How long until the inbox is empty again?

Question

An email inbox is either empty or has mail at the top of each hour, and the next hour depends only on this one: - If empty now, it has mail next hour with probability 0.7 (and stays empty with probability 0.3). - If it has mail now, it is empty next hour with probability 0.3 (and still has mail with probability 0.7). The inbox is empty now. On average, how many hours until it is empty again (its expected return time to the empty 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 empty. For a two-state chain the share is \frac{b}{a+b}, with a the leaving probability and b the arriving probability. Leaving empty: a = 0.7; arriving at empty (from has-mail): b = 0.3: \pi{	ext{empty}} = \frac{0.3}{0.7 + 0.3} = \frac{0.3}{1.0} = 0.3. Invert: \frac{1}{\pi{	ext{empty}}} = \frac{1}{0.3} = \frac{10}{3} \approx 3.33 	ext{ hours}. So the inbox is empty again after about 3.33 hours on average. Takeaway. Once you have a state's long-run share, its expected return time is jus…

How long until the inbox is empty again?

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