## 5 Performance Evaluation
### Quality of Service and Impact of Node Density
From these definitions, we have Equation (7) defining the rate (\(\kappa\)) of change between the lowest and the highest delay as follows:  
\[\kappa(\tau)=\frac{E2 ED(\tau)_{\max}-E2 ED(\tau)_{\min}}{PDR(\tau)_{\max}-PDR( \tau)_{\min}}, \tag{7}\]  
where \(\kappa\to 0\) when E2 ED is constant, but \(\kappa\rightarrow\infty\) when PDR is constant, regardless E2 ED. In addition, reliability requirements is an important term in QoS, since it is necessary to measure the impact of node density or \(\tau\). The same experimental design was taken from [PERSON] [15] and we compared this proposal with the algorithms developed by [PERSON] et al. [31] and [PERSON] et al. [32]. The comparison between E2 ED and PDR was made in two ways. The first, in terms of reliability requirements and node density, is shown in Figure 7a,b, respectively. The lower is the PDR, the better is the performance, while the E2 ED works otherwise, i.e., when it increases the performance of the algorithm, it also increases this metric. On the one hand, Figure 7a shows a comparison of our proposal in terms of the reliability requirements. When PDR is increased in REER or DETR, they tend to increase their E2 ED, while the RRDLA algorithms and our proposal have a lower E2 ED, regardless of the number of packets that are delivered, that is both algorithms keep their recovery reliability on the IIoT network. On the other hand, Figure 7b shows a comparison of the present proposal in terms of the node density. When PDR is increased in REER or DETR, they tend to significantly increase their E2 ED, while the RRDLA and the present proposal have a low E2 ED regardless of the number of sensor nodes connected to the network. Thereby, both RRDLA and the present proposal reduce the delay and increase the reliability of delivering the packets adequately, regardless of the sensor nodes connected to the IIoT network. Therefore, the best results are obtained by our proposal anr the RRDLA algorithm. This happens because both protocols minimize the delay for data transfer from the node sensors when the packet receiver retrieves it. In summary, using all the information in Figure 7, the present proposal is slightly better than the other algorithms, exceeding them by at least 2%. Hence, propagation delay in the IIoT network is highlighted. Both the present proposal and RRDLA remain stable regardless of the node density or QoS. Other algorithms such as DETR and REER obtain much larger delays than this proposal. The node density impact and quality of service in \(\kappa(\tau)\) are shown in Table 2. The best result obtained by this proposed is highlighted in **bold**, while the second best result is obtained by RDDLA algorithm, in _italics_. Finally, when a sensor node shares data in the IIoT network, it takes less time to reach the receiver when the proposed algorithm is used. Hence, the recovery propagation of the original packet before its lifetime expires is significantly increased. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Algorithm** & \(\beta\) **(ND)** & \(\beta\) **(QoS)** \\ \hline _RDDLA_ & _0.0833_ & _0.0909_ \\ DETR & 1.1667 & 0.2143 \\ REER & 3.000 & 0.4000 \\
**This proposal** & **0.0417** & **0.0783** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ND and QoS impact for \(\beta(\tau)\).