# A Finite State Method in the Performance Evaluation of the Bernoulli Serial Production Lines
## 3 Mathematical Modeling
### Performance Measures  
Performance measures are of particular interest for the PSE and Markov chain framework. It is, therefore, of great significance to develop simple expressions for the production rate (\(PR\)), work-in-process (_WIP_) for each buffer, as well as the probabilities of blockage (\(BL\)) and starvation (\(ST\)) for each machine [2]. The production rate, \(PR\), is the average number of parts produced by the last machine of the production line. Therefore, it is an intersection of events [\(m_{M}\) is in state up] and [\(b_{M-1}\) is not empty], i.e.:  
\[PR=P\big{(}\{m_{M}\text{ is in state up}\}\cap\{b_{M-1}\text{ is not empty}\}\big{)}. \tag{9}\]  
Equation (9) can  
Similarly, the work-in-process as the average number of parts contained at buffer \(b_{i}\) is equal to:  
\[\mathit{WIP}_{i}=\mathit{WIP}_{e}=\sum_{i_{1}=0}^{N_{1}}\sum_{i_{2}=0}^{N_{2}} \sum_{i_{3}=0}^{N_{3}}\dots\sum_{i_{M-1}=0}^{N_{M-1}}i_{e}P_{i_{1}i_{2}i_{3}\dots i _{M-1}}. \tag{14}\]  
Again, using Equation (5), the work-in-process can be expressed as:  
\[\mathit{WIP}_{e}=P_{000\dots 0}\sum_{i_{1}=0}^{N_{1}}\sum_{i_{2}=0}^{N_{2}} \sum_{i_{3}=0}^{N_{3}}\dots\sum_{i_{M-1}=0}^{N_{M-1}}\left(i_{e}\prod_{e=1}^{M -1}\frac{P_{i_{e}}^{(e)}}{P_{0}^{(e)}}\right). \tag{15}\]  
Substitution of Equation (8) into Equation (15) after algebraic manipulation and by using identity \(\sum\limits_{i_{e}}^{N_{e}}P_{i_{e}}^{(e)}=1\) yields:  
\[\mathit{WIP}_{e}=\sum_{i_{e}=0}^{N_{e}}i_{e}P_{i_{e}}^{(e)}=\mathit{WIP}_{i}. \tag{16}\]  
Consequently, the work-in-process associated with the \(i^{\text{th}}\) buffer depends solely on the elementary eigenvector \(\left\{P^{(e)}\right\}\) obtained using the corresponding finite state element.  
A machine blockage is commonly formulated using union and intersections of several events, i.e.:  
\[\mathit{BL}_{i}=P(\{m_{i}\text{ is up}\}\cap\{m_{i+1}\text{ is down}\}\cap\{b_{i}\text{ is full}\})+ \tag{17}\]  
The probability of blockage, \(\mathit{BL}_{i}\), of the machine \(m_{i}\), Equation (17), can be expressed using the finite state elements as:  
\[\mathit{BL}_{i}=p_{i}(1-p_{i+1}+\mathit{BL}_{i+1})\sum_{i_{1}=0}^{N_{1}}\sum_ {i_{2}=0}^{N_{2}}\sum_{i_{3}=0}^{N_{3}}\dots\sum_{i_{M-1}=0}^{N_{M-1}}P_{i_{1 }i_{2}i_{3}\dots(i_{e}=N_{e})-i_{M-1}}, \tag{18}\]  
which after inserting Equations (5) and (8), yields:  
\[\mathit{BL}_{i}=p_{i}(1-p_{i+1}+\mathit{BL}_{i+1})P_{N_{e}}^{(e=i)}. \tag{19}\]  
Since the last machine can never be blocked, i.e., \(\mathit{BL}_{M}=0\), the probability of the blockage of the penultimate machine equals:  
\[\mathit{BL}_{M-1}=p_{M-1}(1-p_{M})P_{N_{e}}^{(M-1)}. \tag{20}\]  
The probability of starvation, \(\mathit{ST}_{i}\), of the machine \(m_{i}\) can be determined using the intersection of events:  
\[\mathit{ST}_{i}=P(\{m_{i}\text{ in state up}\}\cap\{b_{i-1}\text{ is empty}\}). \tag{21}\]  
Using the finite state elements, Equation (20) can be formulated as:  
\[\mathit{ST}_{i}=p_{i}\sum_{i_{1}=0}^{N_{1}}\sum_{i_{2}=0}^{N_{2}}\sum_{i_{3}=0 }^{N_{3}}\dots\sum_{i_{M-1}=0}^{N_{M-1}}P_{i_{1}i_{2}i_{3}\dots(i_{e}=0)\dots i _{M-1}}. \tag{22}\]  
Again, using Equations (5) and (8), the probability of starvation of the \(i^{\text{th}}\) machine is equal to:  
\[\mathit{ST}_{i}=p_{i}P_{0}^{(e=i-1)}. \tag{23}\]As the first machine cannot be starved, the probability of starvation of the machine \(m_{1}\) takes a zero value.  
Finally, it is important to point out that a complete reconstruction of the stationary distribution of the considered line is not necessary. It is sufficient to evaluate only the stationary distributions at the finite state element level and to collect the first and the last components to determine the production rate as well as the probabilities of blockage and starvation. Evaluation of the work-in-process, however, requires more detailed data on all components of the considered finite state element. The FSM approach is, therefore, very convenient for practical and daily factory-floor applications.