## 3 Parallel Heterogeneous Servers with a Common Waiting Queue  
Model B includes servers without any prior queue. Next, we considered model E, Figure 4, where the servers have a common queue of limited capacity in front of them, and thus, the incoming arrivals are lost only when the queue is full. The method of assignation is ordered entry. The whole system has a total capacity of \(m+k\), where \(k\) is the capacity of the queue and \(m\) is the number of servers. For the case of \(m=3\), the transition rate diagram of Figure 4 represents the states of the system, where the state >(3) + \(k\) is the state where the loss probability is accounted for (\(p_{b}\)). In the birth and death transitions, the death of state b falls into the (3) + \(k-1\) state because as soon as any customer is serviced in any  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **\(\lambda\)** & **(\(\mu\)1, \(\mu\)2, \(\mu\)3)** & **Loss from Simulation** & **Powerflow M/Mi/k (k = 3)** \\ \hline \multirow{8}{*}{[21]} & & (1.2, 1, 0.8) & 0.0109 & 0.01086 \\  & & (1.2, 0.8) & 0.0118 & 0.01179 \\  & 0.5 & (1.2, 0.8, 1) & 0.0119 & 0.01189 \\  & & (0.8, 1.2, 1) & 0.0140 & 0.01402 \\  & & (1, 0.8, 1.2) & 0.0140 & 0.01397 \\  & & (0.8, 1, 1.2) & 0.0 server, a customer enters the server and a new allocation in the queue of capacity k becomes available. Its transition rate matrix \(k=3\) is A in (6). In order to evaluate the performance of the model, Table 3 shows the comparison of the throughput of model E vs. the simulation results, where the dimensionless throughput is \(R^{*}=R/\Sigma\mu=\lambda(I-p_{f}-p_{b})/\Sigma\mu\). There was complete agreement in the results and the proper representation of the parallel loss system with a buffer using model E. Model B and model E of multiclass or heterogeneous servers are presented for \(m=3\) and they were accurate. The accuracy of this methodology can be used for large systems, such as M/Mi/m/m + k, at the cost of the description of intermediate states in the state transition matrix and computational effort because the size of the transition matrix is of the order \(2^{\text{m}}\). Nevertheless, the construction for a higher queue capacity \(k\) is a much simpler birth and death sequence of extra states in the queue. We developed the algorithm to automate the generation of elements of the transition matrix for an arbitrary number of servers \(m\). Even though the coding of the algorithm was out of the scope of this study, it is worth mentioning some basic insights. The matrix is sparse and can be formulated as the sum of three rate matrices: rates out (negative) from the state (allocated in the diagonal of the matrix), rate into the state (positive) from the death of other states, and rate to the state (positive) from other state births. The states are also automatically enumerated from the binary number representation (m digits) of every server state (0-1, idle or busy). Even when programmed in a matrix-efficient code, like Matlab, this exact transition matrix construction of size order \(2^{\text{m}}\) is heavy for a moderate system size m. Therefore, these transition matrix models are of particular interest for systems with a moderate number of heterogeneous servers, where the accurate assessment of individual servers' performance differences is an objective. Figure 4: Model E transition rate diagram of 3 parallel heterogeneous entry-ordered servers with a waiting queue. For the sake of presenting the model generalization for any size of the system, Figure 5 includes the operating curves \(R^{*}=R/\Sigma\mu\) vs. offered arrival rate \(\lambda\), as calculated for parallel systems of \(m\) = 6, 8, and 10 servers, in each case with queue capacity \(k=m/2\) and \(\Sigma\mu=8\). The service rates are in descending arithmetic progression from the first server \(\mu_{1}=1.5(\Sigma\mu/m)\) for faster to slower (\(f\)-\(t\)-\(s\)) and ascending to the last server with \(\mu_{m}=1.5(\Sigma\mu/m)\) in the slower-to-faster (\(s\)-\(t\)-\(f\)) arrangement.