Statisitics of Availability

The following examples are taken from Dr. Kishor Trivedi's book, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd edition, John Wiley & Sons, 2001. These examples come from chapter 10.

Go directly to the examples:

We consider the two-state continuous-time Markov chain (Figure 10.7). In this model, the parameters to be estimated are failure rate

and repair

Figure 10.7: Two-state CTMC
rate

or MTTF and MTTR.The steady-state availability is computed as

where

is the ratio

.
Assume we observed n failure events and repair events, the total failure time is S_n and the total repair time is Y_n. The maximum-likelihood estimator of

and a 100(1-

)% confidence interval for

is given by

The maximum-likelihood estimator of

and a 100(1-

)% confidence interval for

The ratio

is estimated by

and a 100(1-

)% confidence interval for

is given by (

_L,

_U), where

The MLE estimate of A is

.Since the availability A is a monotonically decreasing function of

, the 100(1-

)% confidence interval for A is

Next, we consider the 100(1-

)% upper one-sided confidence interval for A, (A_L,1). A_L is given by

(10.20) Example 10.27

Now assume for a certain system that we observed only one failure event and one repair event, so that n=1,S₁=999 h, and Y₁ = 1 h. The point extimate fo steady-state Availability is

To obtain 95% confidence intervals for A, we can use an F distribution with (2,2) degrees of freedom. Noting that

and

we compute the 95% confidence interval for A as (0.9624, 1). Now assume for a certain system that we observed ten failure events and ten repair events, so that n =10, S₁₀=9990 h, and Y₁₀ = 10 h. The point estimate of availability is unchanged:

However, we use an F distribution with (20,20) degrees of freedom to calculate the confidence interval for A:

thus, the confidence interval for A is narrowed to (0.9975,0.9996). Next we derive the upper one-sided confidence interval for A .In the first case (n=1),we calcuate

So the 95% upper one-sided confidence interval for A is (0.9813,1). In the second case (n =10),we calculate

thus the confidence interval for A is narrowed to (0.9979,1).

Example 10.28

In this example,we investigate how to achieve "5 nines" availability with 95% confidence.First,it is obvious that the point estimate of A should be above "5 nines":

Second, because we have seen the width of the confidence interval narrows as the number of samples increases, we need a sufficient number of samples to make A_L > 0.99999. Consider the 95% upper one-sided confidence interval. From the equation (10.20), we have

We plot the number of samples n against the lower boundary of the interval A_L for different point estimates

in Figure 10.8. We observe that to achieve 95% upper one-sided confidence interval as (0.99999,1), the least number of samples required is

In other words,the lower the point estimate of availability,larger must be the number of samples from which this estimate is computed in order for the given availability confidence interval to be ascertained.

Lower Boundary of One-sided Confidence Interval, A_L

Figure 10.8: Number of samples n versus lower boundary, A_L