How often is the photo booth full?

Question

A photo booth holds up to 2 people. Each minute at most one thing happens: a person enters (if there is room) or a person leaves (if any are inside). The chances depend on how many are inside: - Enter chance: 0.6 from 0 people, 0.2 from 1 person. - Leave chance: 0.4 from 1 person, 0.8 from 2 people. In the long run, what fraction of time is the booth full (both spots taken)?

Accepted Answer

The count moves up or down by one, a birth-death chain, so detailed balance holds: upward flow from level i equals downward flow from level i+1, \pii 	imes (	ext{enter chance at } i) = \pi{i+1} 	imes (	ext{leave chance at } i+1). Write each share as a multiple of \pi0: \pi1 = \pi0 \cdot \frac{0.6}{0.4} = 1.5\,\pi0, \pi2 = \pi1 \cdot \frac{0.2}{0.8} = 1.5 \cdot 0.25\,\pi0 = 0.375\,\pi0. Normalize so the three shares add to 1: \pi0\,(1 + 1.5 + 0.375) = 2.875\,\pi0 = 1 \implies \pi0 = 	frac{8}{23}. So \pi2 = 0.375 	imes 	frac{8}{23} = 	frac{3}{23}. The booth is full three twenty-thirds o…

How often is the photo booth full?

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