# An Efficient Relayed Broadcasting Based on the Duplication Estimation Model for IoT Applications
## 4 Simulation Study
### The Effect of the Duplication-Index-Decision Factor  
In this section, we observe the effect of \(\alpha\) (duplication-index-decision factor) on the performance of DRBF. Recall that as shown in Figures 4 and 6a, as \(\alpha\) decreased, the duplication ratio increased and the transmission of the duplicate frame was aggressively suppressed. Figure 9 shows four performance indices; \(\overline{F}_{\mathit{bal}}\) and \(\overline{T}_{\mathit{dis}}\), for various values of \(\alpha\) ranging from 0.5 to 1 and \(N_{\mathit{node}}\) ranging from 60 to 140.  
First, we observed \(\overline{F}_{\mathit{bal}}\) shown in Figure 9a. When \(\alpha\) increased from 0.5 to 0.8, \(\overline{F}_{\mathit{val}}\) increased almost linearly from approximately 550 to 965. However, the rate of increase in \(\overline{F}_{\mathit{bal}}\)  
\begin{table}
\begin{tabular}{p{85.4 pt} p{284.5 pt}} \hline \hline
**Symbol** & **Description** \\ \hline \(\overline{F}_{\mathit{val}}\) & The average number of valid frames per node, i.e., frames received in each node without corruption or duplication. \\ \hline \(\overline{F}_{tx}\) & The average number of frames transmitted per node \\ \hline \(\overline{F}_{\mathit{dup}}\) & The average number of duplicate frames received per node without corruption. \\ \hline \(\overline{T}_{\mathit{dis}}\) & The traffic dissemination time defined as the time from the initial broadcast of the source node until all of the nodes stop relaying. \\ \hline \(\overline{R}_{tx}\) & The total number of frames transmitted per second in the whole network. \\ \hline \(G(X)\) & The performance value of ADBRF divided by that of DRBF in terms of performance index \(X\) \\ \hline \(G_{RQ}(X)\) & The performance value of ADBRF\({}_{\mathtt{RQ}}\) divided by that of DRBF\({}_{\mathtt{RQ}}\) in terms of performance index \(X\) \\ \hline \(R_{\mathit{val}}(p\%)\) & The fraction of nodes that received valid frames more than \(p\%\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: The performance indices and their descriptions.  
exceeded 0.8. In particular, when \(\alpha\geq 0.9\) and \(N_{node}\geq 100\), \(\overline{F}_{val}\) did not change much and remained at around 965. Next, we observed from Figure 9b how \(\overline{T}_{dis}\) changed depending on \(\alpha\) and \(N_{node}\). As \(\alpha\) increased, \(\overline{T}_{dis}\) increased and its slope also mostly increased. Moreover, the difference between the \(\overline{T}_{dis}\)s with different values of \(N_{node}\) was negligible when \(\alpha\) was 0.5; but it was remarkably magnified as \(\alpha\) increased. For example, when \(N_{node}\) was between 60 and 140, \(\overline{T}_{dis}\) was approximately between 0.98 s and 0.99 s in the case of \(\alpha=0.5\), but it expanded to between 3.0 s and 4.5 s in the case of \(\alpha=1\). We can summarize the results in Figure 9 as follows:  
* When the value of \(\alpha\) is small (e.g., 0.5), the increase in \(\alpha\) is considerably effective at improving the reliability in terms of \(\overline{F}_{val}\) while somewhat degrading the performance of fast traffic dissemination in terms \(\overline{T}_{dis}\).
* When the value of \(\alpha\) is large (e.g., more than 0.9), the increase in \(\alpha\) slightly increases \(\overline{F}_{val}\) but notably increases \(\overline{T}_{dis}\).
* It is desirable to set \(\alpha\) close to 1 for applications that are primarily intended to ensure the reliability strictly (Apart from increasing \(\alpha\), the reliability of flooding can be further improved by adopting a proper coding technique in the application layer (e.g., erasure coding or digital fountain [43])), whereas it is better to set \(\alpha\) as a relatively small value if the quality-of-service (QoS) of applications largely depends on the latency.