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Conditional Reliability for a System

 

 

Engineers are often asked to estimate the reliability of not just new systems, but also of systems that have already spent some time in the field. If one wants to do this estimation using a bottom-up approach with the help of a reliability block diagram in BlockSim (i.e., estimating system reliability using component-level information of the system), then care must be taken as to how that calculation is performed in order to avoid incorrect results.

In some cases, one can compute the conditional reliability of a system using BlockSim's QCP via one of two methods: 1) Using the conditional reliability function with all blocks set to a current age of zero, or 2) Using the reliability function with all blocks set to a current age of T. In other cases, the conditional reliability can be computed using only the first method. In this article we will discuss some simple systems and the methods that are applicable for each case.

Conditional Reliability Function

Conditional reliability is defined as the probability that a component or system will operate without failure for a mission time, t, given that it has already survived to a given time, T. Mathematically, this is expressed as:

Equation

Conditional Reliability for a Single Component System

For a single component system, one can compute the conditional reliability by using either of the following methods:

Block properties show the current age at time 0 hours.

BlockSim QCP shows the conditional reliability to be 69.629%

Block properties show the current age at 500 hours.

BlockSim QCP shows the conditional reliability to be 69.629%

Conditional Reliability for a System of N Components in Series

For the case of a series system of N new components:

Equation

So, for the case of the series system of N components, the conditional reliability of the system is:

Equation

Next, consider the case of a series system of N components where each component has a current age of T. The system reliability equation is:

Equation

This is identical to the system conditional reliability equation for a series system of N components. Therefore, one can compute the conditional reliability for a series system by using either of the following methods:

Block properties show the current age at time 0 hours.

BlockSim QCP shows the conditional reliability to be 44.29%

Block properties show the current age at 500 hours.

BlockSim QCP shows the conditional reliability to be 44.29%

Conditional Reliability for a System of N Components in Parallel

For the case of a parallel system of N components:

Equation

So, for the case of a parallel system of N components, the conditional reliability of the system is:

Equation

Next, consider the case of a parallel system of N components where each component has a current age of T. The system reliability equation is: 

Equation

The equations for the conditional reliability for a parallel system of N components and the reliability for a parallel system of N components with age T are not equivalent. Using the reliability function with blocks with a current age of T will not provide the conditional reliability of the system for a parallel configuration.

For example, consider the special case of a parallel system of 2 identical components. Then the system conditional reliability becomes:

Equation

The system reliability equation for a system of 2 identical components, each with a current age of T, is given by:

Equation

Let T = 500 hours and t = 300 hours. Assuming both components follow a Weibull distribution with β = 1.5 and η = 1000 hours, then RB(T) = 70.2189% and RB(t + T)) = 48.8927%. So:

Equation

and

Equation

Therefore, changing the current age of the blocks to T and computing the system reliability for a mission time of t will give incorrect results for the conditional reliability for this simple parallel system. Instead, one should compute the conditional reliability for a parallel system by the following method:

Block properties show the current age at time 0 hours.

BlockSim QCP shows the conditional reliability to be 81.07%

Conclusion

There are two ways to compute for the conditional reliability of a series system; however, for any system with a parallel or more complex configuration, one of the two methods will give incorrect results. Therefore, it is recommended that the conditional reliability for a system is always computed using the conditional reliability function in the QCP, rather than changing the current age of the components and using the reliability function.

 
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