#### 6.1. Performance Evaluation by Computer Simulation

In this section, the proposed and standard schemes with two-hop extension are evaluated based on communication distance by computer simulations. The computer simulator was built by us with MATLAB. The main simulation parameters are listed in

Table 3 and refer to our previous work [

20,

21,

22].

Table 4 shows the preset parameters of Weldon’s ARQ protocol at the

$i$th transmission for Scheme 1 [

20,

21,

22]. The computer simulation assumes that there is no error in SHR and PHR. That is, only the characteristics of PSDU are evaluated. In computer simulations of the compared schemes, Data A was transmitted using the default mode with (63, 51) BCH code in IEEE Std. 802.15.6 and the error control scheme utilizing the (63, 55) Reed-Solomon code in IEEE Std. 802.15.4a with ordinary ARQ, whereas Data B was transmitted using the high QoS mode with (126, 63) shortened BCH code and type-II hybrid ARQ, and then the error control scheme utilizing the concatenated code consisting of the (63, 55) Reed–Solomon code and the convolutional code whose constraint length is three and coding rate is 1/2 in IEEE Std. 802.15.4a with ordinary ARQ [

17,

19]. In these computer simulations, the IEEE model CM 3 is applied as a channel model, which is targeted for wearable WBAN and includes multi-path fading [

33]. Then, a hospital room case in the IEEE model CM3 is utilized as a path loss model [

33]. The path loss is expressed as follows:

Here,

$a$ and

$b$ are linear fitting coefficients,

$d$ is the communication distance (millimeter, mm) between a transmitter and a receiver, and

$N$ is a normally distributed variable with zero mean and standard deviation

${\sigma}_{N}$. Details about these parameters can be found in the literature [

33]. Using

$PL\left(d\right)$, the signal to noise ratio (SNR) at a receiver can be expressed as follows:

where

${P}_{t}$ is transmission power and

${N}_{thermal}$ is thermal noise. The average path loss is shown in

Figure 10. It is assumed that the channel condition does not change until the two-hop relay is completed or the two-hop relay fails beyond the maximum number of retransmissions.

In addition, each case of the proposed scheme in each hop is summarized in

Table 5.

Then, energy efficiency

$\eta $ is derived from our previous work [

22] as follows:

Here,

${E}_{link,A\to B}$ is the energy consumption of the communication link at each hop and

${P}_{succ}$ is the transmission success ratio,

${T}_{TOT}$ is the total duration of packet transmission,

${T}_{ACK}$ is the duration of ACK,

${L}_{PSDU,i}$ is the length of PSDU,

${N}_{tx}$ is the number of transmission,

${P}_{tx,RF}$ is the transmitter RF power consumption,

${P}_{tx,circ}$ is the transmitter circuitry power consumption,

${P}_{rx}$ is the receiver power consumption, and

${\epsilon}_{enc}$ and

${\epsilon}_{dec}$ are the encoding and decoding energies, respectively [

34,

35,

36,

37].

Figure 11,

Figure 12 and

Figure 13 show the performance results when the distance of the first hop

${d}_{1st}$ is changed from 10 centimeters (cm) to 3 m and the distance of the second hop

${d}_{2nd}$ is constant (

${d}_{2nd}$ = 40 cm). In this scenario, it can be said that the performance in the range in which the WBAN mainly operates (10 cm~1.5 m) and certain limitations of the WBAN system (1.5 m~2.3 m) are evaluated. PDFR means the ratio at which the two-hop relay failed beyond the maximum number of retransmissions. As can be seen, the proposed scheme satisfies the QoS requirements for data A and B as shown in

Table 1, while IEEE Std. 802.15.6 and 15.4a do not. Hence, the proposed method can improve PER of Data A more, while it can improve the energy efficiency and the number of transmissions of Data B more. Conversely, Data B has better performances with respect to both standard schemes. The reason is that those standard schemes are not basically designed so that any QoSs can be satisfied. Hence, it can be considered that the performances of each mode of IEEE Std. 802.15.6 and error control schemes of IEEE Std. 802.15.4a were simply expressed. Also, that is one of problems of these standard schemes. Cases 2 and 3 show better energy efficiency and average number of transmissions than Case 1, because the coding rate of Case 2 and Case 3 is set appropriately for the channel SNR and the number of retransmissions is reduced by utilizing Scheme 2, while Case 1 uses only Scheme 1 and it requires a larger number of retransmissions. However, there is not a large difference between Cases 2 and 3 because

${d}_{2nd}$ is short and the error correcting capability of coding rate

${r}_{c}$ = 8/9 at the first transmission can reduce bit errors sufficiently. That is, there is no large difference between Schemes 1 and 2 with respect to the second hop.

