## 1 Introduction
With direct application to conveyor systems, they found that the optimal order for maximum throughput while minimizing the overflow is the faster-to-slower server arrangement. [PERSON] and [PERSON] (1981) [22] showed that the overflow (loss) is minimal for \(\mathrm{M}/\mathrm{Mi}/\mathrm{c}\) under the faster service rule. [PERSON] and [PERSON] (1986) [23] studied approximate expressions of the heterogeneous server \(\mathrm{G}/\mathrm{G}/\mathrm{n}\). With a focus on heterogeneous systems with retrial, surveys by [PERSON] and [PERSON] (1979) [24] and [PERSON] (2007) [25] are noted for simulation works. [PERSON] (1983, 1984) [26; 27] studied the analytical model of two heterogeneous servers with recirculation, while [PERSON] (1987) [28] studied the asymptotic performances of random access and an ordered entry \(\mathrm{G}/\mathrm{M}/\mathrm{K}/\mathrm{O}\) with approximate expressions. Recently, [PERSON] et al. (2019) [29] presented a revision focused on the operation and not on modeling. Even though closed-loop or recirculating systems have been studied much less than ordinary loss systems, the recirculating flow represents additional difficulties in system modeling. Using stochastic models, [PERSON] and [PERSON] (1979) [24] analyzed a conveyor with one loading station. [PERSON] (1982) [30] modeled the recirculating conveyor systems with a single loading station and a single unloading station as GI/M/1/1 using the superposition of the recirculating rate with the new arrivals. He approximated the joint interarrival distribution as a hyperexponential with an iterative method. [PERSON] and [PERSON] (2000) [31] modeled a recirculating system as a queue network based on an approximate method of solving tandem finite queues with blocking by [PERSON] and [PERSON] (1988) [32]. [PERSON] and [PERSON] (2005) [33] used conditional probabilities to approach the recirculating overflow and considered multiple unloading stations. [PERSON] and [PERSON] (2006) [34] revised the formulation of [PERSON]'s model in a two-station multiserver model with balking and reneging. Retrial queues or recirculating systems are also of particular interest in modeling service systems, like call centers ([PERSON] et al. (2003) [6]) or in merging systems with recirculation ([PERSON] et al. (2018) [35]). Together with the analytical models, time-consuming simulation allows for the evaluation of the system performance of complex systems, alleviating the difficulties of heterogeneous systems. Proper simulation can also provide experimental verification of analytical models. Today's growing computing capabilities are paving the way for the increasing use of simulation. Nevertheless, the possibilities of analytical models can be better valued in optimization analysis, transient behavior, or system control, in particular when quick solutions are better than long simulation runs, with a particular interest in the decision making of evolving flexible systems. Heterogeneous systems have no accurate analytical models available, like homogenous ones. We contributed to reducing this gap in this study with a set of new scalable models that gives accurate quantification of the loss of the parallel system with heterogeneous servers of limited capacity, and even including recirculation. It allows for the calculation of performance metrics based on the state probabilities solved from the transition matrix. We developed a detailed justification with a novel approach to evaluate the loss of the system applied to parallel heterogeneous servers and we also verified its agreement with simulation results with differences under 0.5%, followed by the analysis of system performance. The model contributes to overcoming the low accuracy of other loss estimations that are only based on the probability of a busy system from conventional transition state diagrams, whose states only enumerate the size of the system. Its application to recirculating systems also gives a good approximation, within 1% across most of the arrival rate range.