How often is the tiny lot full?

Question

A tiny parking lot has 2 spaces. Each minute at most one thing happens: a car arrives (if there is room) or a parked car leaves (if any are present). The chances depend on how many cars are there: - Arrival chance: 0.4 from 0 cars, 0.3 from 1 car. - Departure chance: 0.6 from 1 car, 0.5 from 2 cars. In the long run, what fraction of time is the lot full (both spaces taken)?

Accepted Answer

The car 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{arrival chance at } i) = \pi{i+1} 	imes (	ext{departure chance at } i+1). Write each share as a multiple of \pi0: \pi1 = \pi0 \cdot \frac{0.4}{0.6} = 	frac23\,\pi0, \pi2 = \pi1 \cdot \frac{0.3}{0.5} = 	frac23 \cdot 	frac35\,\pi0 = 	frac25\,\pi0. Normalize so the three shares add to 1: \pi0\,\big(1 + 	frac23 + 	frac25\big) = \pi0 \cdot 	frac{31}{15} = 1 \implies \pi0 = 	frac{15}{31}. So \pi2 = 	frac25 	imes 	frac{15…

How often is the tiny lot full?

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