How long until every backup generator dies?

Question

A data center has 4 backup generators wired in parallel, power is lost only when the last generator dies. Each generator's lifetime is independent and exponential with mean 10 years (rate \lambda = 0.1 per year). What is the expected time until all four have failed?

Accepted Answer

A parallel system survives until the last component fails, so its lifetime is the maximum of the four exponential lifetimes. Here memorylessness makes the calculation clean by thinking in stages. - Start with 4 generators running. The first failure is the minimum of 4 exponentials, which is exponential with rate 4\lambda, so it takes \frac{1}{4\lambda} on average. - After one fails, 3 remain; by memorylessness they are "as good as new." Time to the next failure: \frac{1}{3\lambda}. - Then \frac{1}{2\lambda} with 2 left, and finally \frac{1}{\lambda} for the last survivor. Add the stage times:…

How long until every backup generator dies?

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