Availability Analysis Reference Example

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images and more targeted search.

As of January 2024, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest references at BlockSim examples and BlockSim reference examples.

This example validates the results for availability analysis BlockSim.

Reference Case

The data set is from example 11.3 on page 259 in the book, An Introduction to Reliability and Maintainability Engineering, by Dr. Charles E. Ebeling, McGraw-Hill, 1997.

Data

A two-component system’s point and interval availability for a 10 hour mission and the steady-state availability for both series and parallel configurations are calculated. The components share the same failure rate and repair rate distributions. The failure and repair rates both follow exponential distributions with a failure rate of 0.1 failures per hour and a repair rate of 0.2 repairs per hour.

Components of the System	Number of Failures, [math]\displaystyle{ \lambda_{f} }[/math] (per hour)	Number of Repairs, [math]\displaystyle{ \lambda_{r} }[/math] (per hour)
Component 1	0.1	0.2
Component 2	0.1	0.2

Result

The point availability is formulated in Equation 11.12 on page 257 in the reference book as:

[math]\displaystyle{ P_{1}(t) = \frac{r}{\lambda + r} + \frac{\lambda}{\lambda + r}e^{-(\lambda+r)t}\,\! }[/math]

The interval availability is formulated in Equation 11.13 on page 258 in the reference book as:

[math]\displaystyle{ A_{t2-t1} = \frac{r}{\lambda+r} + \frac{\lambda}{(\lambda+r)^{2}(t_{2}-t_{1})}\left [e^{-(\lambda+r)t_{1}} - e^{-(\lambda+r)t_{2}} \right ]\,\! }[/math]

The system availability for n independent components in series, each having a component availability of [math]\displaystyle{ A_{i}(t)\,\! }[/math], is given in Equation 11.15 on page 259 in the reference book as:

[math]\displaystyle{ A_{s}(t) = \prod _{i=1}^{n}A_{i}(t)\,\! }[/math]

The system availability for n independent components in parallel, each having a component availability of [math]\displaystyle{ A_{i}(t)\,\! }[/math], is given in Equation 11.16 on page 259 in the reference book as:

[math]\displaystyle{ A_{s}(t) = 1- \prod _{i=1}^{n}(1-A_{i}(t))\,\! }[/math]

Plugging in the numbers to the given equations, the point and interval availability for a 10 hour mission, and steady-state availability are calculated below.

For a series configuration:

[math]\displaystyle{ \begin{align} A_{s}(10) = (0.684)^{2} = 0.468\\ A_{s,0-10} = (0.772)^{2} = 0.596\\ A_{s} = (0.667)^{2} = 0.445\\ \end{align}\,\! }[/math]

And for a parallel configuration:

[math]\displaystyle{ \begin{align} A_{s}(10) = 1- (1- 0.684)^{2} = 0.900\\ A_{s,0-10} = 1- (1- 0.772)^{2} = 0.948\\ A_{s} = 1- (1- 0.667)^{2} = 0.889\\ \end{align}\,\! }[/math]

Results in BlockSim

Availability Analysis Reference Example

Navigation menu

Search