Reliability R(t) is the probability that an item will perform 

• a required function without failure
• under stated conditions
• for a stated period of time.

R(t) is a decreasing function over time, because as time passes, more units would fail. Alternatively, if we focus on a distinct unit, we would interprete R(t) as the probability of survival of that unit over time.
The next picture shows a representation of how R(t) could look like:
At time t = 0, no units have failed and the Reliability (probability of survival) is 1. For t > 0, more and more units would fail,  Therefore, the Reliability would tend towards 0 for t -->00. 
R(t) is always a monotonously decreasing function over time.
The most powerful tool for determining reliability is the reliability block diagram.

Up

2. Hazard Rate

Down

The hazard rate h(t) is the first derivative of R(t). h(t) is the change in probability of survival over time. Unlike R(t), h(t) is not necessarily a monotonously decreasing function over time, however, in most practical cases it is.

If we consider two timepoints, t₁ and t₂, then the change in probability between t₁ and t₂ is the probability that a unit will fail between t₁ and t₂. Alternatively speaking, the same change in probability can be interpreted as the fraction of units of the initial population failing between t₁ and t₂.  If we make (t₁ - t₂) small enough, then we obtain the hazard rate h(t).

The negative symbol is neccessary in order to make h(t) a positive figure.
The next picture shows h(t) for the same R(t) of the picture before:


In contrast to R(t), the range of h(t) is between 0 and a specific value determined by the nature of R(t). Furthermore, as mentioned above, h(t) need not necessarily be monotonous as the picture might suggest. 
In reliability analysis, the hazard rate is of less interest.

Up

3. Failure Rate

Down

Remember that the hazard rate is the rate of failure occurences with reference to the initial population size (at t =0). 
What we want is the rate of failure occurences with reference to the current population. This is called the failure rate Lambda(t). The following exemplary statements makes the difference between clear:

Failure rate Lambda(t):   1% of all functioning units are expected to fail within the next hour.
Hazard rate h(t):               0,4% of all  units that were functional at t=0 are expected to fail within the next hour.

--> In order to obtain the failure rate we must simply divide h(t) by R(t):

As with the hazard rate h(t), Lambda (t) need not necessarily be monotonous. Its range is between 0 and a specific value determined by the nature of R(t). The next picture shows Lambda(t) for the same R(t) and h(t) of the pictures before:

Next Topic