, protocols are designed on the assumption of reciprocity between uplink and downlink channels. However, affected by the imbalance of the RF circuitries in practice, the reciprocity between the uplink and downlink channels cannot be maintained, which will cause a performance decrease. 2. It used to be widely believed that sensor networks with smaller average hop counts perform better than those with larger average hop counts. Nevertheless, we find out that when a collector chooses a parent collector with a much smaller _DtA_ value, it may actually reduce the transmission success rate. That is because the child and parent nodes are too far from each other and the bit error rate is relatively high. 3. In the theoretical analysis, we ignore the fact that packet length affects the transmission success rate.