# A Performance Prediction Method Utilizing Time-Dependent Subsystem Transfers between Family Systems
## 4 Time-Dependent Subsystem Transfers between Family Systems  
By using the key performance of subsystems as identification parameters, the family system is divided and a sample collection of subsystems is established. By combining the Markov process' property of no aftereffect, multiple intervals of key performance are determined, enabling the establishment of a time-related inter-family transfer matrix for subsystems.  
The division of family systems involves selecting the key performance indicators for subsystems. Based on factors such as expert experience, design requirements, and computational complexity, appropriate hierarchical divisions can be carried out by considering the interval between optimal performance and failure state values. The performance values during division can exhibit either equal or unequal differences. This example takes the division of five family systems as an example, from the best-state family system to the failure-state family system, which are \(ST_{1}^{t}\), \(ST_{2}^{t}\), \(ST_{3}^{t}\), \(ST_{4}^{t}\), and \(ST_{5}^{t}\).  
Establish the transition matrix:  
At the initial time \(t_{0}\), the number of subsystems belonging to the five family systems is \(ST_{1}^{t_{0}}\), \(ST_{2}^{t_{0}}\), \(ST_{3}^{t_{0}}\), \(ST_{4}^{t_{0}}\), and \(ST_{5}^{t_{0}}\).  
At the next time \(t_{1}\), the number of subsystems belonging to the five family systems is \(ST_{1}^{t_{1}}\), \(ST_{2}^{t_{1}}\), \(ST_{3}^{t_{1}}\), \(ST_{4}^{t_{1}}\), and \(ST_{5}^{t_{1}}\).  
The quantity transferred from family system \(ST_{1}^{t}\) to the five family systems \(ST_{1}^{t}\), \(ST_{2}^{t}\), \(ST_{3}^{t}\), \(ST_{4}^{t}\), and \(ST_{5}^{t}\) is \({}^{1}ST_{1}^{t_{1}}\), \({}^{1}ST_{2}^{t_{1}}\), \({}^{1}ST_{3}^{t_{1}}\), \({}^{1}ST_{4}^{t_{1}}\), and \({}^{1}ST_{5}^{t_{1}}\), respectively; the number of family systems transferred from family system \(ST_{2}^{t}\) to \(ST_{1}^{t}\), \(ST_{2}^{t}\), \(ST_{3}^{t}\), \(ST_{4}^{t}\), and \(ST_{5}^{t}\) is \({}^{2}ST_{1}^{t_{1}}\), \({}^{2}ST_{2}^{t_{1}}\), \({}^{2}ST_{3}^{t_{1}}\), \({}^{2}ST_{4}^{t_{1}}\), and \({}^{2}ST_{5}^{t_{1}}\), respectively; similarly, the number of family systems \(ST_{3}^{t}\), \(ST_{4}^{t}\), and \(ST_{5}^{t}\) transferred to \(ST_{1}^{t}\), \(ST_{2}^{t}\), \(ST_{3}^{t}\), \(ST_{4}^{t}\), and \(ST_{5}^{t}\) can be obtained.  
Therefore, the transfer matrix is as follows:  
\[T_{t_{0}\to t_{1}}=\left[\begin{array}{cccc}\frac{ 1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t }T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1} }}\\ \frac{1}{S_{2}^{t}T_{2}^{t_{1}}}&\frac{2}{S_{1}^{t}T_{2}^{t_{1}}}&\frac{2}{S_{ 1}^{t}T_{2}^{t_{1}}}&\frac{2}{S_{1}^{t}T_{2}^{t_{1}}}&\frac{2}{S_{1}^{t}T_{1}^ {t_{1}}}{S_{2}^{t_{0}}}\\ \end{array}\right] \tag{10}\] \[T_{t_{0}\to t_{1}}=\left[\begin{array}{cccc}\frac{ 1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t }T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1} }}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}\\ \end{array}\right]\] \[T_{t_{0}\to t_{1}}=\left[\begin{array}{cccc}\frac{ 1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t }T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}&\frac{1}{S_{1}^{t}T_{1}^{t_{1} }}&\frac{1}{S_{1}^{t}T_{1}^{t_{1}}}\\ \end{array}\right]\]  
The elements in row a and column b of matrix \(T_{t_{0}\to t_{1}}\) represent the probability of a subsystem belonging to family system a transition to family system b from time \(t_{0}\) to time \(t_{1}\). The probability range is \([0,1]\). The possibility of having zero subsystems in a family system at a certain time implies that the number of each family system will inevitably be zero in the next transition. At this point, the probability is not calculated through division, but is directly marked as 0.