## 4 Proposed Framework
### Modelling and Verification
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Process Discovery Algorithm** & \begin{tabular}{c} **Replay** \\ **Algorithm** \\ **1** \\ \end{tabular} & \begin{tabular}{c} **Replay** \\ **Algorithm** \\ **2** \\ \end{tabular} & \begin{tabular}{c} **Replay** \\ **Algorithm** \\ **3** \\ \end{tabular} & \begin{tabular}{c} **Replay** \\ **Algorithm** \\ **4** \\ \end{tabular} & \begin{tabular}{c} **Replay** \\ **Algorithm** \\ **5** \\ \end{tabular} & \begin{tabular}{c} **Replay** \\ **Algorithm** \\ **6** \\ \end{tabular} & \
\begin{tabular}{c} **Replay** \\ **Algorithm** \\ **7** \\ \end{tabular} &
\begin{tabular}{c} **Replay** \\ **Algorithm** \\ **8** \\ \end{tabular} \\ \hline Alpha Miner & 0.105 & 0.092 & 0.053 & 0.202 & 0.091 & 0.096 & 0.101 & 0.104 \\ Discover using Decomposition & 0.1 & 0.101 & 0.104 & 0.105 & 0.098 & 0.107 & 0.102 & 0.106 \\ Mine for Heuristic Net & 0.161 & 0.271 & 0.154 & 0.274 & 0.252 & 0.187 & 0.238 & 0.275 \\ Mine Transition System & 0.266 & 0.101 & 0.105 & 0.107 & 0.132 & 0.103 & 0.24 & 0.103 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Property \(P=?\) [\(F\leq 10~{}(s=x)\)]. The estimation of such a property is essential, as it facilitates a quantifiable understanding of the resource expenditure needed to transition within the system, thus providing crucial insights into the process model's performance. By quantifying the cost to reach specific states, we are better equipped to evaluate the model's performance under different conditions and make informed decisions. The summary presented in Table 5 offers an analytical exploration of the property \(P=?\)\([X\left(s=x\right)]\), a construct defined to answer the question _what is the likelihood that the subsequent state following the initial state is \(x\)?_ In this property, we utilise the _NEXT_ operator, denoted by \(X\), which signifies the next reachable state from the current position in the sequence. The state labelled \(x\) is an arbitrary state of interest within the model, subject to the specific requirements or investigative focus of the analysis. This table, therefore, provides a quantitative estimation, offering a calculated probability that the model will transition into specified state \(x\) immediately after leaving the initial state. Specific state \(x\) is selected based on the research objectives or the specific characteristics of the process under investigation. #### 5.2.2 Estimation with Confidence Level of 95%

Tables 6-9 present the findings derived from our probabilistic analysis of the properties under investigation. Each of these results provides an estimation with a confidence level of 95%. To ensure a robust set of data, a series of 100 simulations were carried out for each property. Specifically, Table 6 provides a detailed analysis of the projected results, differentiating between a variety of process discovery algorithms and their associated replay algorithms. The focus of our evaluation is the property \(P\ =?\)\([F\leq 10\ (s=x)]\). This property queries the _probability of attaining target state \(x\) within a predetermined temporal boundary of 10 units_. A 95% confidence level offers a slightly broader margin for uncertainty compared with the 99% confidence level for the same property, as presented in Table 2. This might be suitable for scenarios where the implications of the outcome are less critical.