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8. Assessing Product Reliability
8.2. Assumptions/Prerequisites
8.2.1. How do you choose an appropriate life distribution model?

Multiplicative degradation argument

The lognormal model can be applied when degradation is caused by random shocks that increase degradation at a rate proportional to the total amount already present A brief verbal description of the multiplicative degradation argument (leading to a derivation of the lognormal model) was given under Uses of the Lognormal Distribution Model. Here a formal derivation will be outlined because it gives insight into why the lognormal has been a successful model for many failure mechanisms based on degradation processes.

Let \(y_1, \, y_2, \, \ldots, \, y_n\) be measurements of the amount of degradation for a particular failure process taken at successive discrete instants of time as the process moves towards failure. Assume the following relationships exist between the \(y\)'s: $$ y_i = (1 + \epsilon_i)y_{i-1} \, , $$ where the \(\epsilon_i\) are small, independent random perturbations or "shocks" to the system that move the failure process along. In other words, the increase in the amount of degradation from one instant to the next is a small random multiple of the total amount of degradation already present. This is what is meant by multiplicative degradation. The situation is analogous to a snowball rolling down a snow covered hill; the larger it becomes, the faster it grows because it is able to pick up even more snow.

We can express the total amount of degradation at the \(n\)-th instant of time by $$ x_n = \left( \prod_{i=1}^n (1 + \epsilon_i) \right) x_0 \, , $$ where \(x_0\) is a constant and the \(\epsilon_i\) are small random shocks. Next we take natural logarithms of both sides and obtain: $$ \mbox{ln } x_n = \sum_{i=1}^n \mbox{ln }(1 + \epsilon_i) + \mbox{ln } x_0 \approx \sum_{i=1}^n \epsilon_i + \mbox{ln } x_0 \, . $$ Using a Central Limit Theorem argument we can conclude that ln \(x_n\) has approximately a normal distribution. But by the properties of the lognormal distribution, this means that \(x_n\) (or the amount of degradation) will follow approximately a lognormal model for any \(n\) (or at any time \(t\)). Since failure occurs when the amount of degradation reaches a critical point, time of failure will be modeled successfully by a lognormal for this type of process.

Failure mechanisms that might be successfully modeled by the lognormal distribution based on the multiplicative degradation model What kinds of failure mechanisms might be expected to follow  a multiplicative degradation model? The processes listed below are likely candidates:
  1. Chemical reactions leading to the formation of new compounds
  2. Diffusion or migration of ions 
  3. Crack growth or propagation
Many semiconductor failure modes are caused by one of these three degradation processes. Therefore, it is no surprise that the lognormal model has been very successful for the following semiconductor wear out failure mechanisms: 
  1. Corrosion
  2. Metal migration
  3. Electromigration
  4. Diffusion
  5. Crack growth
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