## 5 System Model in Two-Hop Case  
### System Model and QoS Requirement  
It is assumed that a sensor node (\(N1\)) includes multiple sensors that produce different data types that are transmitted via a relaying node (\(N2\)) to the target hub (\(H\)) (Figure 6). Here, \(tr_{A\to B}\) is the number of transmissions from nodes A to B and \(q_{A\to B}\) is the maximum number of transmissions from nodes A to B. If bit errors are detected, the system retransmits until the maximum number of retransmissions is reached. Then, the transmission is considered to have failed if the data from a sensor node do not reach the target hub. Figure 5: Flowchart of the proposed automatic repeat request (ARQ) protocol [20; 21; 22]. Figure 6: System model. The average number of transmissions from node A to node B \(\overline{tr_{A\to B}}\) is expressed as follows:  
\[\overline{tr_{A\to B}}=\sum\limits_{i=1}^{q_{A\to B}-1}i\prod\limits_{j=1}^{i}P_{ fail,j-1}\Big{(}1-P_{fail,i}\Big{)}+q_{A\to B}\prod\limits_{j=1}^{q_{A\to B}}P_{fail,j-1} \tag{1}\]  
Here, \(P_{fail,i}\) is the probability of transmission failure in the \(i\)th transmission. In this study, \(P_{fail,i}\)[21] is the same as PER because packet collisions in the MAC layer are not considered. Hence, bit errors are taken into account due to noise and multipath fading. Then, the average number of transmissions in a two-hop case is expressed as follows:  
\[\overline{tr_{2 hop}}=\left\{ \begin{array}{l}\frac{q_{N1\to N2} \left(P_{fail,1 st}=1\right)}{\overline{tr_{N1\to N2}}+\overline{tr_{N2\to H}} \left(P_{fail,1 st}\
eq 1\right)}\\ (*\sum_{i=1}^{-1}f(i)=0,\,\sum_{i=1}^{0}f(i)=0).\end{array} \right. \tag{2}\]  
Here, \(P_{fail,1 st}\) is the probability of transmission failure at the first hop. In this study, two data (Data A and Data B) with different types of QoS requirements are considered. Here, it is assumed that a low PER is desired for Data A and high energy efficiency is important for Data B as an example [20; 21; 22]. As the first reason for selecting those QoSs, this paper particularly focused on an error controlling scheme utilizing a hybrid ARQ, and then PER and the energy efficiency are very important parameters for evaluation of such a scheme. Secondly, those two parameters are related to a trade-off. Hence, we also aimed to show the relationship in some evaluations. Data A is assumed to be a physiological parameter with a low data rate, for example blood pressure, SpO2, or temperature, and Data B to be a waveform, such as an ECG output [20; 21; 22]. The transmission order and error control process of different types of data packets depend on such QoS requirements. The characteristics of different data types are summarized in Table 1[13; 14; 22]. Then, each \(q_{A\to B}\) is set as shown in Table 2. The maximum number of retransmissions is four in high QoS mode in the IR-UWB (impulse radio ultra-wideband) PHY of the standard. However, the default mode in the IR-UWB PHY, the narrowband PHY, and the Human Body Communication PHY in IEEE Std. 802.15.6 do not define a maximum number of retransmissions. Thus, in the current study, this parameter was set according to the QoS requirements of the data in our previous work [20; 21; 22]. Figure 7 shows examples of each \(\overline{tr_{2 hop}}\). \(P_{fail,1 st,i}\) and \(P_{fail,2 nd,i}\) denote the probability of transmission failure at the \(i\)th transmission of the first and second hop, respectively. With Data A, \(\overline{tr_{2 hop}}\) increases steeply under high \(P_{fail,2 nd,i}\) conditions, especially in low \(P_{fail,1 st,i}\) cases, because \(tr_{N2\to H}\) increases towards \(q_{max}\) as \(P_{fail,2 nd,i}\) increases. On the other hand, \(\overline{tr_{2 hop}}\) increases gradually with Data B because \(q_{N1\to N2}\) and \(q_{N2\to H}\) are constant. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Data Types** & **Data A** & **Data B** \\ \hline User priority & 5 & 6 \\ PER & \(\leq 10^{-2}\) & \(\leq 10^{-1}\) \\ Energy efficiency & low & high \\ \hline \hline \end{tabular}
\end{table}