# A Study Comparing Waiting Times in Global and Local Queuing Systems with Heterogeneous Workers
## 4 Simulation Results
### Heterogenous Workers Simulation Results
The impact is more significant in the heterogeneous case than in the homogeneous case, where the statistical multiplexing is higher. It explains why the utilization for 9 and 12 workers was reduced. Figure 9a-c presents the waiting times. When running the GHT fastest (most left bar in each group and category), the waiting time for any task, no matter to whom it was assigned, is the same. For example, when there are six workers in the slow (Figure 9a), medium (Figure 9b), and fast (Figure 9c) categories, the average waiting times for tasks are 0.19, 0.186, and 0.179, respectively. This follows the explanation provided in Section 4.1, which states that there is no distinction between workers from different categories in the global queue system. This is because all tasks are in the same queue until the workers retrieve them. The waiting times for GHT fastest are also lower than those for all other policies. As was expected, the WJSQL is more effective than the JSQ random TB. It is better than the JSQ's fastest TB algorithm, which is only for slow workers and when the number of workers is 3, 6, and 9. As the number of workers increases, JSQ's fastest TB for medium and fast workers outperforms WJSQL and is like the global case. This finding is significant because a simple JSQ is more practical to deploy than WJSQL. The probability of this scenario increases when there are more workers. Additionally, when there are many workers, waiting times and queue sizes decrease, resulting in more idle workers. In such cases, the global or WJSQL has no advantage over the JSQ Fastest TB. WJSQL is more effective than the JSQ random TB and JSQ fastest TB algorithms for managing slow workers and reducing waiting times for assigned tasks. To achieve this, the system must monitor worker performance and apply appropriate policies when workers work slower. Simulation Results at Load 0.93: Figure 10a-c for utilization and Figure 11a-c for waiting times present the waiting times (vertical bars) and queue sizes (dashed lines) for heterogeneous systems with 3, 6, 9 and 12 workers and mean inter-processing times of (1, 0.5, 0.25) for the slow, medium, and fast groups. The calculated load is \(\rho\) = 0.93. The system is loaded, and the queue sizes are more extended--there are no idle workers most of the time. Figure 10: The utilization of the workers in the system with different sized groups at load 0.93. (**a**) Slow group; (**b**) medium group; and (**c**) fast group. Figure 11: The mean waiting time in the system with different sized groups at load 0.93. (**a**) Slow group; (**b**) medium group; and (**c**) fast group. The performance of the global queue for fast workers is not as good as before. This is because the waiting times are longer than all other local policies, indicating that fast workers cannot maximize their potential when the queue is global. In the local queues, the queues represented by the dashed lines are larger than usual but not as large as expected, given the high load of 0.93. The queue of the fast workers when the routing policy is WJSQL is longer than the others, which indicates the good classification of the processing rate and a finer, more accurate assignment to the fast worker. The WJSQL, when the load is 0.93, performs better than the JSQ fastest for the slow and medium workers and also when the number of workers is high. As shown for load 0.8, the waiting times of the tasks assigned to fast workers in a local queue JSQ fastest TB system are lower than those for the WJSQL, highlighting the advantage of JSQ fastest TB. When the load is 0.93, and there are more workers in a system, the waiting times are reduced but not so sharply as for load 0.8. This is demonstrated for all systems except JSQ random TB, especially when the systems are LH and LHT. Figure 12a,b is focused on the local queue architecture and compares the average waiting times in each group of workers when the architecture is local with JSQ fastest TB, WJSQL, and JSQ random TB. The LH line shows the waiting time for the local homogeneous case.