## 4 Illustrated Case
### Performance Analysis of Multistate FMS  
In this subsection, the performance of FMS can be obtained by using the Lz-transform method in which the states' probabilities of CNC machine tools at Station 1, 2 and that of devices in the MHS are calculated by system (10) and (14) under the designed initial state, respectively. Thus, in accordance with Equation (12) and parameters shown in Table 1, the Lz-transform for all three identical CNC machine tools at Station 1 have the same rewritten form (21). From here for simplification we substitute \(p_{mi}\) with \(p_{i}\). \[L_{2}\Big{(}\mathbf{g}_{m}^{S1}(t)\Big{)}=p_{1}^{S1}(t)z^{6}+p_{4}^{S1}(t)z^{5 }+\Big{(}p_{2}^{S1}(t)+p_{3}^{S1}(t)+p_{5}^{S1}(t)+p_{6}^{S1}(t)\Big{)}z^{0}. \tag{21}\]  
We will use \(\Lambda^{S1}\) to denote \(\Big{(}p_{2}^{S1}(t)+p_{3}^{S1}(t)+p_{5}^{S1}(t)+p_{6}^{S1}(t)\Big{)}\). Consequently, the mean instantaneous performance of one CNC machine tool at Station 1 can be easily derived from the modified Lz-transform in the following form:  
\[E(t)=\sum_{\mathbf{g}_{i}^{S1}\geq 0}p_{i}^{S1}(t)\mathbf{g}_{i}^{S1},i=1, \ldots,\,6. \tag{22}\]  
Output performance that is manufactured by the CNC machine tools at Station 1 is depicted in Figure 6.  
Figure 5: Failure criticality importance of each non-working state. Using Equation (13) and the rewritten Lz-transform for one CNC machine tool at Station 1, one can obtain the Lz-transform, associated with the whole of Station 1 as  
\[ \begin{array}{ll}L_{z}\big{\{}G^{\rm S1}(t)\big{\}}=\Omega_{fpar}&\Big{(}L_{z} \big{\{}\mathfrak{g}_{1}^{\rm S1}(t)\big{\}},L_{z}\big{\{}\mathfrak{g}_{2}^{\rm S 1}(t)\big{\}},L_{z}\big{\{}\mathfrak{g}_{3}^{\rm S1}(t)\big{\}}\Big{)}\\ &=P_{1}^{\rm S1}(t)z^{18}+P_{2}^{\rm S1}(t)z^{17}+P_{3}^{\rm S1}(t)z^{16}+P_{4 }^{\rm S1}(t)z^{15}+P_{5}^{\rm S1}(t)z^{12}\\ &+P_{6}^{\rm S1}(t)z^{11}+P_{7}^{\rm S1}(t)z^{10}+P_{8}^{\rm S1}(t)z^{6}+P_{9 }^{\rm S1}(t)z^{5}+P_{10}^{\rm S1}(t)z^{0}.\end{array}  \tag{23}\]  
The meaning of notations are  
\[ \begin{array}{ll}P_{1}^{\rm S1}(t)=p_{1}^{\rm S1}(t)p_{1}^{\rm S1}(t)p_{1}^{ \rm S1}(t);\\ P_{2}^{\rm S1}(t)=3p_{1}^{\rm S1}(t)p_{1}^{\rm S1}(t)p_{4}^{\rm S1}(t);\\ P_{3}^{\rm S1}(t)=3p_{1}^{\rm S1}(t)p_{4}^{\rm S1}(t)p_{4}^{\rm S1}(t);\\ P_{4}^{\rm S1}(t)=p_{4}^{\rm S1}(t)p_{4}^{\rm S1}(t)p_{4}^{\rm S1}(t);\\ P_{5}^{\rm S1}(t)=3p_{1}^{\rm S1}(t)p_{1}^{\rm S1}(t)A^{\rm S1};\\ P_{6}^{\rm S1}(t)=6p_{1}^{\rm S1}(t)p_{4}^{\rm S1}(t)A^{\rm S1};\\ P_{7}^{\rm S1}(t)=3p_{4}^{\rm S1}(t)p_{4}^{\rm S1}(t)A^{\rm S1};\\ P_{8}^{\rm S1}(t)=3p_{1}^{\rm S1}(t)A^{\rm S1}\Lambda^{\rm S1};\\ P_{9}^{\rm S1}(t)=3p_{4}^{\rm S1}(t)A^{\rm S1}\Lambda^{\rm S1};\\ P_{10}^{\rm S1}(t)=\Lambda^{\rm S1}\Lambda^{\rm S1}\Lambda^{\rm S1}.\end{array} \]  
Similarly, the Lz-transform for the two identical CNC machine tools at Station 2 have the same rewritten form as  
\[L_{z}\big{\{}\mathfrak{g}_{q}^{\rm S2}(t)\big{\}}=p_{1}^{\rm S2}(t)z^{12}+p_{ 4}^{\rm S2}(t)z^{10}+\Lambda^{\rm S2}z^{0},\;q=1,2, \tag{24}\]  
where \(\Lambda^{\rm S2}\) represents the \(\big{(}p_{2}^{\rm S2}(t)+p_{3}^{\rm S2}(t)+p_{5}^{\rm S2}(t)+p_{6}^{\rm S2}(t) \big{)}\). Therefore, the Lz-transform associated with the whole of Station 2 can be derived from Equation (19) as  
\[ \begin{array}{ll}L_{z}\big{\{}G^{\rm S2}(t)\big{\}}=\Omega_{fpar}&\Big{(}L_{ z}\big{\{}\mathfrak{g}_{1}^{\rm S2}(t)\big{\}},L_{z}\big{\{}\mathfrak{g}_{2}^{ \rm S2}(t)\big{\}}\Big{)}\\ &=P_{1}^{\rm S2}(t)z^{24}+P_{2}^{\rm S2}(t)z^{22}+P_{3}^{\rm S2}(t)z^{20}+P_{ 4}^{\rm S2}(t)z^{12}+P_{5}^{\rm S2}(t)z^{10}\\ &+P_{6}^{\rm S2}(t)z^{0}.\end{array}  \tag{25}\]  
Figure 6: Output of product manufactured by one machine at Station 1.