 Reliability HotWire

Issue 62, April 2006

Reliability Basics

### Component Reliability Importance in System Reliability Analysis

Once the reliability of a system has been determined, engineers are often faced with the task of identifying the component(s) that cause the most problems to the system in order to prioritize improvements in the design and channel resources and efforts of system improvement to the areas that will have the most impact on the system's performance. In simple systems such as a series system, it is easy to identify the weak components. In more complex systems, however, this becomes quite a difficult task. Identifying the weakest component is an exercise that is based on understanding both the reliability of each component and the roles the components play reliability-wise in the system, which is determined by their location in the reliability block diagram (RBD). For complex systems, the analyst needs a mathematical approach that will provide the means of identifying and quantifying the importance of each component in the system. (Note: the cost of improving the reliability of the component is not considered. Cost of improvement is covered in the Reliability Allocation section of the online Life Data Analysis Reference.) In this article, BlockSim is used to generate plots that assist in understanding the reliability importance of each component in the analyzed system.

Using reliability importance measures is one method of identifying the relative importance of each component in a system with respect to the overall reliability of the system. The reliability importance, IR, of component i in a system of n components is given by Leemis . Eqn. (1)

Where:

• Rs is the system reliability

• Ri is the component reliability

The value of the reliability importance given by Eqn. (1) depends on both the reliability of a component and its corresponding position in the system. It can be observed that for a simple series system the rate of increase of the system reliability is greatest when the least reliable component is improved.

Let us consider a simple example of three static blocks in series. The blocks have the following reliability values. By increasing the reliability of a component i by DRi , the system reliability increases. In this example, Component #1 has the largest reliability importance in the system relative to the other two components, as shown in the next figure. Figure 1: Rate of change of system reliability when increasing the reliability of each component

The same conclusion can be drawn by using Eqn. (1) and obtaining the reliability importance in terms of a value for each component.

Using BlockSim, the reliability importance values for these components can be calculated with Eqn. (1). Using the plot option and selecting the "Static Reliability Importance" plot type, the plot shown next can be obtained. Note that the time input required to create this plot is irrelevant for this example because the components are static. Figure 2: Static Reliability Importance plot

The values in the y-axis of the above plot are derived using Eqn. (1). The reliability equation for this series system is given by: Eqn (2)

Taking the partial derivative of Eqn. (2) with respect to R1 yields: Thus, the reliability importance of Component # 1 is IR1=0.72. The reliability importance values for Components 2 and 3 are obtained in a similar manner. Since Component # 1 is the weakest component, efforts should be focused on improving it before other components.

The same concept can be extended to time-dependent (time varying reliability) components. That is, if Rs(t) = R1(t) R2(t) R3(t), then one could compute at any time x or  This is quantified in Eqn. (3). Eqn. (3)

In turn, this can be viewed as either a static plot (at a given time) or as time varying plot, as illustrated in the next figures. Specifically, Figures 3, 4 and 5 present the analysis for three components configured reliability-wise in series following a Weibull distribution with β = 3 and η1 = 1,000, η2 = 2,000 and η3 = 3,000. Figure 3 shows a bar chart of while Figure 4 shows the in BlockSim's "tableau chart" format. In this chart, the area of the square is , thus bigger areas indicate more importance. Lastly, Figure 5 shows the (t) vs. time. Figure 3: Static Reliability Importance plot at t = 1,000. Figure 4: Static Reliability Importance tableau plot at t = 1,000 Figure 5: Reliability Importance vs. time plot

The above analysis can be applied to complex systems. Let us consider the following system's RBD. The next table shows the components' reliability distribution models. Figure 6 illustrates (t = 100). It can be seen that even though B, C and F have a much lower rate of occurrence, they are much more significant at 100 hours. By 500 hours, (t = 500), the effects of the lower reliability components become greatly pronounced and thus they become more important, as can be seen in Figure 7. Finally, the behavior of (t) can be observed in Figure 8. Note that not all lines are plainly visible in Figure 8 due to overlap. Figure 6: Plot of (t = 100) Figure 7: Plot of (t = 500) Figure 8: Plot of (t)

References:

Leemis, L.M., Reliability - Probabilistic Models and Statistical Methods, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1995 