## 2 Methodology of Parallel System Modeling
### System of Heterogeneous Parallel Servers
In the Erlang model of identical servers, \(p_{\textit{busy}}=B(3,r)\) using (2); for model A with three servers, \(p_{\textit{busy}}=p_{111}=p_{f}\) from (3); and in the case of model B, with three servers and the  
Figure 2: Model B transition rate diagram of 3 parallel heterogeneous ordered entry servers. loss server, \(p_{busy}=p_{111}+p_{>111}=p_{f}+p_{b}\) using (4). Model A is a representation of a loss system that evaluates the throughput from the offered load \(\lambda\). The direct calculation based on the PASTA property with the effective offered load \(\lambda_{\textit{eff}}=\lambda(1-p_{b})\) was only a rough approximation for heterogeneous parallel servers, as the experimental results in Table 1 show for model A. \[ \begin{array}{cccccccc}\lambda_{\textit{eff}}=c\\ \left[\begin{array}{cccccccc}\lambda&-&\lambda&\mu&0&\mu&0&0&0\\ \lambda&-(\lambda+\mu)&0&\mu&0&\mu&0&0\\ 0&0&-(\lambda+\mu)&0&0&\mu&0&0\\ 0&\lambda&\lambda&-(\lambda+\mu)&0&0&0&\mu&0\\ 0&0&0&0&-(\lambda+\mu)&0&\mu&0\\ 0&0&0&0&0&-(\lambda+\mu)&0&0&\mu&0\\ 0&0&0&0&0&0&-(\lambda+\mu)&0&\mu&0\\ 0&0&0&0&0&0&-(\lambda+\mu)&0&\mu&0\\ 0&0&0&\lambda&0&\lambda&-(\lambda+\mu)&0&\mu&0\\ 1&1&1&1&1&1&1&-(\lambda+\mu)&-(\lambda+\mu)&0\\ \end{array}\right]\\ \end{array}  \tag{4}\] \[L=\sum\limits_{i=0}^{k+1}i\cdot p_{i}=0\cdot p_{000}+1\cdot(p_{001}+p _{010}+p_{100})+2\cdot(p_{110}+p_{101}+p_{011})+3\cdot p_{f}+3\cdot p_{b}\] (5) \[U=L/3\]  
The results indicate that the description of the system states and their probabilities should take into account not only the states of the servers but also the associated allocation of all the arrivals and the different independent states in order to be considered a product form the set of state probabilities that sum to 1. In addition to the server's busy state (111), the lost customers enter the loss server. In the transitions diagram of model B, \(>\)(111) = \(b\) is a state of full size for the servers. The system transits into the state \(>\)(111) with the rate \(\lambda\) when the arrivals overflow the servers, and the net input into the servers is \(\lambda_{\textit{eff}}\), while the rate through the loss server is \(\lambda-\lambda_{\textit{eff}}\). Based on the PASTA property, the split of the Poisson rate for all the servers, including the loss server, remains a Poisson distribution [1]. Based on [36], the net output rate through the loss server, which includes a queue of infinite capacity, is also \(\lambda-\lambda_{\textit{eff}}\), providing that the service rate of the loss server is greater than \(\lambda-\lambda_{\textit{eff}}\). This loss server has infinite capacity and we associates an arbitrarily sufficient high service rate \(>\)(\(\lambda-\lambda_{\textit{eff}}\)) to comply with it. Finally, the total output of the whole system is \(\lambda\), with the sum of the servers and loss server together. The system transitions into the loss server with probability \(p_{b}\) and rate \(\lambda\) after the system is full (111), and the system leaves the state at the rate of the sum of the service rates of the three servers; therefore, as soon any server releases a customer after a service, the number of customers in the servers becomes m-1, as represented in the transition states diagram. Note that by including the loss server in model B, the system is not a loss system anymore, and thus, the calculation of the states associated with the offered rate is accurate. The probability of state \(>\)(111) = \(b\) is the percentage of time that the arrivals fall into the loss server, but this always happens after the servers are occupied. In model B, the PASTA property was applied to the three real servers defined in the transition diagram: the offered rate to the servers is \(\lambda(1-p_{>111})=\lambda(1-p_{b})\) before the application of the PASTA property. After applying the PASTA property to the three-server input, the effective rate through the servers becomes \(\lambda_{\textit{eff}}=\lambda(1-p_{>111}-p_{111})=\lambda(1-p_{b}-p_{f})\) by discounting the percentage of time the servers occupy \(p_{111}\).