## 4 Feasibility Test of the API Protocol via Simulations
### Performance Analysis of the API Protocol
#### 4.2.2 Initialization Time  
First, we investigate the initialization time of the API and the FI. In Section 3, the time duration of a sub-round is fixed as \(\tau\), and thus that of a round is represented as a multiple of sub-rounds. In(i.e., the first round), the maximum number of sub-rounds for initialization is \(Q_{1}\) (i.e., the number of sub-rounds). \(Q_{1}\) is determined by multiplying the number of slave nodes by the initialization rate (i.e., the application requirement). If the application requirement is not satisfied in \(R_{1}\), \(Q_{2}\) sub-rounds will be conducted for initialization in \(R_{2}\). If this is generalized to the case that the initialization is completed in \(R_{k}\), the initialization time can be expressed as \(\sum_{i=1}^{k}Q_{i}\). Thus, the initialization time can be also expressed as a multiple of sub-rounds. This implies that a sub-round can be used as a unit to represent the initialization time metric. Figure 7 illustrates the initialization time according to the simulation experiment index in the case of the BER of \(3\times 10^{-3}\) and the initialization rates of 20%, 60%, and 100%. The initialization time is distributed around the average value, as shown in Figure 7. As the initialization rate increases, the initialization time also increases in order to initialize more slave nodes. In addition, the initialization time is linearly proportional to the initialization rate. When the initialization rate is 20%, the average initialization time takes about 45 sub-rounds. In the case of the initialization rates of 40%, 60% 80%, and 100%, the corresponding average initialization times are 90 (=\(2\times 45\)), 137 (\(\approx 3\times 45\)), 179 (\(\approx 4\times 45\)), and 225 (=\(5\times 45\)) sub-rounds, respectively. The linearity of the initialization time with respect to the initialization rate can be remarkably seen in Figure 7(b). This linearity is originated from the property that the API selects as many slave nodes as the initialization rate. Figure 7: The initialization time of the API and the FI. (**a**) The API with the initialization rate 20%. (**b**) The API with the initialization rate 60%. (**c**) The FI with the initialization rate 100%. This linearity persists until the BER deteriorates to a point where the communication between the master and slave nodes is very unreliable. Under the simulation conditions, when the BER value is \(7\times 10^{-3}\), the linearity is broken, as illustrated in Figure 8a. This result can be predicted via simple calculation. The packet error rate (PER) can be expressed as \(\text{PER}=1-(1-BER)^{L}\), where \(L\) is the number of bits of a packet. By applying the PER equation, when the BER is \(7\times 10^{-3}\) and \(L\) is 100 bits, the PER is approximately 0.5. From the simulation results, when the PER exceeds 50%, communication failure between a master node and a slave node is obvious. In this case, the initialization satisfying the application requirement cannot be realized, even if the initialization process is performed up to the maximum number of initialization repetitions. If the length of the packet is adjusted, the value of BER, which causes communication failure, will be also changed. This linearity can be considered as a characteristic of the API that adaptively initializes nodes according to the initialization rate. Using this linearity, it is possible to roughly estimate the initialization time of the API according to the initialization rate in a specific channel state (e.g., BER). Figure 9 shows the comparison of the initialization time between the API and the FI. Because the FI needs to initialize all slave nodes, its initialization time performance is unrelated to the initialization rate. Therefore, as the initialization rate increases, the initialization time of the API approaches that of the FI, which is shown in Figure 9.  
Figure 8: The initialization time of the API according to BER and the initialization rate.