## 1 Introduction  
Parallel server systems with limited capacity are present in many operational situations of interest in manufacturing [1], but also in transportation or traffic [2], service [3], or computer systems [4]. Their study using Markovian network models requires accurate loss quantification to properly evaluate their performance, and the ordinary model of these systems consists of enumerating all the states for the size of the system in order to calculate the state probabilities that sum to 1. Among them, there is interest in the probabilities of the full and empty states that allow for estimating the maximum expected capacity or idleness. In particular, the analysis of the parallel systems with homogeneous or identical servers facilitates analytical explicit expressions of the state probabilities. Therefore, based on the PASTA property [5], the probability of the full-size system is established in manuals as the estimation of the arrival rate percentage that becomes lost due to blocking when the system is busy, and thus, the effective throughput is the difference from the offered load rate. For parallel systems with identical servers, the Erlang systems of Poisson arrivals and service rates are well known and of great interest for service systems, like call centers [6]. In addition, the important property of performance equivalence of M/G/c/c and M/M/c/c [3] gives extra opportunities for analysis and decision making with service time distributions other than the exponential one. Even though parallel homogeneous systems have been studied in depth (Erlang models), many real systems have heterogeneous servers, where non-uniform human behavior [7] or task variety is present. The practice of ordered entry involves assigning work upon arrival to the first idle server in the order of arrangement. This is a frequent natural assignation, for instance, in conveyor systems or call centers. In this context, some former studies are in more direct connection with the present work. Initial research of parallel heterogeneous systems can be found in [8], who analyzed ordered entry with the heterogeneous systems with the random allocation of service. His work assumed no queue in front of the servers, i.e., \(\mathrm{M}/\mathrm{Mi}/\mathrm{n}\). His error analysis of the heterogeneous vs. homogeneous system concluded that a heterogeneous system cannot be replaced by one of the equal servers, whose mean service rate is the average of the mean service rate of each server. The authors of [9; 10] studied a parallel system with homogeneous servers and ordered entry with application to conveyor systems. [PERSON] (1970, 1971) [11; 12] compared homogeneous and heterogeneous Markovian queuing systems and obtained the steady-state probabilities of a heterogeneous three-server system, i.e., \(\mathrm{M}/\mathrm{Mi}/\mathrm{3}\). Later, some other researchers dealt with the analysis of the server arrangement for performance optimization. [PERSON] (1983) [13] and [PERSON] (1987) [14] studied the optimal allocation of buffers in front of the servers but without the quantification of blocking. [PERSON] (1986) [15] and [PERSON] and [PERSON] (2007) [16] paid attention to minimizing the overflow of the system. Further extensions in parallel heterogeneous system studies can be found in [PERSON] et al. (1994) [17], who gave a comprehensive view of solution methods for the epoch. [PERSON] and [PERSON] (2014) [18] dealt with the semi-Markov process of GI/M/n/0 and loss quantification; more recently, [PERSON] et al. (2020) [19] presented an exact formulation but with an approximate resolution for ordered entry. Specifically, for systems using the ordered entry process, one can also refer to the work of [PERSON] (1976) [20], who concluded the need for a multi-dimensional birth-and-death process to represent the system. [PERSON] and [PERSON] (1977) [21] studied the state probabilities of a parallel system of heterogeneous multiservers (multichannel) with ordered entry discipline and no waiting line, i.e., \(\mathrm{M}/\mathrm{Mi}/\mathrm{n}\).