Reliability terminology

The bathtub curve is typically used as a visual model to illustrate the three key periods of product failure rate and not calibrated to depict a graph of the expected behavior for a particular product family. It is rare to have enough short-term and long-term failure information to actually model a population of products with a calibrated bathtub curve, so estimations are made using reliability modeling.

There are three primary phases of semiconductor product lifetime:

• Early life failure rate (or infant mortality): This phase is characterized by a relatively higher initial failure rate, which decreases rapidly.
• Normal life: This phase consists of a relatively constant failure rate, which remains stable over the useful lifetime of the device. The failure rate is described in units of "FITs", or alternatively as a "Mean Time Between Failures" (MTBF) in hours.
• Wearout phase: This represents the point at which intrinsic wear-out mechanisms begin to dominate and the failure rate begins increasing exponentially. The product lifetime is typically defined as the time from initial production until the onset of wear-out.

For a given sample size n, there will be m failures after t hours
Operating hours – If ‘n’ operated for ‘t’ hours before the failure-count ‘m’ was noted, then 

FIT – Failures in Time, number of units failing per billion operating hours. You can use TI’s Reliability Estimator to get a FIT rate for most TI parts.

DPPM – Defective Parts Per Million, also known as number of failing units per million shipped.

MTTF (Mean Time To Fail) = (t1+t2+t3+….tm)/m

It is the average time for a failure to occur. MTTF is used in context of non-repairable systems.

T50 (Median Time To Fail) = Time for 50 percent of units to fail.

Half the fails happen before T50; the other half after T50. Used mostly in statistical treatment of failure distributions. If the fail times are normally distributed, then T50 is the same as MTTF.

MTBF (Mean Time Between Fails) = [t1 + (t2- t1) + (t3 – t2) ….(tm – tm-1) ]/m = tm/m

MTBF is the average time between successive failures. MTBF is used for repairable systems. It is truly a Mean Up-time Between Failures since it does not include the time to repair.

Probability distributions are graphical or mathematical representations of the failing fraction of units with time. For a limited sample of discrete failures, this distribution is commonly shown as a histogram. The profile shape of this distribution is represented mathematically by a Probability Distribution Function (PDF).

Probability density function f(t): 
This function represents the probability of failure at a specific time t, as f(t).Δt
Area f(t).Δt can also predict the expected number of fails at a specific time t.
Cumulative distribution function F(t):
It represents the cumulative number of failures up to a given time ‘t’.

Failure rate is the conditional probability of failure at time t, i.e. probability of fail at time t, given that the unit has survived untill then.

It can also be expressed as the number of units failing per unit time, in a time-interval between t and t+ΔT, as a fraction of those that survived to time t.

As shown in the figure, the change of fail-rate with time starts out high during the early life of the product and declines rapidly. During the useful life phase, the fail-rate is constant. As the materials degrade and reach wear-out, the fail rate keeps increasing with time.

The probability of survival to time t. Expressed another way, it is the fraction of units surviving to time t.

Total fraction failing and surviving must add to 1.

R(T) + F(T) = 1

Based on definition of f(t), F(t), R(t) and l(t), previously described

When the failure-rate l(t) is constant, reliability function becomes an exponential distribution

For Constant Failure Rates, as in the normal life part of the bathtub curve, exponential distributions are useful to model fail probabilities and lifetimes.

The Weibull distribution is a continuous probability distribution created by Waloddi Weibull. In reliability, it is used for time-varying fail rates. In practice, the fail probabilities are modeled by a 3-parameter Weibull Distribution:

η,β,γ, are parameters to be determined by stress-testing units to failure.

In a large number of cases, only two parameters are necessary for modeling reliability, and the Weibull distribution simplifies to:

β is known as the ‘Weibull Slope’ and η is called the ‘Characteristic Life’ of the distribution.

The three sections of the bathtub curve – early fail, useful life, and wear-out – often have different shapes for failure distributions, as illustrated in the figure.

Weibull distribution is a versatile mathematical function that can represent all three sections of the bathtub curve, typically using only two adjustable parameters – β and η.

This is used commonly for reliability modeling.