## 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|贝叶斯网络代写贝叶斯网络代考| SIS的性能参数计算

$$\mathrm{PFD}{\mathrm{avg}}=\frac{1}{T} \int_0^T \operatorname{PFD}(t) \mathrm{d} t=\lim {N_{\Delta i} \rightarrow \infty} \frac{1}{N_{\mathrm{TI}} \cdot \mathrm{TI}} \sum_{j=1}^{N_{\mathrm{TI}}} \sum_{i=1}^{N_{\Delta t}} \frac{\operatorname{PFD}\left(t_i^j\right)+\operatorname{PFD}\left(t_{i+1}^j\right)}{2}$$

$$\mathrm{PFS}{\mathrm{avg}}=\frac{1}{T} \int_0^T \operatorname{PFS}(t) \mathrm{d} t=\lim {N_{\Delta t} \rightarrow \infty} \frac{1}{N_{\mathrm{TI}} \cdot \mathrm{TI}} \sum_{j=1}^{N_{\mathrm{TI}}} \sum_{i=1}^{N_{\Delta t}} \frac{\operatorname{PFS}\left(t_i^j\right)+\operatorname{PFS}\left(t_{i+1}^j\right)}{2}$$

## 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

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