How long between charging sessions?

Question

A phone is either charging or unplugged each hour, and the next hour depends only on this one: - If charging now, it becomes unplugged next hour with probability 0.5 (and stays charging with probability 0.5). - If unplugged now, it becomes charging next hour with probability 0.1 (and stays unplugged with probability 0.9). The phone is charging now. On average, how many hours until it is charging again (its expected return time to the charging 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 charging. For a two-state chain the share is \frac{b}{a+b}, with a the leaving probability and b the arriving probability. Leaving charging: a = 0.5; arriving at charging (from unplugged): b = 0.1: \pi{	ext{charging}} = \frac{0.1}{0.5 + 0.1} = \frac{0.1}{0.6} = \frac{1}{6}. Invert: \frac{1}{\pi{	ext{charging}}} = \frac{1}{1/6} = 6 	ext{ hours}. So the phone returns to charging every 6 hours on average. Takeaway. Expected return time is one over the stationary probability. A state that o…

How long between charging sessions?

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