How often does a flaky server come back online?

Question

A server is either up or down, checked each minute. If it is up this minute, it stays up next minute with probability 0.7 (and goes down with probability 0.3). If it is down, it stays down with probability 0.4 (and comes back up with probability 0.6). Starting from an "up" minute, what is the expected number of minutes until it is next up again? (Count the return to an up-minute.)

Accepted Answer

The clean way uses the fact that, for an irreducible Markov chain, the mean return time to a state is the reciprocal of that state's stationary probability: E[	ext{return to up}] = \frac{1}{\pi{	ext{up}}}. So find \pi{	ext{up}}. For a two-state chain with up	odown probability a = 0.3 and down	oup probability b = 0.6, the stationary "up" fraction is \pi{	ext{up}} = \frac{b}{a + b} = \frac{0.6}{0.3 + 0.6} = \frac{0.6}{0.9} = \frac{2}{3}. Therefore E[	ext{return to up}] = \frac{1}{2/3} = \frac{3}{2} = 1.5 	ext{ minutes}. Cross-check by direct conditioning. From an up-minute, next minute …

How often does a flaky server come back online?

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