Figure 14,

Figure 15 and

Figure 16 show the performance results for fixed communication distance in two hops

${d}_{2hops}={d}_{1st}$+

${d}_{2nd}$ (1.5 m) and varying the

${d}_{1st}$ and

${d}_{2nd}$ values. For

${d}_{1st}$ = 1.5 m, data are transmitted using only a single hop. Thus, the proposed scheme satisfies the QoS requirements for Data A and B, while both standard schemes approach do not, like in the first scenario. Also, when comparing the standard schemes and the proposed scheme, the performances of both standards are worse than the proposed one. For example, Data A of the proposed scheme satisfies PDFR <

$\text{}{10}^{-2}$, while that of both standards do not satisfy PDFR <

${10}^{-1}$. This is because the correcting capability of error correcting codes used in those standards is lower than that of the proposed scheme. In other words, the standard schemes do not have sufficient correcting capability in a hop with poor channel conditions. Comparing Case 1 and Case 2, it is understood that Case 2 has better characteristics. The reason is that Case 2 can select a coding rate suitable for the channel condition by using Scheme 2 at the second hop. On the other hand, regarding Case 1, since Scheme 1 is used at both hops, it is considered that a hop having a bad channel condition is greatly affected. Then, Case 3 shows the best performance because Scheme 2 is used at both hops. In addition, all systems except Case 2 of the proposed scheme show the best performance when the communication distance of the first hop equals that of the second hop because

${d}_{1st}$ or

${d}_{2nd}$ becomes long (unlike the previous condition) and the long-distance communication influences performance in other cases.

#### 6.2. Theoretical Analysis of Constant ${d}_{2hops}$

Here, we present a theoretical analysis when

${d}_{2hops}$ is fixed because this scenario appears to show the optimal point in

Figure 14,

Figure 15 and

Figure 16. The reason for the optimized performances, except for Case 2, when

${d}_{1st}$ =

${d}_{2nd}$ =

${d}_{2hop}/2$ is described in this section.

The probability of transmission failure in the two-hop case

${P}_{fail,2hop}$ is expressed using the probability of transmission failure at each hop

${P}_{fail,1st}$,

${P}_{fail,2nd}$ as follows:

Here,

${P}_{fail,1st}{\left({d}_{1st}\right)}^{\prime}$ =

$\frac{\mathrm{d}{P}_{fail,1st}\left({d}_{1st}\right)}{\mathrm{d}\left({d}_{1st}\right)}$ and

${P}_{fail,2nd}{\left({d}_{1st}\right)}^{\prime}$ =

$\frac{\mathrm{d}{P}_{fail,2nd}\left({d}_{1st}\right)}{\mathrm{d}\left({d}_{1st}\right)}$. The communication distance in two hops

${d}_{2hops}$ is defined as follows:

${P}_{fail,2hop}$ is differentiated by

${d}_{1st}$ as follows:

Here, the case that (30) = 0 is considered. Equation (30) is modified by the following equation:

Both sides of (31) are integrated by

${d}_{1st}$ as follows:

Here,

${d}_{c}$ where

${P}_{fail,1st}\left({d}_{c}\right)={P}_{fail,2nd}\left({d}_{c}\right)$ is considered. Under this condition,

$C=0$. Thus, (32) is rewritten as follows:

${d}_{1st},\overline{t{r}_{N1\to N2}},$ and

$\overline{t{r}_{N2\to H}}$ that satisfies (33) are considered. Here, it is assumed that

${d}_{1st}={d}_{2nd}=\frac{{d}_{2hop}}{2}$. Under this condition, (33) is satisfied when

$\overline{t{r}_{N1\to N2}}=\overline{t{r}_{N2\to H}}$ in the computer simulations (except for Case 2) because, when

$\overline{t{r}_{N1\to N2}}\ne \overline{t{r}_{N2\to H}}$, (27) is modified as follows:

When (26) is satisfied,

or

However, due to (27)–(29), (35) and (36) are not satisfied. Thus, it can be said that

$\overline{t{r}_{N1\to N2}}=\overline{t{r}_{N2\to H}}$. On the other hand, for Case 2, (33) is not satisfied when

$\overline{t{r}_{N1\to N2}}=\overline{t{r}_{N2\to H}}$ because

${P}_{fail,1st,i}(\frac{{d}_{2hop}}{2})\ne {P}_{fail,2nd,i}(\frac{{d}_{2hop}}{2})$. From

Figure 14,

Figure 15 and

Figure 16, it can be observed that all cases (except Case 2) achieve optimal performance under the above conditions; thus,

${d}_{1st}={d}_{2nd}=\frac{{d}_{2hop}}{2}$.