Issue 68, October 2006

How to Simplify Complex RBDs With a Commonly Used Distribution

System reliability analysis relies on deriving a system reliability equation (cumulative density function, cdf, and/or probability density function, pdf) from the Reliability Block Diagram (RBD) of a system. The derived equations can be used to calculate various reliability metrics. The pdf and cdf can easily become complex equations as the RBDs become more complex and the distributions of the blocks in the RBDs become more varied.

Often, reliability analysts need to approximate a system's RBD with a simple distribution. There are many reasons for this need, such as:

• An RBD might be a subsystem of a more complex system. Including diagrams of all of the subsystems might complicate the analysis of the system. Therefore, approximating the complex subsystems using one distribution for each subdiagram would help in simplifying the analysis.
• Customers who use your system in their applications or designs may want to know which distribution best fits your system's reliability in order to be able to treat it as a black box modeled with a commonly used distribution. Therefore, after completing your RBD construction, you can approximate the system’s model with a simple distribution that can be reported to your customer.

In this article, we explain an approach for simplifying RBDs with a simple representative common distribution using BlockSim 6 and Weibull++ 7.

Introduction:
System reliability analysis requires that a model be obtained to describe the system's reliability-wise configuration. Let us consider the following system’s RBD.

Figure 1 – Example RBD

The blocks in the above RBD follow the following distributions:

A: Weibull with β = 1.5; η = 1000
B: Exponential with μ =10000
C: Normal with μ =250; σ = 50
D: Weibull with β = 1.5; η = 1000

The reliability equation that describes the above system can be derived using rules that govern system reliability calculations. It can also be obtained using BlockSim 6 as follows:

Figure 2 – System Reliability Equation for the System in Figure 1

Once the reliability equation has been obtained, the various life distributions representing the blocks in the system can be incorporated into the equation. The system's reliability equations then becomes :

[Click to Enlarge]

The above equation expresses the reliability function, R(t), of the system. This is related to the unreliability function, F(t), by R(t)=1-F(t). F(t) is also called the cumulative density function (cdf). F(t). The derivative of F(t) gives us the probability density function (pdf) f(t).

Once the cdf and pdf equations are known, other reliability metrics calculations become possible. For example, the failure rate of the RBD mentioned above can be obtained as follows:

Approximating RBDs with a Common Distribution
Simulation can be used as an alternative approach to the analytical approach described above for obtaining the pdf equation of a system.

Since R(t) is a monotonically decreasing function, there is always only one t value that can result in a certain R value. Further, R(t) is always a value between 0 and 1 (0 and 100%). If R(t) is equated to a random number, r, where r lies between 0 and 1, then a single time value can be derived by solving for the t value such that R(t) = r using the equation of R(t), such as the one in Figure 2. Relying on this simple idea, we can generate a list of n random times to failure values (t₁, t₂, … t_n), that corresponds to random reliability values (r₁, r₂, …, r_n). This approach can also be described using the following algorithm.

For i=1 to n

• Generate a random value from a uniform distribution between 0 and 1, r_i=Unif (0,1)
• Solve for t_i, such that R(t_i)=r_i

Using the list of random failure times, t_i, a distribution (pdf) can be fit to the data using life data analysis methods.

The pdf and its equivalent cdf can then be used to calculate the various common reliability metrics.

Using BlockSim 6
The above simulation analysis can be performed in BlockSim 6. First, the RBD of interest should be constructed (refer to Figure 1). Let this be referred to as "Lower."

Then another RBD (we call it "Upper") should be constructed having just one block. This block can then be set to represent the first RBD (Lower) as a subdiagram using the Subdiagram page of the General tab in the Block Properties window.

BlockSim 6 offers two options to make the connection between the upper and the lower RBDs:

1. Using reliability/maintainability characteristics from Blocks in linked diagram. This option allows the upper diagram to use all the properties in the lower subdiagram.
2. Using reliability/maintainability characteristics from Subdiagram block properties and using the Approximate Failure Distribution button to allow the lower RBD to be simplified with a simple distribution.

When you click the Approximate Failure Distribution button, the following window appears.

This window can be used to fit your distribution of choice. For example, you can choose to fit the 2-parameter Weibull to the lower RBD (select the Weibull distribution and click Compute). The fitted parameters are obtained as shown next.

This tool also offers a Distribution Wizard designed for assisting in the selection of the distribution that best fits the generated random failure times.

Using Weibull++
We can take this analysis a little further and transfer the generated failure times to Weibull++ 7 in order to view the data and perform a more advanced life data analysis. The data transfer can be achieved by checking the Show Weibull++ box and entering the Number of Sampling Points.

Once the data set has been transferred to Weibull++ 7, various life data analyses can be performed, such as selecting an analysis method, Regression or Maximum Likelihood Method (MLE); comparing different distributions; assessing goodness of fit; plotting, etc.

The following is a plot of the data fitted with a 2-parameter Weibull distribution.

Conclusion
This article presented a methodology that relies on using simulation to generate failure times data for an RBD in order to fit a simple distribution to the RBD. This is useful when an RBD needs to be simplified with a single common distribution.

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