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统计代写|贝叶斯网络代写Bayesian network代考|Calculation of Performance Parameters of SIS

Probability of failure on demand (PFD), average probability of failure on demand ( $\mathrm{PFD}{\text {avg }}$ ), probability of failing safely (PFS), and average probability of failing safely ( $\mathrm{PFS}{\text {avg }}$ ) are main target failure measures of SISs operating in a low-demand mode. PFD is the probability that a SIS fails to perform its intended safety function during a potentially dangerous condition, which is called a dangerous failure. IEC 61508 focuses only on PFD and ignores the issues about system safe failure PFS [41]. The current work studies both dangerous failure and safe failure using the proposed MSBN method. PFD $_{\text {avg }}$ and PFS $_{\text {avg }}$ are the average values of PFD and PFS in a period of time, respectively, and the expressions are obtained by means of the compound trapezium rule as follows:
$$
\mathrm{PFD}{\mathrm{avg}}=\frac{1}{T} \int_0^T \operatorname{PFD}(t) \mathrm{d} t=\lim {N_{\Delta i} \rightarrow \infty} \frac{1}{N_{\mathrm{TI}} \cdot \mathrm{TI}} \sum_{j=1}^{N_{\mathrm{TI}}} \sum_{i=1}^{N_{\Delta t}} \frac{\operatorname{PFD}\left(t_i^j\right)+\operatorname{PFD}\left(t_{i+1}^j\right)}{2}
$$

$$
\mathrm{PFS}{\mathrm{avg}}=\frac{1}{T} \int_0^T \operatorname{PFS}(t) \mathrm{d} t=\lim {N_{\Delta t} \rightarrow \infty} \frac{1}{N_{\mathrm{TI}} \cdot \mathrm{TI}} \sum_{j=1}^{N_{\mathrm{TI}}} \sum_{i=1}^{N_{\Delta t}} \frac{\operatorname{PFS}\left(t_i^j\right)+\operatorname{PFS}\left(t_{i+1}^j\right)}{2}
$$
where $t_i^j$ is the time of the $i$ th time slice in the jth proof test interval TI. $N_{\mathrm{TI}}$ is the combination of time interval TI in total time $T$, and $T$ should be divisible by TI. $N_{\Delta t}$ is the number of small time intervals $\Delta t$ in time interval TI, and TI should be divisible by $\Delta t$. Therefore, the number of time slices for MSBNs in a time interval TI should be $N_{\Delta t}$ plus one.

The most important measure of safety system performance, SIL, can be determined in terms of average probability of a dangerous failure on demand of the safety function [4]. Four discrete integrity levels are associated with SIL: SIL 1, SIL 2, SIL 3, and SIL 4. Higher SIL level means the associated safety level is also higher; consequently, the probability that a system will fail to perform properly is lower.

统计代写|贝叶斯网络代写Bayesian network代考|Effects of Common Cause Weight on the Model Precision

To identify the effects of common cause weight $w$ on the model precision, the following parameters are provided, including failure rate of a single channel, $\lambda=2.0 \times$ $10^{-6} \mathrm{~h}^{-1}$, total time, $T=4038 \mathrm{~h}$, safe failure fraction (SFF), $R_{\mathrm{S}}=0.5$, self-diagnostic coverage for safe (dangerous) failure, $C_{\mathrm{S}}\left(C_{\mathrm{D}}\right)=0.9$, undetected common cause failure fraction, $\beta=0.02$, detected common cause failure fraction, $\beta_{\mathrm{D}}=0.02$, mean time to repair, MTTR $=8 \mathrm{~h}$, mean time to system restoration, MTSR $=24 \mathrm{~h}$, and number of time slices, $N=4097$. When the common cause weight $w$ is $0,0.2,0.4$, $0.6,0.8$, and 1, the PFD, PFS, PFD avg $^{\text {PFS }}$ avg for 2002D and 2003 architectures are plotted, as provided in Fig. 7. As can be seen from the three-dimensional figures, PFD and PFS almost have no change with the increase of common cause weight $w$. As time $t$ increases, PFD and PFS also increase. The two-dimensional figures show that PFD $_{\text {avg }}$ and PFS $_{\text {avg }}$ have little change despite the increase of common cause weight $w$. Above all, common cause weight has little effects on the four target failure measures.

