## 2 Mathematical Modeling
### Discussion  
Based on the compared performance measures, it may be noted that FSM approximately yields the same results as the analytical approach regardless of line arrangement. That is, the accuracy of the approximation is not affected by the relative positions of the machines. This conclusion is of great significance, as it supports the extension of the approach to more complex and realistic cases of assembly systems including multiple assembly machines and secondary flows. An absolute error of the results obtained using FSM concerning the analytical solution is presented in Table 2, conditioned to the total number of system states, \(D\). In each case, the absolute error is determined as an absolute value of the difference between the analytically determined performance measure and its counterpart determined using FSM. For example, the absolute error in the case of the production rate, \(E_{PR}\), equals \(|PR_{AN}-PR_{FSM}|\). From Table 2, it can be seen that the FSM approach does demonstrate some discrepancies in the lowest range of the buffer capacities, however, these diminish with the capacity growth due to more reliable discretization of the system's state space. In other words, the accuracy of the results increases with the increase in the number of system states, \(D\) (Table 2). This fact justifies the application of FSM in the case of large-scale and transition-rich Markov chains that quickly become unattainable for the analytical approach. Thus, similarly to the cases of the Bernoulli serial and splitting lines [3; 14], the FSM proved to be a CPU-efficient framework, yielding results in a matter of seconds compared to the analytical solution requiring hours and even days to return the required steady-state probability distributions. Thus, the FSM can be pointed out as a production system modeling framework, which is particularly suitable for tackling design issues such as bottleneck identification, improvability considerations, or lean design, especially as it maintains the functional relationship between the production system's properties and the respective performance measures. Similar methods addressing the same issues are also present in the current literature and practice including the aggregation procedure and computer simulations. The aggregation procedure [5] is based on an iterative algorithm reducing the system's state space into simpler two- or three-machine systems, depending on a particular problem. When compared to the aggregation procedure, the finite state method stands out as a method operating in a complete system state space (i.e., it is not reduced), which enables the reconstruction of the vector of steady-state probability distribution, used later for the definition of key performance indicators. In addition, the functional relationship between production system properties (machine reliabilities and buffer capacities) and key performance indicators is preserved through the vector of steady-state probability distribution. This is also a key advantage of the finite state method concerning computer simulations, often related to demanding modeling and interpretation of results [46; 47]. It has to be pointed out that some authors have stated the equality between assembly systems and systems involving the splitting of material flow through the reversibility  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(M\) & \(N\) & \(D\  
property [12]. Thus, a conclusion could be drawn that it is suffice to master, for example, splitting lines, in order to also evaluate the assembly systems. However, a closer comparison of transition matrices in both cases (i.e., Equations (5)-(8)) with equivalent ones valid in a reversed flow case (splitting line) presented in [3], proves that, in general, this is not the case. The main reason for this is the structure of the transition matrix, which takes a more complex form in the latter case. Additionally, the Bernoulli splitting line involves flow rates transferring material flow to a certain secondary direction.