Weibull Distribution
The Weibull distribution is a probability distribution most known for its ability to model time between events. Both time to failure and time to repair fall into this category. It has a few key advantages that make it well suited to this type of modeling:
There is no chance of a negative result. The way a Weibull distribution is constructed, all values below zero have zero probability. This is useful for because it does not allow for the time between two events to be negative, which would be unrealistic.
By modifying the different parameters, distributions of many different shapes can be achieved. This makes the Weibull extremely versatile in its ability to model all sorts of events.
ChiAha's Weibull distribution uses the three parameter model:
Parameter 1 is the scale parameter, which represents the spread that 63% of the TTF values occupy.
Parameter 2 is the shape parameter, which is how the failure rate data is distributed.
Parameter 3 is the location parameter, which shifts the distribution along the x-axis.
Parameter 4 is not used.
Parameter 2 can be interpreted as follows:
A value less than 1 represents a decreasing failure rate over time. This happens if there are significant pre-mature failures leading to a high probability of failure very early in a cycle.
A value of 1 shows that a failure rate that is constant over time. This typically suggests random external events are causing interrupts or failures. This reduces the Weibull to an exponential distribution.
A value greater than 1 indicates that the failure rate increases with time. This is typically seen as some component of the process changes over time in some way. This could manifest as physical wear or a drift in the process over time that increases the likelihood of failure as time moves on. These are typically described as wear-out failure modes. As the value grows above 1, the Weibull distribution reduces from a lognormal distribution to approaching a normal distribution.
The example below shows the probability density function for 4 Weibull distributions. Each distribution has a parameter 1 value of 1.0 and a range of parameter 2 values from 0.5 to 5.0. This helps illustrate the many different forms that a Weibull can take with parameter modification.
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