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LETTERS TO THE EDITORS

# A Tale of Tails

To the Editors:

Brian Hayes's column "Fat Tails" (Computing Science, May-June) reminded me of a class I taught last semester on the reliability of electronic devices, in which my goal was to explore how electronic components break.

I wanted to explain various reliability problems (such as the breakdown of gate dielectrics in modern integrated circuits) as a "stochastic process terminated by a threshold." In the context of dielectric breakdown, this process would entail the random generation of defects until a percolation path shorts the gate dielectric.

I thought of the simplest modification of the classical one-dimensional random-walk problem, in which I would terminate the random walk with an absorbing point and then explore the arrival-time distribution at the absorption point. This I thought would be an example of a "stochastic process with a threshold."

Specifically, I defined an infinite grid, set the absorption point at grid location 0 and injected particles at grid point N. After injection, the particle hops to the left or right with equal probability of 1/2 until it reaches the grid location 0—and I noted the number of steps required to reach this point and then inject another particle.

To my utter surprise, however, I soon noticed that the average number of steps taken to reach the absorption point continued to increase with the number of particles injected. I then discovered that the arrival-time distribution is also a power law and has a "fat tail"—just as was discussed in the article.

The implication is interesting: Before shipping integrated circuits, semiconductor companies test a few at accelerated-aging conditions to find the average failure time and then extrapolate to normal operating conditions and to millions of circuits to ensure that the product will have a given lifetime. If the law of averages does not hold, this extrapolation becomes meaningless, and the average lifetime could be better than expected!