## 2 Problem Description and Assumptions
### Steady-State Probability Density Function  
According to the series-parallel production system shown in Figure 1, whether each unit needs maintenance is considered by using the DSSP method [24]. The division results are shown in Figure 2. For a piece of production equipment, the key components are different, and the threshold of each unit described in Figure 2 is different. The deterioration state space is divided into six maintenance groups according to Figure 2. (1) \(M\) (0) represents that no unit needs maintenance. (2) \(M\) (1) represents that only unit 1 needs maintenance. (3) \(M\) (1,2) represents that unit 1 and unit 2 need maintenance. (4) \(M\) (1,3) represents that unit 1 and unit 3 need maintenance. (5) \(M\) (2,3) represents that unit 2 and unit 3 need maintenance. (6) \(M\) (1,2,3) represents that unit 1, unit 2, and unit 3 need maintenance. Since each unit has different importance in the equipment, the maintenance combination of only maintenance unit 2 and only maintenance unit 3 does not exist.  
It is assumed that the deterioration of each unit follows a gamma distribution. That is, the deterioration increment \(\Delta x\) between two consecutive time units of unit \(t\) follows \(\Gamma(\alpha_{i},\beta_{i})\), and the increment \(\Delta x^{t}\) of \(t\) units of time follows \(\Gamma(\alpha_{i}t,\beta_{i})\)[24]. The probability density function can be expressed as:  
\[f_{i}(x)=\frac{\beta^{\alpha_{i}t}x^{\alpha_{i}t-1}e^{-\beta_{i}x}}{\Gamma( \alpha_{i}t)},x>0 \tag{1}\]  
Each unit in the device has a state transition, as shown in Figure 3. When unit \(i\) was maintained in the previous cycle, the initial state of the cycle is 0. When unit \(i\) was not maintained in the previous cycle, the initial state of the cycle is \(y_{i}\). After a period of deterioration, the state of unit \(i\) is \(x_{i}\).  
Figure 2: Deterioration state partition diagram of three-unit series–parallel equipment.  
Combined with the above six maintenance groups and Figure 3, the probability of different maintenance groups and the probability of state deterioration to \((x_{1},x_{2},x_{3})\) are obtained, as shown in Table 2. The probability density function of the joint distribution of system state \(s(x_{1},x_{2},x_{3})\) is simplified to \(s(X)\). \(Y=(y_{1},y_{2},y_{3})\) and \(s(Y)\) represent the steady-state probability density function.  
According to Table 2, the steady-state probability density function expression of the system can be described as follows:  
\[\begin{array}{ll}s(Y)&=\int_{0}^{\min(x_{1},D_{p}^{(1)})}\int_{0}^{\infty} \int_{0}^{\min(x_{3},D_{p}^{(3)})}s(Y)f_{1}(x_{1}-y_{1})f_{2}(x_{2}-y_{2})f_{3 }(x_{3}-y_{3})dy_{1}dy_{2}dy_{3}\\ &+\int_{0}^{\min(x_{1},D_{p}^{(1)})}\int_{0}^{\min(x_{2},D_{p}^{(2)})}\int_{D_{ p}^{(3)}}^{\infty}s(Y)f_{1}(x_{1}-y_{1})f_{2}(x_{2}-y_{2})f_{3}(x_{3}-y_{3})dy_{1}dy_{2}dy_{ 3}\\ &+f_{1}(x_{1})\int_{D_{p}^{(1)}}^{\infty}\int_{0}^{\min(x_{2},D_{p}^{(2)})}\int _{0}^{\min(x_{3},D_{p}^{(3)})}s(Y)f_{2}(x_{2}-y_{2})f_{3}(x_{3}-y_{3})dy_{1}dy_ {2}dy_{3}\\ &+f_{1}(x_{1})f_{2}(x_{2})\int_{D_{p}^{(1)}}^{\infty}\int_{D_{p}^{(2)}}^{\infty }\int_{0}^{\min(x_{3},D_{p}^{(3)})}s(Y)f_{3}(x_{3}-y_{3})dy_{1}dy_{2}dy_{3}\\ &+f_{1}(x_{1})f_{3}(x_{3})\int_{D_{p}^{(1)}}^{\infty}\int_{0}^{\min(x_{2},D_{p }^{(2)})}\int_{D_{p}^{(3)}}^{\infty}s(Y)f_{2}(x_{2}-y_{2})dy_{1}dy_{2}dy_{3}\\ &+f_{2}(x_{2})f_{3}(x_{3})\int_{0}^{\min(x_{1},D_{p}^{(1)})}\int_{D_{p}^{(2)} }^{\infty}\int_{D_{p}^{(3)}}^{\infty}s(Y)f_{1}(x_{1}-y_{1})dy_{1}dy_{2}dy_{3}\\ &+f_{1}(x_{1})f_{2}(x_{2})f_{3}(x_{3})\int_{D_{p}^{(1)}}^{\infty}\int_{D_{p}^{ (2)}}^{\infty}\int_{D_{p}^{(3)}}^{\infty}s(Y)dy_{1}dy_{2}dy_{3}\\ \end{array} \tag{2}\]