## 3 Parallel Heterogeneous Servers with a Common Waiting Queue
Then, we examined how the pattern of the servers influenced the performance of the system (Figure 6). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Arrival Rate** & **Service Rates** & **Simul** & **Mod E** & **Model E vs. Simul** \\ \(\Lambda\) & **(\(\mu\)1, \(\mu\)2, \(\mu\)3)** & **Mean** & **Half** & **R\({}^{*}\)** & **[\%]** \\ \hline \(1\) & & 0.3323 & 0.0005 & 0.3326 & \(-\)0.09  
With a slight imbalance, some extra throughput was obtained when the order from faster to slower was applied, but when the imbalance was high--see, for instance, (2, 0.75, 0.5)--the benefit could only be obtained under medium to low utilization (low arrival rate). The system underperformed for the balanced service rates (1, 1, 1) with the same average for high utilization. When the utilization tended to saturation (see Figure 6 over 90% utilization), the throughput was quite similar, attenuating the effect of the server's arrangement. However, for a given average arrival rate of the mean service time, the maximum throughput was reached by homogenous (1, 1, 1) or slightly heterogeneous servers. A high imbalance in service rates reduced the throughput. As a rough estimation, reductions up to the order of 5% in \(R^{*}=R/\Sigma\mu\) can be observed in Figure 6, depending on the pattern of multiclass servers that caused significant heterogeneity for the performance and not only the average of service rates. Next, the performance improvement by adding a queue was evaluated using model E for different values of the maximum queue capacity (Figure 7). For \(k=0\), model E was equivalent to model B. The operative curves of Figure 7 show that the size of the buffer increased the throughput, but more relevant, for heterogeneous service rates, the performance could reach higher results with imbalanced service rates arranged from slower to faster, i.e., (0.25, 0.75, 2), than the balanced distribution (1, 1, 1) since this provided a buffer of enough capacity. In the case of heterogeneous servers without any queue, ordering the servers from faster to slower gave the best results for the throughput. When heterogeneous servers were combined with a prior queue, the performance was slightly better with assignation from slower to faster service rates when there was no queue (k = 0), but in combination with a minimum queue, the results became similar, and with sufficient capacity (k = 4), the performances of heterogeneous and homogeneous systems were indistinguishable (see Figure 8). A properly dimensioned queue before the multiclass parallel servers can provide improvements in performance, overwhelming the heterogeneity of service times. This applies to traffic problems, like call centers, where the use of an interactive voice response to the arriving customers can handle a waiting queue to be serviced. Former research established priority for the faster agents in [37], as was also mentioned in [14; 16], or in the problem of fair agents [7]. Our results show that a proper queue or buffer can compensate for heterogeneity. Figure 6: Model B (_m = 3_) utilization vs. throughput for different service rate patterns (\(\mu_{1}\), \(\mu_{2}\), \(\mu_{3}\)). Conversely to manufacturing systems of tangible products, where the cycle time of customers in the system can be secondary after throughput, in service manufacturing or pure service systems, the sojourn time in the system can be a priority and fundamental metric. For the system under study with \(m=3\), multiclass servers with a buffer of maximum size \(k\) before them, the time in the system, also called the sojourn time or cycle time \(CT\), is calculated using [PERSON]'s law (7) and is represented in Figure 8. At low \(\lambda\), the better results (minimum \(CT\)) were obtained with faster-to-slower server priority, while in the rest of the range, the better configuration was the homogeneous arrangement of servers. While the queue capacity improved throughput by mitigating loss with a high offered load, it also increased \(CT\).