# 统计代写|贝叶斯网络代写Bayesian network代考|TAMS22

## 统计代写|贝叶斯网络代写Bayesian network代考|DBNs for Proof Test Interval Phase

The proof test interval is a long period of time; the DBNs include many time slices, denoted by different lower indexes, such as $t-1$ and $t$. A unified DBN structure model for proof test interval phase is constructed, as shown in Fig. 2. The DBNs contains three layers from the top down, that is, failure cause, single channel state, and system state layers. The node IC in failure cause layer denotes the independent cause failure for a single channel, and the node CC denotes the common cause failure for multiple channels. Common cause failure can be referred to as common mode failure and dependent failure. Moreover, common cause failure is the result of an event. These events with dependencies cause a coincidence of failure states of components in two or more separate channels of a redundancy system, leading to the failure of defined systems to perform its intended function. The node $\mathrm{CH}$ in single channel state layer denotes the state of a single channel simultaneously affected by independent cause failure and common cause failure. Therefore, the IC nodes are connected to corresponding $\mathrm{CH}$ nodes and the CC node is connected to all of the $\mathrm{CH}$ nodes via arcs. The node $\mathrm{S}$ in system state layer denotes the state of an entire system consisting of all channels. The nodes IC, CC, and $\mathrm{CH}$ have five states, that is, detected failure (DD), and dangerous undetected failure (DU). For node S, the states SD and SU are combined to form a safety state (SS). Thus, node S has four states, that is, NS, SS, DD, and DU. In adjacent time slices, the causal relationships of IC nodes are illustrated by connecting the corresponding nodes via inter-slice arcs (black solid line or red dash line) and other nodes via auxiliary inter-slice arcs (black dot dash line).

The DBNs with different architectures of KooM and KooMD systems are determined by the number of channels $M$. An $M$ equal to 1 represents the architectures of 1001 and 1001D. The entire DBNs are reduced to the models in red dashed line in Fig. 2 because the common cause failure for the two architectures is meaningless. An $M$ equal to 2 can represent the architectures of $1002,2002,1002 \mathrm{D}$, and $2 \mathrm{oo} 2 \mathrm{D}$. An $M$ equal to 3 can represent the architectures of 1003,2003 , and others. The DBN structures for systems with same channels are identical; however, the parameter model must be different, which means that DBNs for different architectures with same number of channels are determined by conditional probability tables (CPTs) of nodes.

## 统计代写|贝叶斯网络代写Bayesian network代考|CPTs of IC Node in Proof Test Interval Phase

The CPTs of IC node of KooM and KooMD systems are illustrated using a flowchart, indicating the time-dependent rules of failure and repair, as provided in Fig. 4. The rules are defined according to practical condition of engineering. When the current state of IC node is NS, the IC node degrades to $\mathrm{SD}, \mathrm{SU}, \mathrm{DD}$, and DU states exponentially. When the current state of IC node is SD or DD, and if it causes the safe failures of the system, the IC node translates into NS state exponentially with the repair rate $\mu_{\mathrm{SR}}$; otherwise, the IC node translates into NS state exponentially with the repair rate $\mu_{\mathrm{TR}}$. When the current state of IC node is $\mathrm{SU}$, and if the parent nodes of IC cause the safe failure of the system, IC node translates into NS state exponentially with the repair rate $\mu_{\mathrm{SR}}$; if the number of detected failure of parent nodes of IC is equal or larger than 1, the IC node translates into NS state exponentially with the repair rate $\mu_{\mathrm{TR}}$; otherwise, it maintains $\mathrm{SU}$ state. When the current state of IC node is DU, and if the parent nodes of IC cause the safe failure of the system, the IC node translates into NS state exponentially with the repair rate $\mu_{\mathrm{SR}}$; if the number of detected failure of parent nodes of IC is equal or larger than 1, the IC node translates into NS state exponentially with the repair rate $\mu_{\mathrm{TR}}$; otherwise, it maintains a DU state.

The transition relationships of CC node for KooM architecture are provided in Table 1. The CC node has five states: NS, SD, SU, DD, and DU; the failure rates have the relationship that $\lambda_{\mathrm{C}}=\lambda_{\mathrm{SDC}}+\lambda_{\mathrm{SUC}}+\lambda_{\mathrm{DDC}}+\lambda_{\mathrm{DUC}}$. The rules are defined according to practical condition of engineering. When the current state of $\mathrm{CC}$ node is NS, it degrades to SD, SU, DD, and DU states exponentially. When the current state of CC node is SD or SU, it causes the safe failure of the system. Moreover, the system is translated into NS state exponentially with the repair rate $\mu_{\mathrm{SR}}$. When the current state of CC node is DD, it causes the dangerous detected failure of the system, and the CC node translates into to NS state exponentially with the repair rate $\mu_{\mathrm{TR}}$. When the current state of $\mathrm{CC}$ node is DU, it causes the dangerous undetected failure of the system because the self-diagnosis of the system cannot detect the dangerous undetected failure, and the CC node maintains a DU state. To calculate the SIL of a specified system, the safety-case authors should collect and obtain the rates of common mode failure of components using statistical approaches or from failure database directly first. Subsequently, the SIL can be calculated using the proposed MDBNs methodology.

