# A Coordination Technique for Improving Scalability of Byzantine Fault-Tolerant Consensus
## 4 Evaluation
### RQ2: What Is the Correlation between the Trouble Transactions and Performance?  
The second research question is for finding out how much our approach can contribute to the performance improvement of the BFT-based consensus algorithm. In the real-world, all blockchain nodes issue transactions, respectively, and it may rarely happen that all transactions in a transaction pool are equal. According to [PERSON] and [PERSON]'s experiment [32], transaction propagation time in the Bitcoin network composed of 344 nodes took 35 min on average, which means that transactions in each node's transaction pool are commonly different. In this research question, we build an environment where each node has many trouble transactions as in the real-world by controlling the interval of the transaction issue time and computed correlation between the elapsed time and the proportion of trouble transactions as \(\tau\).  
Experimental setting for _RQ2_. To simulate the environment, we started with the experimental setting for RQ1 with the same hardware specification and issued one million transactions by controlling interval of transaction issue time from every 15 to 1 ms. Then, we measured a total elapsed time and obtained an average elapsed time for each transaction by dividing the total elapsed time by the number of transactions, as shown in Table 1. In addition to this, we observed the proportion of the trouble transaction \(\tau\) in the consensus coordinator to obtain the correlation between trouble transactions and the elapsed time.  
Experimental results for _RQ2_. Table 1 shows the result of the experiment. In the table, Columns 15 ms, 10 ms, 5 ms, 2 ms and 1 ms denote the time interval of the transaction issue and we only selected some of the representative time intervals. We computed \(\tau\) by averaging the proportion of trouble transactions in a transaction pool collected from each node for every \(time_{p}\sim time_{c}\) period (i.e., 10 s). Based on the \(\tau\), we highlight cells with the same colors associated with its \(\tau\) value (see _Color Legend_ in the table). When the number of nodes is 16, and a transaction is issued every 15 ms, our approach obtained _0.099_ s for each of the average time interval (see the _italic_ in the table). Depending on decreasing the transaction issue time interval, the average elapsed time and \(\tau\) were increased. In addition, an increase in the number of nodes causes an increase in the average elapsed time and \(\tau\).  
From the result, we established the correlation between \(\tau\) and the average elapsed time for a transaction \(Avg.Elap.Time_{tx}\) based on the transaction issue time interval \(tit\) and the number of nodes \(N(Node)\) as Equation (1). The equation implies that the average elapsed time is \(\delta\) times proportional to the proportion of the trouble transaction \(\tau\). In addition, \(\tau\) is inversely proportional to the transaction issue interval \(tit\) and it has logarithmic relation with the number of nodes \(N(Node)\). In the experiment, we obtained \(\delta=2.5\), \(\alpha=2\), and \(\beta=0.1\), which indicates that the proportion of the trouble transaction strongly affects the average elapsed time for each transaction.  
\[Avg.Elap.Time_{tx}\approx\delta*\tau\approx\alpha\frac{1}{tit}+\beta log(N( Node)) \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**N(Node)** & **15 ms** & **10 ms** & **5 ms** & **2 ms** & **1 ms** \\ \hline
**16** & 0.099 & 0.120 & 0.244 & 0.330 & 0.406 \\
**28** & 0.186 & 0.253 & 0.316 & 0.405 & 0.521 \\
**40** & 0.232 & 0.438 & 0.533 & 0.753 & 0.835 \\
**52** & 0.462 & 0.813 & 1.100 & 1.308 & 1.672 \\
**64** & 0.714 & 1.000 & 1.355 & 1.915 & 2.511 \\
**76** & 0.713 & 1.805 & 2.106 & 2.781 & 3.155 \\ \hline
**Color Legend** & \(\tau=0.1x\) & \(\tau=0.2x\) & \(\tau=0.3x\) & \(\tau=0.4x\) & \(\tau>0.5x\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The average elapsed time per transaction and the proportion of trouble transactions (\(\tau\)).