To identify the effects of imperfect proof test and repair on the model precision, the following parameters are provided, including failure rate of a single channel, $\lambda=$ $2.0 \times 10^{-6} \mathrm{~h}^{-1}$, total time, $T=8760 \mathrm{~h}$, safe failure fraction $(\mathrm{SFF}), R_{\mathrm{S}}=0.5$, selfdiagnostic coverage for safe (dangerous) failure, $C_{\mathrm{S}}\left(C_{\mathrm{D}}\right)=0.9$, undetected common cause failure fraction, $\beta=0.02$, detected common cause failure fraction, $\beta_{\mathrm{D}}=0.02$, mean time to repair, MTTR $=8 \mathrm{~h}$, mean time to system restoration, MTSR $=24 \mathrm{~h}$, and number of time slices, $N=4097$. The values of eight variables, namely $\zeta, \delta$, $\theta, \sigma, \alpha, \varepsilon, \mu$, and $\gamma$, reflect the degree of imperfect proof test and repair, and the difference in performance of SISs after the proof test. However, the eight variables have many combinations; thus, researching all of the combinations is rarely practical. Four typical combinations of variables, A, B, C, and D, are researched. The PFD, PFS, PFD $_{\text {avg }}$, PFS $_{\text {avg }}$ for $2002 \mathrm{D}$ and 2003 architectures are plotted, as provided in Fig. 8. The variables in group A represent that the proof test coverage is $100 \%$, and the repair in proof test phase is perfect. The variables in group B represent that the proof test coverage is $100 \%$, and no repair is in proof test phase. The variables in proof test is repaired perfectly; nevertheless, the detected failure by self-diagnosis proof test is repaired perfectly; nevertheless, the detected failure by self-diagnosis remains. The variables in group D represent that the proof test coverage is 0 , and the and $P F S_{\text {avg }}$ for group A are the best, whereas the curves for group $D$ are the worst, for group B and $\mathrm{C}$ are in between best and worst. The results agree with the practical engineering situation.

统计代写|贝叶斯网络代写Bayesian network代考|BAYES-2022

统计代写|贝叶斯网络代写贝叶斯网络代考| SIS的性能参数计算

按需故障概率(PFD)、按需故障平均概率($\mathrm{PFD}{\text {avg }}$)、安全故障概率(PFS)和安全故障平均概率($\mathrm{PFS}{\text {avg }}$)是SISs在低需求模式下运行的主要目标故障度量。PFD是SIS在潜在危险条件下未能执行其预期安全功能的概率,这被称为危险故障。IEC 61508只关注PFD,忽略了关于系统安全故障PFS[41]的问题。目前的工作使用所提出的MSBN方法同时研究了危险失效和安全失效。PFD $_{\text {avg }}$和PFS $_{\text {avg }}$分别为PFD和PFS在一段时间内的平均值,利用复合梯形规则得到表达式:
$$
\mathrm{PFD}{\mathrm{avg}}=\frac{1}{T} \int_0^T \operatorname{PFD}(t) \mathrm{d} t=\lim {N_{\Delta i} \rightarrow \infty} \frac{1}{N_{\mathrm{TI}} \cdot \mathrm{TI}} \sum_{j=1}^{N_{\mathrm{TI}}} \sum_{i=1}^{N_{\Delta t}} \frac{\operatorname{PFD}\left(t_i^j\right)+\operatorname{PFD}\left(t_{i+1}^j\right)}{2}
$$

$$
\mathrm{PFS}{\mathrm{avg}}=\frac{1}{T} \int_0^T \operatorname{PFS}(t) \mathrm{d} t=\lim {N_{\Delta t} \rightarrow \infty} \frac{1}{N_{\mathrm{TI}} \cdot \mathrm{TI}} \sum_{j=1}^{N_{\mathrm{TI}}} \sum_{i=1}^{N_{\Delta t}} \frac{\operatorname{PFS}\left(t_i^j\right)+\operatorname{PFS}\left(t_{i+1}^j\right)}{2}
$$
其中$t_i^j$是第j个证明测试区间TI中第$i$个时间片的时间。$N_{\mathrm{TI}}$是总时间$T$中的时间间隔TI的组合,$T$应该能被TI整除。$N_{\Delta t}$是时间间隔TI中的小时间间隔$\Delta t$的数量,TI应该能被$\Delta t$整除。因此,在TI时间间隔内msbn的时间段数应为$N_{\Delta t}$ + 1。


