How long the server stays up

Question

A server crashes as a Poisson process at rate \lambda = 0.4 per day. What is the probability that the server runs for more than 3 days without a crash?

Accepted Answer

The time until the next crash is exponential, so the chance the server survives longer than a time t is P(T t) = e^{-\lambda t}. With \lambda = 0.4 per day and t = 3 days, P(T 3) = e^{-0.4 	imes 3} = e^{-1.2} \approx 0.301. So about a 30\% chance of three crash-free days. Takeaway. Reliability "survival past time t" is exactly the exponential tail e^{-\lambda t}, with \lambda the crash rate.

How long the server stays up

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