# A Study Comparing Waiting Times in Global and Local Queuing Systems with Heterogeneous Workers
DES experiments are designed to compare various aspects of queuing architecture and local queue routing. Considering market behavior analysis [3], we chose three categories of processing rates as follows: slow, medium, and fast. We conducted tests on systems with 3, 6, 9, and 12 workers, equally divided into category groups when the systems' loads were 0.8 and 0.93, respectively. We defined accurate parameters to compare the homogenous and heterogeneous simulation environments for a certain system load. We adjusted the arrival rate parameters according to the processing rate categories and number of workers (Section 4.2). Our simulation results provide per-category waiting times and worker utilization, not only the total mean waiting time. The few papers that compare both systems did not provide these details. Our findings are as follows:  
1. Global queue architectures (GH, GHT) ensure consistent average task waiting times across all workers, regardless of speed settings. This happens because all tasks are in the same queue until the workers retrieve them, and it occurs for all numbers of workers or system load. 2. While GH and GHT outperform LH and LHT, on average, the push mechanisms that control the assignment processes and heterogeneity-aware algorithms improve the local queue system waiting times and load balance. It is significant and better than the global queues when tasks are assigned to medium and fast workers. Due to the control mechanism, slow worker's assignments are easier to control. 3. We observed a connection between the duration of waiting times and the proportion of TB events. In particular, the waiting times for tasks assigned to slow and medium workers tend to be excessively long when using a simple JSQ with random TB in a local queue architecture. 4. To conclude, the average waiting times (W) for GH, GHT, LH, and LHT for loads at 0.8 and 0.93 are ranked in the following order: \(\bullet\)\(W_{\text{GH}}\sim=W_{\text{GHT}}^{\text{Slow}}\), \(W_{\text{GHT}}^{\text{Medium}}\), \(W_{\text{GHT}}^{\text{Fast}}\sim<W_{\text{LH}}<W_{\text{LHT}}^{\text{Medium}}<W _{\text{LHT}}^{\text{Slow}}\) \(\bullet\)\(W_{\text{LHT}}^{\text{Fast}}\sim<W_{\text{GH}}\) The proportions of the TB events have a similar order. On the one hand, \(W_{\text{GH}}\sim<W_{LH}\) follows previous works, but the behavior of the waiting times of tasks assigned to different categories when using GHT and LHT shows that heterogeneity-aware routing policies are required. 5. For both loads of 0.8 and 0.93 and the same number of workers, JSQ's fastest TB outperforms WJSQL and is like the waiting times in the global architecture. This finding is significant because simple JSQ is more practical to deploy, and the WJSQ is sometimes considered optimal. The WJSQL is a version of SED that we developed and tested. WJSQL learns the workers' processing rates and assigns the task to the faster worker in the shortest expected time. The WJSQL is more effective than the JSQ random TB and JSQ fastest TB routine policies for managing slow workers and reducing waiting times for their assigned tasks. The paper is organized in the following way. Section 2 discusses the related work. Section 3.1 describes the global system, and Section 3.2 describes the local system. Section 3.3 describes the methodologies and simulation setup for the homogenous and heterogeneous environments. Section 4 describes the simulation results. Section 4.1 compares global and local systems with homogeneous workers (GH vs. LH). Section 4.2 demonstrates the correlation between the number of tie-breaking events and the waiting times when a random selection is used. Section 4.3 describes heterogeneity-aware routing policies for the local queue system, and Section 4.4 compares global and local systems with heterogeneous workers (GHT, LHT). Section 5 summarizes and concludes the paper.