## 6 Model Validation and Performance Evaluation
### Criteria to Minimize Delay and Energy Consumption  
The average delay (\(t_{absorption}\)) required to terminate a data collection round using RFSA and FSA, expressed in the average number of frames, is represented in Figure 3 as a function of the frame length (from five to 50 slots per frame). We have considered \(n=25\), 50 and 100 end-devices. As could be expected, the results show that the value of \(t_{absorption}\) decreases exponentially and tends to a constant value when the frame length increases. Indeed, the longer the frame length, the lower the probability of collision among end-devices and the higher the number of successful transmissions per frame. As can be observed, the value of \(t_{absorption}\) is considerably greater in FSA than in RFSA. The fact that, in RFSA, an end-device that succeeds in transmitting its first data packet reserves the slot with success for the next \(L-1\) frames leads to a lower probability of collision in subsequent frames, since the number of contending end-devices is also reduced, yielding a shorter delay. The differences between the value of \(t_{absorption}\) using RFSA and FSA become more apparent when the number of end-devices is higher or when the frame length is shorter, and thus, contention is greater in each frame. The average delay (\(\overline{T_{CR}}\)), expressed in seconds, and the average energy consumed by the coordinator in a data collection round (\(\overline{E_{coord}}\)) are represented in Figure 4, respectively. They have been evaluated as a function of the frame length (from five to 50 slots) for RFSA and FSA considering \(n=25\), 50 and 100 end-devices. It is worth noting that, as could be expected from the delay and energy models of  
Figure 3: Average delay (in frames) required to terminate a data collection round over the frame length. RFSA, the values of \(\overline{T_{CR}}\) in Equation (12) and \(\overline{E_{coord}}\) in Equation (13) are highly correlated, showing a similar trend. As can be observed in Figure 4, \(\overline{T_{CR}}\) and \(\overline{E_{coord}}\) tend to infinity in both RFSA and FSA when the frame length is lower than \(n/8\) and \(n/4\), respectively. This is a direct consequence of the very high value of \(t_{absorption}\), due to the high probability of collision among the end-devices when the frame length is very low. In addition, the values of \(\overline{T_{CR}}\) and \(\overline{E_{coord}}\) in FSA are always greater than in RFSA. This is due to the fact that the values of \(t_{absorption}\) are considerably greater in FSA, as shown in Figure 3.  
There exists an optimum frame length \(m_{opt}\) that minimizes the value of both \(\overline{T_{CR}}\) and \(\overline{E_{coord}}\). The value of the optimum frame length has been obtained numerically. First, we have computed the average delay and the average energy consumption over the frame length, and then, we have searched for the frame length that provides the lowest values of delay and energy consumption. We have repeated the process for different values of the number of end-devices in order to confirm the validity of the optimum value, which depends on the number of end-devices, being \(m_{opt}\simeq n/5\) for RFSA and \(m_{opt}\simeq n/2\) for FSA. For example, when \(n=100\) end-devices, we have minimum average delay, \(\overline{T_{CR}}\simeq 50\) s, and minimum average energy consumption of the coordinator, \(\overline{E_{coord}}\simeq 3.2\) J, at \(m\simeq 20\) slots when using RFSA; and \(\overline{T_{CR}}\simeq 90\) s and \(\overline{E_{coord}}\simeq 6\) J at \(m\simeq 50\) slots when using FSA. Therefore, RFSA provides a delay reduction and energy savings of approximately 45% with respect to FSA when using optimum frame lengths in both access protocols. For frame lengths greater than \(m_{opt}\), the values of \(\overline{T_{CR}}\) and \(\overline{E_{coord}}\) increase proportionally to the frame length in RFSA and FSA.