# Inter-Blockchain Communication Message Relay Time Measurement and Analysis in Cosmos +
## 5 Results and Analysis
### Results of Analysing IBC Msg Relay Time  
The histogram of experimental values is skewed to the left, showing a deviation from the normal distribution, as presented in Figure 16. To address this, we assumed that the related graph had a lognormal distribution because values cannot be smaller than 0. We modified the natural logarithm on the experimental, focusing on a visually similar trend for the log-transformed message relay time and the lognormal distribution, as shown in Figure 17.  
Numerical tests such as the Kolmogorov-Smirnov test, the Shapiro-Wilk test, and the Anderson-Darling test were used to test whether the previously mentioned graph follows a normal distribution. The null hypothesis said that the data follows a normal distribution, whereas the alternative hypothesis stated that it did not. The Kolmogorov-Smirnov test yielded a \(p\)-value of 0.839, suggesting that we cannot reject the null hypothesis because it is above the significance level of 0.05. Similarly, the Shapiro-Wilk test produced a \(p\)-value of 0.073, which was likewise over the significance level, implying that the null hypothesis was accepted. Furthermore, the Anderson-Darling test supported the null hypothesis because the estimated statistic of 0.362 was less than the critical values at the selected significance  
Figure 15: IBC Message Relay Time. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Message Relay Time Interval** & **Min** & **Max** & **Mean** & **Median** & **Var** & **Std** \\ \hline G to T time & 0.0 & 79.308 & 19.623 & 16.732 & 123.483 & 11.112 \\ T to R time & 3.0 & 72.0 & 13.515 & 13.0 & 16.449 & 4.055 \\ R to A time & 5.0 & 94.0 & 22.310 & 18.0 & 172.626 & 13.138 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: IBC Time Statistics Summarized by Message Relay Time Intervals. levels [0.574, 0.653, 0.784, 0.914, 1.088]. As a result, we can conclude that the experimental data are consistent with a lognormal distribution. The standard normal distribution table was used to obtain the values corresponding to the probability of 68.3%, 95.4%, and 99.7% for Z-scores of 1, 2, and 3, respectively, because the IBC message relay time distribution follows a lognormal distribution. The correlation coefficients were calculated to analyze the relationship between IBC Msg Relay Time and the time for each subsequent session. The relay of IBC Acknowledgement transactions during the R to A time interval exhibited the highest correlation of all the time intervals. Several numerical tests assessed whether the acquired findings followed a normal distribution, including the Kolmogorov-Smirnov, Shapiro-Wilk, and Anderson-Darling tests. The null hypothesis assumed a normal distribution, while the alternative hypothesis indicated a deviation from normality. The Kolmogorov-Smirnov test results show that the null hypothesis cannot be rejected because the computed \(p\)-value of 0.839 is above the significance level of 0.05. Similarly, the Shapiro-Wilk test produces a \(p\)-value of 0.073, more than the significance criterion. As a result, the null hypothesis cannot be rejected in this test either. Furthermore, as indicated in Table 4, the Anderson-Darling test does not give enough evidence to reject the null hypothesis because the calculated statistic of 0.362 is less  
Figure 16: KDE of IBC Msg Relay Time and Normal Distribution. Figure 17: Log-taken IBC Msg Relay Time and Lognormal Distribution. than the crucial values of [0.574, 0.653, 0.784, 0.914, 1.088]. Based on these findings, it is possible to conclude that the experimental data follow a lognormal rather than a normal distribution. Because the IBC message relay time distribution is lognormal, the probabilities corresponding to Z-values of 1, 2, and 3 are often employed and reflect about 68.3%, 95.4%, and 99.7% of the data, using standard normal distribution standards. Table 5 displays the IBC message relay time range per stage. A correlation study explored the relationship between the IBC Msg Relay Time and the time by section.