# A Finite State Method in the Performance Evaluation of the Bernoulli Serial Production Lines
## 3 Mathematical Modeling
### A Finite State Method  
Let us first take a look at the properties of a stationary distribution (eigenvector associated with the eigenvalue equal to 1) in the case of a three-machine line with arbitrary properties. Limitation to the case of a three-machine line of specific properties will be used here without loss of generality to illustrate the basic property of the stationary distribution. A similar conclusion can be drawn if a different arrangement of the line or a line with another number of machines is considered. Therefore, assume for simplicity that \(p_{1}=0.6\), \(p_{2}=0.7\), and \(p_{3}=0.8\), and that \(N_{1}=N_{2}=10\), Figure 2a. The corresponding stationary distribution can be calculated without significant efforts using the analytical approach [14]. The obtained distribution of the stationary probabilities across the state space of the Bernoulli serial line is presented in Figure 2b including \((N_{1}+1)\times(N_{2}+1)=121\) possible system states. Proportional relationships between stationary probabilities of the considered system can easily be detected. Moreover, it turns out that such proportionality is a common property of Markov chains describing the behavior of the serial Bernoulli production lines. This property is used as the basis to develop FSM as an analytically-based method capable of breaking the dimensionality issues. The eigenvector of the considered three-machine line (see Figure 2) can be decomposed into two classes of states. State 0 and states which are a multiple of \(N_{1}+1=11\) stand for combinations of buffer occupancy \(N_{1}=N_{2}=0\), \(N_{1}=1\) and \(N_{2}=0\), N\({}_{1}=2\) and \(N_{2}=0\),..., \(N_{1}=10\) and \(N_{2}=0\). These combinations represent the external class of the states. States between 0 and 11, as well as states between multiples of \(N_{1}+1=11\) cover combinations \(N_{1}=0\) and \(N_{2}=1\), \(N_{1}=0\) and \(N_{2}=2\),..., \(N_{1}=0\) and \(N_{2}=10\), \(N_{1}=1\) and \(N_{2}\)=1,..., \(N_{1}=10\) and \(N_{2}=10\). These combinations stand for the internal class of the states. Further analysis of the considered problem reveals that the proportional relationship between the states in the external class is governed by the stationary probability distribution determined analytically in the case of a two-machine line: \(p_{1}\)-\(N_{1}\)-\(p_{2}\). Similarly, the proportional relationship between the states in the internal class is governed by the stationary probability distribution determined analytically in the case of a two-machine line: \(p_{1}\)-\(N_{2}\)-\(p_{3}\). Therefore, the corresponding eigenvector can be determined using the well-known and simple analytical solution of the two-machine line problem [2] at considerably lower CPU and memory storage costs using two finite state elements: \(p_{1}\)-\(N_{1}\)-\(p_{2}\) and \(p_{1}\)-\(N_{2}\)-\(p_{3}\). Notice that \(p_{1}\) is a common property of both elements and also that \(p_{1}\)=min\(\{p_{1},p_{2},p_{3}\}\). Let us now consider a general case of the serial Bernoulli production line (see Figure 3) with a machine \(m\), \(p_{m}=\min\{p_{i}\}\), \(i=1,\,2,\,\ldots,\,M\). It is clear that there is a total number of \(m-1\) upstream and \(M-m\) downstream machines. Define \(M-1\) finite state elements, \((e)\), such that a complete line is replaced by a set of \(p_{e}\)-\(N_{e}\)-\(p_{m}\) upstream and \(p_{m}\)-\(N_{e}\)-\(p_{e+1}\) downstream elements. Figure 2: (**a**) The considered Bernoulli serial line (\(p_{1}=0.6\), \(p_{2}=0.7\), \(p_{3}=0.8\), \(N_{1}=N_{2}=10\)) including the pertaining state space and (**b**) the distribution of the stationary probabilities across the state space (authors’ work).