**6.1. Example on the gap between the optimal and suboptimal methods**

**Table 3.** Energy (in Joules) Results for *K* = 3

In this section, the proposed methods of Section 5 are compared to the optimal multihop solution of Section 4 (with *H* = *K*) for a low number of SNs (in order for the optimal solution to be tractable). Selecting *K* = 3, all the possible cases are shown in Fig. 3. Hence, the optimal solution will be one of the 16 cases presented in Fig. 3, depending on the fading conditions. The results obtained after implementing the optimal solution and the proposed methods are listed in Table 3. It can be clearly seen that the gap between the suboptimal multihop and clustering results from the optimal solution is very small. In addition, Table 3 shows that the cooperative techniques lead to huge savings compared to the non-collaborative scenario.

Fig. 4 shows, for each of the 16 cases, the percentage of times that this case occurs as the optimal solution. When the distance to the BS is small, Case 1 (no collaboration) seems to be optimal for a significant percentage of the time. However, this percentage decreases as the

**Figure 3.** The 16 possible cases when *K* = 3.

distance to the BS increases. Cases 2-7 form a group of similar cases where the only variation is a permutation of the SNs involved in the connections. As expected, these cases have almost equal probability of being the optimal case for a given value of *d*LR. The same reasoning applies for Cases 8-13 and Cases 14-16. Interestingly, Cases 8-13 were never optimal in the obtained results.

In fact, with Cases 2-7, and considering Case 2 as an example, SN A transmits *ST* bits on the LR, SN C transmits *ST* bits on the SR, and SN B transmits 2*ST* bits (its own data in addition to the data of SN C) on the LR. With Cases 14-16, and considering Case 14 as an example, SN B transmits *ST* bits on the SR, SN C transmits *ST* bits on the SR, and SN A transmits 3*ST* (its own data in addition to the data of SNs B and C) bits on the LR. In Cases 8-13, and considering Case 8 as an example, SN C transmits *ST* bits on the SR, SN B transmits 2*ST* bits (its own data in addition to the data of SN C) on the SR, and SN A transmits 3*ST* (its own data in addition to the data of SNs B and C) bits on the LR. Since the SNs are deployed in a confined area of interest, and since SR transmissions in this case can occur at high rates due to the relative proximity of SNs, Cases 14-16 would generally lead to lower energy consumption than Cases 8-13, since both groups have the same LR energy consumption (due to transmitting 3*ST* on the LR by one SN), but on the SR each of the other two SNs transmits *ST* with Cases 14-16. However, with

**Figure 4.** Percentage of having each of the 16 possible cases as the optimal solution when *K* = 3.

Cases 8-13, SR energy consumption is higher because one SN transmits *ST* while the other transmits 2*ST* on the SR.

Fig. 4 shows that as *d*LR increases, Cases 14-16 become more favored than Cases 1-7. In fact, a large distance to the BS leads to spending most of the energy during LR transmission, since the achievable rates become significantly lower due to the increased distance. Thus, one LR transmission with an SN having favorable LR channel conditions in Cases 14-16 would be more energy efficient than two LR transmissions with Cases 2-7 (or three LR transmissions with Case 1).
