The Reliability distribution function R(t) represents the reliability of a system for a given time “t”. In other words, it gives the probability that the system will survive till time “t” provided it was in working condition at time t = 0. Various Reliability distribution functions are available such as Exponential (also known as the Constant failure rate model or CFR), Weibull, Normal, Log normal, etc.

For correct reliability modelling, it is important to choose an appropriate Reliability distribution function which will truly represent the probability distribution of the time to failure of the system. Choice of the appropriate distribution function is determined by the time to failure data of the system collected from the field, and also considering the distribution function for similar systems used in the past.

Exponential distribution function is by far the simplest and the most commonly used distribution function. Sometimes it is chosen due to its simplicity even when other functions would be the better fit solutions. Look at the results it produces, which also appear to be intuitively correct:

- When 2 or more components are connected in series, the overall failure rate of the system is obtained by simply adding the failure rates of the individual components; remember the formula λ = λ1 + λ2 + … + λn? The resulting system also follows the CFR model.
- Mean time to failure MTTF = 1/λ

These properties look quite obvious. After all, if 2 components are connected in series, meaning thereby that the system will fail if any one component fails; overall failure rate of the system will naturally be equal to the sum of the failure rates of the 2 components. Similarly, the second property, MTTF = 1/λ appears to be intuitively correct too.

Do you think these properties are applicable only to the Exponential distribution, or universally applicable to all distribution functions?

The question arises, can the “constant failure rate model” be applied in all cases? Unfortunately no. This function is suitable in case of random failures such as failures occurring due to external random conditions like shock, surge, etc., but what about the failures occurring due to slow deterioration of the component with time? In such circumstances, a time dependent failure model will be more suitable.

Which is the time dependent failure model most popularly used and which can also model all the three sections of the bathtub reliability curve by suitably choosing the values of its parameters – (i) Falling failure rate due to infant failures, (ii) Constant failure rate – the useful life of the component and (iii) the rising failure rate due to ageing?

Incidentally, if 2 components following the Constant failure rate model are connected in parallel (i.e. the system will be operational if at least one component is working), the MTTF increases from 1/λ to 3/2λ. Do you think the failure rate of the resultant system will be simply 1/MTTF, i.e. 2λ/3, or will it be time dependent?

And lastly, if the Reliability distribution function and the Failure distribution function are exponential as shown in the figure, why is the model called a Constant failure rate model?