衡量安全系统性能的最重要指标SIL可以根据安全函数[4]要求的危险故障的平均概率来确定。四个离散完整性级别与SIL相关联:SIL 1、SIL 2、SIL 3和SIL 4。SIL水平越高,意味着相关的安全水平也越高;因此,系统无法正常执行的概率较低

统计代写|贝叶斯网络代写贝叶斯网络代考|共因权重对模型精度的影响


为了识别共因权重$w$对模型精度的影响,提供了以下参数,包括单通道故障率$\lambda=2.0 \times$$10^{-6} \mathrm{~h}^{-1}$,总时间,$T=4038 \mathrm{~h}$,安全故障分数(SFF), $R_{\mathrm{S}}=0.5$,安全(危险)故障自诊断覆盖率,$C_{\mathrm{S}}\left(C_{\mathrm{D}}\right)=0.9$,未检测共因故障分数,$\beta=0.02$,检测共因故障分数,$\beta_{\mathrm{D}}=0.02$,平均修复时间,MTTR $=8 \mathrm{~h}$,平均系统恢复时间,MTSR $=24 \mathrm{~h}$,以及时间片数量,$N=4097$。当共同原因权重$w$为$0,0.2,0.4$, $0.6,0.8$和1时,绘制2002D和2003体系结构的PFD, PFS, PFD avg $^{\text {PFS }}$ avg,如图7所示。从三维图中可以看出,随着共因权重的增加,PFD和PFS几乎没有变化$w$。随着时间$t$的增加,PFD和PFS也会增加。二维图显示,PFD $_{\text {avg }}$和PFS $_{\text {avg }}$在共因权重$w$增加的情况下变化不大。总之,共因权重对四个目标失效度量的影响很小


为了确定不完全证明测试和修复对模型精度的影响,提供了以下参数,包括单个通道的故障率$\lambda=$$2.0 \times 10^{-6} \mathrm{~h}^{-1}$,总时间$T=8760 \mathrm{~h}$,安全故障分数$(\mathrm{SFF}), R_{\mathrm{S}}=0.5$,安全(危险)故障的自诊断覆盖率,$C_{\mathrm{S}}\left(C_{\mathrm{D}}\right)=0.9$,未检测到的共因故障分数,$\beta=0.02$,检测到的共因故障分数,$\beta_{\mathrm{D}}=0.02$,平均修复时间,MTTR $=8 \mathrm{~h}$,系统恢复平均时间,MTSR $=24 \mathrm{~h}$,时间片数量$N=4097$。其中$\zeta, \delta$、$\theta, \sigma, \alpha, \varepsilon, \mu$、$\gamma$八个变量的值分别反映了证明测试和修复的不完善程度,以及证明测试后SISs的性能差异。然而,这八个变量有很多组合;因此,研究所有的组合很少是实际的。研究了四种典型的变量组合,A, B, C, D。图8绘制了$2002 \mathrm{D}$和2003体系结构的PFD、PFS、PFD $_{\text {avg }}$、PFS $_{\text {avg }}$。A组变量表示证明测试覆盖率为$100 \%$,证明测试阶段的修复是完美的。B组变量表示证明测试覆盖率为$100 \%$,证明测试阶段无修复。对校核试验中的变量进行了完美的修复;然而,通过自诊断证明测试检测到的故障被完美修复;然而,通过自我诊断检测到的失败仍然存在。D组变量表示证明测试覆盖率为0,A组的The和$P F S_{\text {avg }}$曲线最好,$D$组曲线最差,B组和$\mathrm{C}$曲线介于最佳和最差之间。计算结果与工程实际情况基本一致

统计代写|贝叶斯网络代写Bayesian network代考

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