# Unavailability/probabilities over a period of time

1. Mar 10, 2009

### notap

Hey, can anyone give me some help?

I'm trying to calculate the time unavailable of a system due to a component failure over a certain period of time. I know the probability of failure of this component, the time to be considered (assume 1 year) and how long the component is unavailable for due to a failure.

How can i work out how long this component causes the system to be unavailable for?

Thanks,

--notap

2. Mar 10, 2009

### John Creighto

Disclaimer: If you are using this for engineering applications please consult a proper text in saftey instrumented systems as I'm doing this off the top of my head.

Let:
$$P(\Delta t)$$ be the probability of the component not failing in a time period of $$\Delta t$$.

There are $${365 \over \Delta t}$$ time steps in one year.

What we want to find is the expected time the system will be unavailable. That is we are computing the expectation value.
http://en.wikipedia.org/wiki/Expected_value

If the system fails at time t. It is unavailable for f(t)=365-t.

To compute the probability of failure we are summing each possible failure time by the probability it will fail at that time. In sumation form this is written as:

$$<f(t)>=\sum_{n=1}^{365 / \Delta t}}f(n \Delta t)(1-P(\Delta t)^n)$$

This can also be written in integral form (Proof left as exercise).

$$<f(t)>=\int_{0}^{365}}f(t)exp(-t \lambda)$$

where:

$$\lambda=\mathop{\lim }\limits_{\Delta t \to 0} {1-P(\Delta t) \over \Delta t}$$