## 4 Illustrated Case
### Performance Analysis of Multistate FMS
The notations represent:  
\[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\] \[P_{2}^{\text{S2}}(t) =2p_{1}^{\text{S2}}(t)p_{4}^{\text{S2}}(t);\] \[P_{3}^{\text{S2}}(t) =p_{4}^{\text{S2}}(t)p_{4}^{\text{S2}}(t);\] \[P_{4}^{\text{S2}}(t) =2p_{1}^{\text{S2}}(t)\Lambda^{\text{S2}};\] \[P_{5}^{\text{S2}}(t) =2p_{4}^{\text{S2}}(t)\Lambda^{\text{S2}};\] \[P_{6}^{\text{S2}}(t) =\Lambda^{\text{S2}}\Lambda^{\text{S2}}.\]  
Based on Equation (19), the Lz-transform associated with the whole system can be obtained as  
\[L_{z}[G_{FMS}(t)]= \Omega_{fser}\Big{(}L_{z}\Big{\{}G^{\text{S1}}(t)\Big{\}},L_{z} \Big{\{}G^{\text{MHS}}(t)\Big{\}},L_{z}\Big{\{}G^{\text{S2}}(t)\Big{\}}\Big{)} \tag{26}\] \[=\Omega_{fser}\Big{(}P_{1}^{\text{S1}}(t)z^{18}+P_{2}^{\text{S1}} (t)z^{17}+P_{3}^{\text{S1}}(t)z^{16}+P_{4}^{\text{S1}}(t)z^{15}+P_{5}^{\text{ S1}}(t)z^{12}\] \[+P_{6}^{\text{S1}}(t)z^{11}+P_{7}^{\text{S1}}(t)z^{10}+P_{8}^{ \text{S1}}(t)z^{6}+P_{9}^{\text{S1}}(t)z^{5}\] \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]0 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]1  
Due to the FMS possessing the characteristic of providing multiple products or services, it is necessary to carry out the performance analysis based on product or service type. Thus, based on the synchronization rule, Equation (26), and simple algebra, where the powers of \(z\) are found as minimum values of the powers of the corresponding terms, the Lz-transform associated with the FMS can be obtained as  
\[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]2 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]3  
where:  
\[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]4 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]5 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]6 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]7 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]8 \[P_{1}^{\text{S2}}(t) =p_{1}^{\text{S2}}(t)p_{1}^{\text{S2}}(t);\]9 \[P_{2}^{\text{S2}}(t) =2p_{1}^{\text{S2}}(t)p_{4}^{\text{S2}}(t);\]0 \[P_{2}^{\text{S2}}(t) =2p_{1}^{\text{S2}}(t)p_{4}^{\text{S2}}(t);\]1 \[P_{2}^{\text{S2}}(t) =2p_{1}^{\text{S2}}(t)p_{4}^{\text{S2}}(t);\]2 \[P_{2}^{\text{S2}}(t) =2p_{1}^{\text{S2}}(t)p_{4}^{\text{S2}}(t);\]3The mean instantaneous output performance of product per unit time manufactured by the FMS can be derived according to Equation (5) as  
\[ \begin{array}{ll}E(t)=18P_{1}^{\text{FMS}}(t)&+15P_{4}^{\text{FMS}}(t)+12\Big{(} P_{2}^{\text{FMS}}(t)+P_{5}^{\text{FMS}}(t)\Big{)}+10\Big{(}P_{3}^{\text{FMS}}(t)+P_{7}^{ \text{FMS}}(t)\Big{)}\\ &+6\Big{(}P_{3}^{\text{FMS}}(t)+P_{6}^{\text{FMS}}(t)+P_{8}^{\text{FMS}}(t) \Big{)}\\ &+5\Big{(}P_{2}^{\text{FMS}}(t)+P_{6}^{\text{FMS}}(t)+P_{9}^{\text{FMS}}(t) \Big{)}.\end{array}  \tag{28}\]  
The productivity calculations of product made by the FMS are presented in Figure 7.  
The evaluation process and results indicate that the random setup commands had a significant impact on the performance of the FMS. How to predict and design optimal production plans based on the proposed methods for satisfying the demand will be an interesting research topic.