Similar transition relationships of CC node for KooMD architecture are provided in Table 2. The self-diagnosis of the system can translate the dangerous detected failure into safe detected failure. Therefore, when the current state of CC node is DD, the CC node causes the safe failure of the system and translates into NS state exponentially with the repair rate $\mu_{\mathrm{SR}}$.

## 统计代写|贝叶斯网络代写Bayesian network代考|证明测试间隔阶段的DBN

KooM和KooMD系统中不同架构的dbn由通道数量$M$决定。$M$等于1表示1001和1001D的体系结构。整个dbn被简化为图2中红色虚线所示的模型，因为这两个架构的共同原因故障是没有意义的。等于2的$M$可以表示$1002,2002,1002 \mathrm{D}$和$2 \mathrm{oo} 2 \mathrm{D}$的体系结构。$M$等于3可以表示1003、2003和其他的体系结构。信道相同的系统的DBN结构是相同的;但是，参数模型必须不同，这意味着在相同通道数的不同架构下，dbn是由节点的条件概率表(cpt)决定的

## 统计代写|贝叶斯网络代写Bayesian network代考|IC节点在证明测试间隔阶段的cpt

. cpt of IC Node in Proof Test Interval Phase KooM和KooMD系统的IC节点的cpt用流程图表示，显示了故障和修复的时间依赖规则，如图4所示。规则是根据工程实际情况确定的。当IC节点当前状态为NS时，IC节点降级为$\mathrm{SD}, \mathrm{SU}, \mathrm{DD}$, DU状态呈指数级。当IC节点当前状态为SD或DD时，如果导致系统安全故障，则IC节点指数级转换为NS状态，修复率为$\mu_{\mathrm{SR}}$;否则，IC节点以指数形式转换为NS状态，修复率为$\mu_{\mathrm{TR}}$。当IC节点当前状态为$\mathrm{SU}$时，如果IC父节点导致系统安全故障，则IC节点以指数形式转换为NS状态，修复率为$\mu_{\mathrm{SR}}$;如果检测到的IC父节点故障个数大于等于1，则IC节点以指数形式转化为NS状态，修复率为$\mu_{\mathrm{TR}}$;否则，它维护$\mathrm{SU}$状态。当IC节点当前状态为DU时，如果IC的父节点导致系统安全故障，则IC节点指数级转换为NS状态，修复率为$\mu_{\mathrm{SR}}$;如果检测到的IC父节点故障个数大于等于1，则IC节点以指数形式转化为NS状态，修复率为$\mu_{\mathrm{TR}}$; .否则，它保持DU状态 KooM体系结构的CC节点的转换关系如表1所示。CC节点有五种状态:NS、SD、SU、DD、DU;失败率有$\lambda_{\mathrm{C}}=\lambda_{\mathrm{SDC}}+\lambda_{\mathrm{SUC}}+\lambda_{\mathrm{DDC}}+\lambda_{\mathrm{DUC}}$的关系。规则是根据工程实际情况确定的。当$\mathrm{CC}$节点的当前状态为NS时，将按指数级降级为SD、SU、DD和DU状态。当CC节点当前状态为SD或SU时，会导致系统安全故障。此外，系统以指数形式转换为NS状态，修复率为$\mu_{\mathrm{SR}}$。当CC节点当前状态为DD时，会导致系统检测到危险故障，CC节点按指数级转换为NS状态，修复率为$\mu_{\mathrm{TR}}$。当$\mathrm{CC}$节点当前状态为DU时，由于系统自诊断无法检测到危险的未检测故障，CC节点保持DU状态，导致系统处于危险的未检测故障状态。为了计算指定系统的SIL，安全案例作者首先应该使用统计方法或直接从故障数据库中收集和获取部件的共模故障率。随后，可以使用提出的MDBNs方法计算SIL 表2提供了KooMD体系结构中类似的CC节点转换关系。系统的自诊断功能可以将危险的检测故障转化为安全的检测故障。因此，当CC节点的当前状态为DD时，CC节点导致系统安全故障，并指数级转换为NS状态，修复率为$\mu_{\mathrm{SR}}$ .

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