# A Study Comparing Waiting Times in Global and Local Queuing Systems with Heterogeneous Workers
## 2 Related Work
### Global Queue with Homogenous (GH) and Heterogeneous Workers (GHT)  
Single queue architecture, where digital servers pull tasks from a single shared FIFO queue managed by a centralized system, was the dominant architecture until the 1990s. This architecture has been considered one way to reduce the average waiting time without increasing staffing levels because of its pooling benefit. However, research shows that the performance of a global queue system with heterogeneous workers decreased. The following studies deal with an M/Mi/K system where the workers are heterogeneous. [PERSON] [6] analyzed an M/Mi/K system with one queue and \(K\) heterogeneous servers. They found that servers with different service rates cannot be replaced with an equal number of servers with the same rate without increasing the waiting time in the queue. They stated that the more significant the difference between the servers' processing rates, the longer the time waiting in the queue. [PERSON] & [PERSON] [7] showed that in the tele-traffic world, combining multiple arriving streams with different service rates into one system may result in a higher waiting time. [PERSON] & [PERSON] [8] studied the importance of selecting a server in a single queueing system with multiple heterogeneous servers since it changes with decreasing traffic intensity, increasing number of servers, and decreasing variation between interarrivals. [PERSON] et al. [9] developed upper bounds for the average queue length and average waiting time in a single queue M/Mi/K system. They showed that heterogeneous systems can perform better than homogeneous systems when using the Fast Server First (FSF) allocation strategy. However, it can also result in longer waiting times with different numbers of servers. There is no condition under which the average waiting time in a queue of a heterogeneous system is better than that of a homogeneous one. [PERSON] et al. [10] discuss the \"Slow Server Problem\" that arises when a service facility has both fast and slow servers, and the question is how to best utilize the slow server in the presence of a faster one. [PERSON] [11] generalizes the Slow Server Problem to include an additional cost structure. This approach provides a more nuanced analysis of activating an additional (slow) server in a system when already operating with servers of varying speeds might be beneficial. To enable customers to manage their waiting times, several works [12; 13; 14] proposed semi-parallel queueing by treating the global queue as several priority queues according to the processing rates. To conclude, there is no clear understanding of how waiting times behave in a global system with workers with differentprocessing rates. It is, therefore, difficult to compare the performance of such a system with one that has homogeneous workers. The waiting time in this kind of system mainly depends on the number of workers and their workload. However, studies have shown that the waiting times can be reduced if faster workers are given priority using techniques like FSF tie-breaking.