# 统计代写|贝叶斯分析代写Bayesian Analysis代考|Deciding When to Stop Testing

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Deciding When to Stop Testing

In addition to estimating the pfd for a complex system from test data, we also want to determine whether the system meets a reliability target and how much testing we need to do to meet some level of confidence in the system’s reliability. Suppose, for example, that our target reliability requirement is a pfd no greater than 0.01 , i.e. once per hundred demands. The BN model for this is shown in Figure 14.3. It is exactly the same as that in Figure 14.1 but with an additional Boolean node pfd requirement whose NPT is specified as
$$\operatorname{if}(p f d<0.01, \text {,”True”, “False”) }$$
that is, it is “True” if it fails less than the required pfd (0.01 in this case, but of course, this value can be changed to any other pfd requirement).
We can use this model to answer the following type of question:
How much testing (i.e., how many demands) would be needed for us to be, say, $90 \%$ confident that the pfd is less than 0.01 , assuming zero failures observed? This is the sort of question that might be asked by the quality assurance or test manager, and is known as a “stopping rule” as it can be used to determine when to stop testing.
Note that in this case we have both a pdf requirement target $t(0.01$ in this case) and a confidence level $\alpha$ ( 0.9 or $90 \%$ in this case).

We can use the sensitivity analysis feature in AgenaRisk (see Sidebar) to calculate the result as shown in Figure 14.4, which shows that 229 successful tests are needed to meet the confidence target of $90 \%$.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Time to Failure for Continuous Use Systems

The TTF is often expressed as the inverse of the failure rate of the system, which itself is either constant, and characterized by the Exponential probability density function, or varies with time, and represented by some other distribution such as the Gamma or Weibull probability density function. Why these functions? Well, the Exponential distribution models situations where the systems do not wear out or degrade with time but nevertheless have a constant failure rate. However, in practice, for many physical systems, wear out occurs over time and the failure probability increases with time and is therefore not constant. In fact wear-out (or at least degradation in reliability) can occur even in software systems that degrade over time because of the presence of newly introduced design faults.

Once you have made a choice of the appropriate distribution (or selected a number of possible choices and decided to perform hypothesis testing, as described in Chapter 12 , to choose the best one) you need to collect and use failure data either from testing or operational field use. This might come from identical systems to the ones you need to estimate and in this case learning the TTF, or failure rate, simply involves parameter learning as discussed in Chapters 10 and 12. Failure data for continuous systems would be the TTF recorded in seconds, hours, and so forth.

However, for novel systems you will often need to look more widely to ask what similar systems might be able to tell you about the reliability of the system at hand. This approach suggests the use of metaanalysis and hierarchical modeling to mix data from specific systems and between families of similar systems, as described in Chapter 10. Here we need to estimate the failure rate parameters for each family or class of system and for the whole superclass of systems.

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Time to Failure for Continuous Use Systems

TTF 通常表示为系统故障率的倒数，它本身要么是常数，并以指数概率密度函数为特征，要么随时间变化，并以其他分布（例如 Gamma 或 Weibull 概率密度）表示功能。为什么有这些功能？好吧，指数分布模拟了系统不会随时间磨损或退化但仍然具有恒定故障率的情况。然而，在实践中，对于许多物理系统来说，磨损会随着时间的推移而发生，并且故障概率会随着时间的推移而增加，因此并不是恒定的。事实上，由于新引入的设计错误的存在，甚至在随着时间的推移而退化的软件系统中也会发生磨损（或至少可靠性下降）。

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