# A Study Comparing Waiting Times in Global and Local Queuing Systems with Heterogeneous Workers
## 4 Simulation Results  
### Homogenous Workers' Simulation Results (GH vs. LH)  
This section compares the global and local systems' waiting time and worker utilization when the workers were identical, given the same number of workers and load, and when using random TB. The following tables show the marginal difference in utilization, waiting times, and queue size between both architectures for loads 0.8 and 0.93 when there were 3, 6, 9, and 12 homogeneous workers and random TB.  
Tables 1 and 2 demonstrate that all the workers are utilized in the same way. The arrival rate \(\lambda\)i of each worker is similar and equals the workload rate. The results were the same for load 0.93 and conformed to the explanation and formalization in Section 2. For both loads, slightly higher task waiting times were demonstrated for workers in the local queue architecture.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**GH: Global Queues**} & \multicolumn{3}{c}{**LH: Local Queues**} \\  & \multicolumn{3}{c}{**Homogenous Workers**} & \multicolumn{3}{c}{**Homogenous Workers**} \\ \hline \multirow{2}{*}{
\begin{tabular}{c} **Number of** \\ **Workers** \\ \end{tabular} } & **Mean** & **Mean** & **Queue** & **Mean** & **Mean** & **Mean** & **Mean** \\  & **Utilization** & **Waiting Time** & **Size** & **Utilization** & **Waiting Time** & **Size** \\ \hline
3 & 0.799 & 0.454 & 2.548 & 0.8 & 0.549 & 1.025 \\ \hline
6 & 0.798 & 0.184 & 2.059 & 0.801 & 0.269 & 0.503 \\ \hline
9 & 0.8 & 0.103 & 1.729 & 0.8 & 0.169 & 0.316 \\ \hline
12 & 0.799 & 0.066 & 1.482 & 0.801 & 0.123 & 0.231 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Global and local system with homogenous workers and load at 0.8.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**GH: Global Queues**} & \multicolumn{3}{c}{**LH: Local Queues**} \\  & \multicolumn{3}{c}{**Homogenous Workers**} \\ \hline \multirow{2}{*}{
\begin{tabular}{c} **Number of** \\ **Workers** \\ \end{tabular} } & **Mean** & **Mean** & **Mean** & **Queue** & **Mean** & **Waiting Time** & **Mean** \\  & **Utilization** & **Waiting Time** & **Size** & **Utilization** & **Waiting Time** & **Size** \\ \hline
3 & 0.931 & 1.889 & 12.326 & 0.931 & 1.941 & 4.215 \\ \hline
6 & 0.93 & 0.861 & 11.237 & 0.929 & 0.914 & 1.986 \\ \hline
9 & 0.9333 & 0.556 & 10.900 & 0.928 & 0.654 & 1.418 \\ \hline
12 & 0.932 & 0.390 & 10.210 & 0.  
The waiting times for the same load are smaller when the number of workers is increased from 3 to 12. We expected the waiting time to remain the same because the number of servers was scaled up with the arrival rate, so the average server utilization stayed the same. For example, the average waiting time for six workers was 0.861 in the global setting and 0.914 in the local setting. The average waiting time for twelve workers was 0.39 in a global setting and 0.555 in a local setting. These results follow the [PERSON] approximation [33] for estimating the waiting time in an M/M/1 queue with variability in arrival and service processes. We can also see that the differences between waiting times in the global and local architecture are higher when the number of workers increases. To conclude, this section demonstrates that when the workers are homogenous, the waiting times and utilization are the same. The waiting times for LH are slightly higher than those for GH, which is consistent with previous research. The following section focuses on GHT and LHT systems with heterogeneous workers.