# A Performance Prediction Method Utilizing Time-Dependent Subsystem Transfers between Family Systems
## 3 Cluster, Family, and Subsystems
### Construction of Family System  
The establishment and division of the family system plays a pivotal role in predicting the lifespan of a cluster system. The division of the family system is contingent upon inter-subsystem relationships, while the contribution of each subsystem to the load is determined by both the overall load of the cluster system and the number of subsystems. The fundamental concept in the process of partitioning family systems, as per the definition of family systems, is to categorize them based on key performance parameters. This approach is necessitated by the intricate and interconnected subjective and objective environments involved in cluster system operations, which are characterized by parameter variations and uncertainties that render it challenging to ascertain the laws and mechanisms governing specific environmental factors' impact on system functioning. The characterization of subsystems' key performance arises from the complex interplay among diverse environmental factors. Therefore, employing key performance parameters as a basis for classification not only simplifies computations but also enhances credibility. Based on the definition and classification of family systems, it is evident that subsystems within the same family system demonstrate the closest critical performance. In other words, in terms of critical performance, each subsystem within a given family system can serve as a reference for other subsystems in the family. This fundamental theory of cluster systems addresses complex system design and applications, most of which are still in their developmental stages. The ideas aim to tackle challenging problems with limited prior experience to draw from. In summary, the division of family systems plays a crucial role in the reliability analysis of cluster systems. This section introduces the k-means clustering method, which employs subsystems' key performance as an identification parameter for partitioning the family system. The cluster system description is \(C=\{\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\ldots,\mathbf{s}_{n}\}\); this means that there are \(n\) subsystems in the cluster system. Meanwhile, each subsystem has \(m\) key features, recorded as an m-dimensional feature vector \(\mathbf{s}_{i}=(s_{i1},s_{i2},\ldots,s_{im})\). Using clustering algorithms, cluster system \(C\) is divided into \(k\) disjusted family systems, written separately as \(\{\bar{F}_{l}|l=1,2,\ldots\ldots,k\}\), where \(F_{l^{\prime}l^{\prime}\cap_{l^{\prime}\
eq i}\bar{F}_{l}=\varnothing\}\) and \(C=\cup_{l=1}^{k}\mathbb{C}_{l}\), and \(\lambda_{j}\in\{1,2,\ldots\ldots,k\}\) is used as the family system label representing subsystem \(s_{j}\), meaning \(s_{j}\in F_{\lambda_{j}}\). Therefore, the k-means clustering results of family system partitioning can be described by n-dimensional vectors with family system labels, meaning \(\mathbf{\lambda}=(\lambda_{1},\lambda_{2},\ldots\ldots,\lambda_{n})\). In simple terms, the k-means algorithm minimizes the square difference of the partitioning results of the family system. \[E=\sum_{i=1}^{k}\sum_{s\in\bar{F}_{i}}\left\|\mathbf{s}-\mu_{i}\right\|_{2}^{2} \tag{4}\]  
where \(\mu_{i}=\frac{1}{|\bar{F}_{i}|}\sum_{s\in\bar{F}_{i}}\mathbf{s}\) is the average vector of \(F_{i}\). Equation (4) intuitively reflects the compactness of the subsystems and mean vectors in the family system. The smaller the \(E\), the higher the similarity of the subsystems in the family system. In summary, the process of using the k-means clustering method to partition family systems under the theory of cluster systems can be divided into five steps, as shown in Figure 1.  
Step 1: Enter the cluster system dataset \(C=\{\mathbf{s}_{1},\mathbf{s}_{2},\ldots\ldots,\mathbf{s}_{n}\}\); then, randomly select \(k\) subsystems as the initial average vector, with a mean of \(\{\mu_{1},\mu_{2},\ldots\ldots,\mu_{k}\}\). Step 2: Calculate the distance between \(s_{j}\) and \(\mu_{i}\) using Equation (5) based on Euclidean distance.