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!

Muhammad Ashraful Alam

Purdue University

West Lafayette, IN

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