## 2 Mathematical Modeling
### Performance Measures
Figure 11: Comparison of the performance measures in the case of line L5C: (**a**) \(N_{1}=N_{2}=N_{3}=N_{4}=N_{5}\) (**b**) \(N_{1}=N,N_{2}={}^{3}/4N,N_{3}={}^{1}/2N,N_{4}={}^{1}/4N.\)  
Figure 12: Comparison of the performance measures in the case of line LSD: (**a**) \(N_{1}=N_{2}=N_{3}=N_{4}=N_{5}\) (**b**) \(N_{1}=N,N_{2}=\sfrac{3}{4}N,N_{3}=\sfrac{1}{2}N,N_{4}=\sfrac{1}{4}N.\)  
The performance measures of the simplest assembly system, composed of three machines and two buffers (line L3), are presented in Figure 5, where agreement between the results, obtained using different approaches, can be seen. Although, the finite state method demonstrates some discrepancies concerning the analytical approach, in the case of uneven distribution and lower range of buffer capacities. Even though these discrepancies are  
\[ST_{A}^{\prime}=ST_{A}-\sum_{h_{1}=0}^{N_{1}}\sum_{h_{2}=0}^{N_{2}}\ldots\sum_ {h_{M-1}}^{N_{M-1}}P_{h_{1}h_{2}\ldots h_{\xi}=0\ldots h_{M-1}}, \tag{15}\]0 Comparison of the performance measures in the case of line L5E: (**a**) \(N_{1}=N_{2}=N_{3}=N_{4}=N\); (**b**) \(N_{1}=N,N_{2}={}^{3}/4N,N_{3}={}^{1}/2N,N_{4}={}^{1}/4N\). more pronounced when compared to, for example, Bernoulli serial or splitting lines [3; 14]), they diminish quickly as the capacity of the buffers increases. The asymptotic values of the performance measures retain the same properties as in the case of Bernoulli serial and splitting lines. Thus, the production rate of line L3 approaches the reliability of the worst machine in the system, which is \(\underset{N\to\infty}{PR}=\min(p_{i})\). Similarly, the work-in-process increases almost linearly when the machine of lower reliability induces a blockage of the machine placed immediately before the respective buffer. Alternatively, the work-in-process quickly converges to a finite limit. The probability of blockage \(BL_{1}\) diminishes quickly as the reliability of machine \(\text{m}_{3}\widehat{\equiv}\text{m}_{\text{A}}\) is greater than the reliability of machine \(\text{m}_{1}\), while \(BL_{2}\) converges to the difference \(p_{2}\)-\(p_{m}\), where \(p_{m}=p_{1}\). Similarly, the probability of starvation \(ST3\) approaches the value \(p_{3}\)-\(p_{m}\), with an increase in buffer capacity. The comparison of results obtained in the case of assembly systems, composed of four machines and three buffers in different arrangements, is presented in Figures 6-8 (lines L4A, L4B, and L4C, respectively). Slight discrepancies between the results of the analytical evaluation and evaluation using the finite state method can be noted, similar to the case of line L3. However, these discrepancies were pronounced to a lesser extent as the systems' states included a greater number of possibilities, thus enabling the finite state method to discretize state spaces more efficiently. As in the previous case, the performance measures approached asymptotic values quickly, while \(\text{\emph{WIP}}_{3}\) grew linearly with the increase in the buffer capacity. The asymptotic values retained the same logic and were equal to \(p_{m}\) in the case of the production rate, \(p_{i}\)-\(p_{m}\) in the case of the probability of blockage (when induced by a subsequent and less reliable machine, otherwise it equals 0), and \(p_{i}\)-\(p_{m}\) in the case of the probability of starvation. The performance measures of assembly systems composed of five machines and four buffers in different arrangements, systems L5A-L5E, are presented in Figures 9-13. The comparison of results, obtained using the finite state method against respective data, determined using the analytical approach, again revealed that the proposed approximative method yielded quite reliable results, with a tendency of an accuracy increase as the system becomes more complex. The asymptotic properties of the performance measures, determined in the case of systems L5A-L5E, remained the same as in the case of line L3 and lines LAA-L4C.