How often is the charging station full?

Question

A charging station holds up to 3 cars. In each minute at most one thing happens: a car arrives (if there is room) or a car finishes and leaves (if any are present). The chances depend on how many cars are currently there: - Arrival chance: 0.6 from 0 cars, 0.4 from 1 car, 0.2 from 2 cars. - Departure chance: 0.3 from 1 car, 0.4 from 2 cars, 0.5 from 3 cars. In the long run, what fraction of time is the station full (all 3 bays occupied)?

Accepted Answer

The count of cars only ever moves up or down by one, so this is a birth-death chain. Such chains satisfy detailed balance: in steady state, the rate of crossing upward from level i to i+1 must equal the rate of crossing back down from i+1 to i. In symbols, with \pii the long-run share of level i: \pii 	imes (	ext{arrival chance at } i) = \pi{i+1} 	imes (	ext{departure chance at } i+1). Apply it level by level, writing every share as a multiple of \pi0: \pi1 = \pi0 \cdot \frac{0.6}{0.3} = 2\,\pi0, \pi2 = \pi1 \cdot \frac{0.4}{0.4} = 2\,\pi0, \pi3 = \pi2 \cdot \frac{0.2}{0.5} = 0.8\,\pi0. T…

How often is the charging station full?

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