How long until the vacuum docks?

Question

A robot vacuum is either Roaming or Near the dock, until it becomes Docked. Each minute: - From Roaming: stay Roaming with probability 0.4, or move Near with probability 0.6. - From Near: drift back to Roaming with probability 0.5, or become Docked with probability 0.5. Docked ends the process. The vacuum starts Roaming. On average, how many minutes until it is Docked?

Accepted Answer

Let h(X) be the expected minutes to Docked from state X; h(	ext{Docked}) = 0. First-step analysis: one minute passes, then add the expected time from the next state. h(	ext{Roam}) = 1 + 0.4\,h(	ext{Roam}) + 0.6\,h(	ext{Near}), h(	ext{Near}) = 1 + 0.5\,h(	ext{Roam}) + 0.5 \cdot 0. From the first equation, 0.6\,h(	ext{Roam}) = 1 + 0.6\,h(	ext{Near}), so h(	ext{Roam}) = 	frac{5}{3} + h(	ext{Near}). Put this into the second: h(	ext{Near}) = 1 + 0.5\,\big(	frac53 + h(	ext{Near})\big) = 1 + 	frac56 + 0.5\,h(	ext{Near}) \implies 0.5\,h(	ext{Near}) = 	frac{11}{6}, so h(	ext{Near}) …

How long until the vacuum docks?

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