## 4 Case Study--Numerical Experiments
### Inter-Facility Material Transfer Process
#### 4.1.3 Utilisation Rates Comparison  
Efficient utilisation of servers is paramount to optimising system performance and ensuring smooth operations in closed queueing networks. The utilisation rate, representing the ratio of time a server is busy to the total time, is a crucial performance indicator for evaluating server allocation strategies. Table 5 presents the comparison of server utilisation rates obtained through the GEM-based analytical method and simulation model. The results demonstrate the accuracy and reliability of the GEM-based analytical method in estimating server utilisation rates, with average differences between the analytical method and simulation models ranging from 1% to 6% in scenario one and from 1% to 7% in scenario 5, except for nodes 6 and 7, where the utilisation rates are close to 1. However, it should be noted that when utilisation rates approach 1, the GEM-based method may exhibit slightly higher deviations from the simulation results. These findings highlight the effectiveness of the GEM-based method in calculating server utilisation rates in finite closed queueing networks while acknowledging the need for caution when utilisation rates are close to 1.  
\[Difference=(Analytical\;meth.\;-\;Simulation\;meth.) \tag{11}\]  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Scenario \#** & \multicolumn{3}{c}{**Scenario 1**} & \multicolumn{3}{c}{**Scenario 5**} \\ \hline
**Node \#** & \begin{tabular}{c} **GEM** \\ **Method** \\ \end{tabular} & \begin{tabular}{c} **Simulation** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **Diff.** \\ **Method** \\ \end{tabular} & \begin{tabular}{c} **GEM** \\ **Method** \\ \end{tabular} & \
\begin{tabular}{c} **Simulation** \\ **Model** \\ \end{tabular} &
\begin{tabular}{c} **Diff.** \\ **Diff.** \\ \end{tabular} \\ \hline Node1 & 63.30\% & 66\% & \(-\)2.70\% & 75.9\% & 79\% & \(-\)3.1\% \\ \hline Node2 & 63.30\% & 66\% & \(-\)2.70\% & 77.8\% & 78\% & \(-\)0.2\% \\ \hline Node3 &  
Analysing the scenarios provides valuable insights into the system dynamics and potential bottlenecks. Scenario 1 demonstrates a relatively stable system, exhibiting lower utilisation rates than scenario 5. In scenario 5, several nodes approach or reach utilisation rates near 1.0, indicative of bottleneck situations where system performance stagnates. Notably, scenario 5 (18 trucks) exhibits different truck quantities and compositions compared to scenario 1 (9 trucks), which has the fewest number of trucks.  
Additionally, consistent observations reveal that the last three nodes consistently exhibit low utilisation rates. This suggests the need for process redesign to balance the workload across both streams rather than exert excessive pressure on certain segments. Potential solutions include:  
* Subdividing specific processes into two sub-processes,
* merging idle stations to consolidate workloads, or
* employing additional servers to alleviate the burden on busy servers.  
These interventions can contribute to better resource utilisation, reduced congestion, and improved overall system efficiency. Further analysis and targeted optimisation strategies should be considered to address these observed bottlenecks and enhance the performance of the queueing network.