How long until the server is back up?

Question

A server is either up or down each hour, and the next hour depends only on this one: - If up now, it goes down next hour with probability 0.1 (and stays up with probability 0.9). - If down now, it comes up next hour with probability 0.3 (and stays down with probability 0.7). The server is up now. On average, how many hours until it is up again (its expected return time to the up state)?

Accepted Answer

The average time to return to a state equals 1 divided by that state's long-run share. Long-run share of up. For a two-state chain the share is \frac{b}{a+b}, with a the leaving probability and b the arriving probability. Leaving up: a = 0.1; arriving at up (from down): b = 0.3: \pi{	ext{up}} = \frac{0.3}{0.1 + 0.3} = \frac{0.3}{0.4} = \frac{3}{4}. Invert: \frac{1}{\pi{	ext{up}}} = \frac{1}{3/4} = \frac{4}{3} \approx 1.33 	ext{ hours}. So the server returns to "up" after about 1.33 hours on average. Takeaway. Expected return time is the reciprocal of the stationary probability. A state tha…

How long until the server is back up?

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