# A Finite State Method in the Performance Evaluation of the Bernoulli Serial Production Lines
## 4 Validation and Application of the Developed Theory  
Validation of the developed FSM is performed in cases of the serial Bernoulli lines L\({}_{1}\)-L\({}_{9}\) with probabilities of the state {up} specified in Table 1. Performance measures were calculated using Equations (12), (15), (18), (19) and (22) for each line using the analytical solution (AN), the aggregation procedure (AGG) and the FSM using ProLab, an in-house software developed by the authors, and PSEToolbox (Production System Engineering Toolbox) [2]. Evaluation of the performance measures was performed for the specified lines as well as for their permutations including even and uneven distribution of the buffer occupancy. However, only selected and the most interesting results are presented here. In addition, to enable a simple graphical presentation of the results, all of the buffer occupancies were selected as fractions or multiplies of the first or the last buffer's capacity. Nevertheless, the considered approach was valid in case of the arbitrary buffer occupancies. Buffer occupancy was specified separately for each considered case. As the first step of the FSM validation, a line L\({}_{1}\) was considered to check the fundamental accuracy of the method against the well known analytical solution of the line composed of two machines and one buffer. The performance measured in the case of the line L\({}_{1}\) and its reverse were evaluated using AN, AGG, and FSM approaches and are compared in Figures 4 and 5. Both AGG and FSM approaches agreed very well with the analytical results in this simplest case, except for some negligible discrepancies due to numerical reasons. An excellent agreement could also be noticed in the asymptotic values approached by the performance curves in all three cases. In addition, a reversibility property of the line was excerpted nicely. The performance measures of the line with three machines were evaluated in several cases. The first case, L\({}_{2}\), took into account the original arrangement of machines (see Table 1) and even buffer capacity distribution along the line. Figure 6 presents the distribution of the performance measures as a function of buffer capacity. A good agreement of both AGG and FSM with the analytical results was \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Line** & **M** & **p\({}_{1}\)** & **p\({}_{2}\)** & **p\({}_{3}\)** & **p\({}_{4}\)** & **p\({}_{5}\)** & **p\({}_{6}\)** & **p\({}_{7}\)** & **p\({}_{8}\)** & **p\({}_{9}\)** & **P\({}_{10}\)** \\ \hline L1 & 2 & 0.6 & 0.9 & & & & & & & \\ L1R & 2 & 0.9 & 0.6 & & & & & & & \\ \hline L2 & 3 & 0.6 & 0.75 & 0.9 & & & & & & \\ L2\({}^{\prime}\) & 3 & 0.6 & 0.75 & 0.9 & & & & & & \\ L2\({}^{\prime}\)R & 3 & 0.9 & 0.75 & 0.6 & & & & & & \\ \hline L3A & 4 & 0.6 & 0.7 & 0.8 & 0.9 & & & & & \\ L3B & 4 & 0.7 & 0.6 & 0.8 & 0.9 & & & & & \\ L3C & 4 & 0.9 & 0.7 & 0.6 & 0.8 & & & & & \\ \hline L4 & 5 & 0.6 & 0.675 & 0.75 & 0.825 & 0.9 & & & & \\ L5 & 6 & 0.6 & 0.66 & 0.72 & 0.78 & 0.84 & 0.9 & & & \\ L6 & 7 & 0.6 & 0.65 & 0.7 & 0.75 & 0.8 & 0.85 & 0.9 & & \\ L7 & 8 & 0.6 & 0.643 & 0.686 & 0.729 & 0.772 & 0.815 & 0.858 & 0.9 & & \\ L8 & 9 & 0.6 & 0.637 & 0.674 & 0.711 & 0.75 & 0.787 & 0.824 & 0.862 & 0.9 & \\ L9 & 10 & 0.6 & 0.633 & 0.666 & 0.699 & 0.732 & 0.765 & 0.798 & 0.831 & 0.864 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of considered serial Bernoulli production lines with probabilities of the state {up} (authors’ work). evident, except for slight discrepancies in PR and ST\({}_{3}\) between AN and FSM solution for the lowest level of buffers' capacity. Uneven distribution of the buffer capacity and perturbed arrangement of the machines were evaluated in the case of the line labeled as L\({}_{2^{\prime}}\). The third case, L\({}_{2^{\prime}R}\), considered a reverse of the L\({}_{2^{\prime}}\) line. The performance measures obtained using AN, AGG, and FSM approaches are presented in Figures 7 and 8 as functions of the buffer capacity.