## 2 Methodology of Parallel System Modeling
### System of Heterogeneous Parallel Servers
The arrival net rate into the loss server is finally \(\lambda(p_{b}+p_{f})\), which is an accurate account of the time percentage that the total arrival rate \(\lambda\) (offered load) does not enter the servers, which was verified in the experiment through the accurate results of Table 1. There is a need to consider every single arrival in terms of the probabilities that arise from the multi-dimensional queue nature of the ordered entry for heterogeneous systems. Meanwhile, Erlang systems of indistinguishable homogeneous servers do not require it. In the timeline of the arrival rates, there are intervals of time when the servers are occupied and no arrivals are trying to enter the system with probability \(p_{f}\), and others where the arrivals try to enter and are rejected to the loss server with probability \(p_{b}\). In both intervals, no entrance of customers into the server system is possible, and thus, they are discounted to get the effective arrival rate into the servers. In model B, every arrival is included in the set of states for accurate accounting, and thus, the system evaluates every customer destination. In general, an ordinary transition matrix that only enumerates the states of the servers, namely, model A, overestimates the throughput (underestimates the probability of occupied servers) for a high offered load by applying the PASTA property (see Table 1). In summary, including the time percentage of the loss state (loss server) provides an accurate throughput calculation of systems of finite capacity, such as model B. The evolution of the probabilities or percentage of time for model B of full servers \(p_{f}\), loss from blocking when servers are busy (blocking) \(p_{b}\), and the system being occupied \(p\_{\_}occ=pb+pf\) are represented in Figure 3. In addition, the probability of an empty system \(p0\) is represented in the results. While \(\lambda\Sigma\mu<1\), the probability of the system being busy \(p_{\_}b\) was lower than the probability of having full servers \(p_{\_}f\). When \(\lambda\Sigma\mu\approx 1\), both were equally probable. It is remarkable that the probability of full servers pf continued to grow with increasing levels of arrival rate up to the point it reached a maximum at about \(\lambda/\Sigma\mu\approx 1.75\). After that maximum, when some extra load was offered, it went directly to the loss server, and \(p_{\_}f\) decreased while \(p_{\_}b\) continuously grew. The percentage of time the servers were occupied with no customers trying to enter the servers, i.e., \(p_{\_}f\), decreased monotonically after that maximum; therefore, for an infinite arrival rate, it was inferred that there was no chance of a full system without loss, and thus, \(p_{\_}f\to 0\), while the probability of loss from blocking, i.e., \(p_{\_}b\), continued to grow. Thus, when an extremely high load was offered, i.e., \(\lambda\rightarrow\infty\), the loss from the blocking probability \(p_{\_}b\) became the only probable situation. Also, from the results in Figure 3, at low \(\lambda\), the probability of an empty system \(p_{\_}0\) was higher than the probability of an occupied system \(p_{\_}occ\). At about \(\lambda/\Sigma\mu\approx 0.6\), both were equally probable, and when \(\lambda/\Sigma\mu>0.7\) (approx. ), the probability of an empty system \(p_{\_}0\) became lower than the probability of occupied servers \(p_{\_}f\). In an attempt to maximize the utilization of servers, increasing the arrival rate \(\lambda\) over that point increased the utilization. Nevertheless, if the increase in cost from losing a customer is higher than the reduction in cost by increasing the utilization, operating the system over that point might not be convenient. Increasing the installed capacity to avoid customer loss instead of increasing server utilization might be a better option. Note that at \(\lambda/\Sigma\mu\approx 0.6\), the perceived saturation of servers was low, i.e., only about 18% of the time the servers were observed to be full (\(p_{\_}f\)) or empty (\(p_{\_}0\)).