Reliability Importance Example

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This example appears in the System analysis reference.

Reliability Importance Measures for Failure Modes

Assume that a system has failure modes [math]\displaystyle{ A\,\! }[/math], [math]\displaystyle{ B\,\! }[/math], [math]\displaystyle{ C\,\! }[/math], [math]\displaystyle{ D\,\! }[/math], [math]\displaystyle{ E\,\! }[/math] and [math]\displaystyle{ F\,\! }[/math]. Furthermore, assume that failure of the entire system will occur if:

• Mode [math]\displaystyle{ B\,\! }[/math], [math]\displaystyle{ C\,\! }[/math] or [math]\displaystyle{ F\,\! }[/math] occurs.
  
• Modes [math]\displaystyle{ A\,\! }[/math] and [math]\displaystyle{ E\,\! }[/math], [math]\displaystyle{ A\,\! }[/math] and [math]\displaystyle{ D\,\! }[/math] or [math]\displaystyle{ E\,\! }[/math] and [math]\displaystyle{ D\,\! }[/math] occur.
  

In addition, assume the following failure probabilities for each mode.

• Modes [math]\displaystyle{ A\,\! }[/math] and [math]\displaystyle{ D\,\! }[/math] have a mean time to occurrence of 1,000 hours (i.e., exponential with [math]\displaystyle{ MTTF=1,000).\,\! }[/math]
  
• Mode [math]\displaystyle{ E\,\! }[/math] has a mean time to occurrence of 100 hours (i.e., exponential with [math]\displaystyle{ MTTF=100).\,\! }[/math]
  
• Modes [math]\displaystyle{ B\,\! }[/math], [math]\displaystyle{ C\,\! }[/math] and [math]\displaystyle{ F\,\! }[/math] have a mean time to occurrence of 700,000, 1,000,000 and 2,000,000 hours respectively (i.e., exponential with [math]\displaystyle{ MTT{{F}_{B}}=700,000\,\! }[/math], [math]\displaystyle{ MTT{{F}_{C}}=1,000,000\,\! }[/math] and [math]\displaystyle{ MTT{{F}_{F}}=2,000,000).\,\! }[/math]
  

Examine the mode importance for operating times of 100 and 500 hours.

Solution

The RBD for this example is shown next:

The first chart below illustrates [math]\displaystyle{ {{I}_{{{R}_{i}}}}(t=100)\,\! }[/math]. It can be seen that even though [math]\displaystyle{ B\,\! }[/math], [math]\displaystyle{ C\,\! }[/math] and [math]\displaystyle{ F\,\! }[/math] have a much rarer rate of occurrence, they are much more significant at 100 hours. By 500 hours, [math]\displaystyle{ {{I}_{{{R}_{i}}}}(t=500)\,\! }[/math], the effects of the lower reliability components become greatly pronounced and thus they become more important, as can be seen in the second chart. Finally, the behavior of [math]\displaystyle{ {{I}_{{{R}_{i}}}}(t)\,\! }[/math] can be observed in the Reliability Importance vs. Time plot. Note that not all lines are plainly visible in the plot due to overlap.