How often are there no alarms?

Question

A monitoring board shows between 0 and 3 active alarms. Each minute at most one thing happens: a new alarm fires (if fewer than 3) or one alarm is cleared (if any are active). The chances depend on the current count: - New-alarm chance: 0.5 from 0, 0.4 from 1, 0.3 from 2. - Cleared chance: 0.5 from 1, 0.6 from 2, 0.9 from 3. In the long run, what fraction of time is the board quiet (zero alarms)?

Accepted Answer

The alarm 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{new-alarm chance at } i) = \pi{i+1} 	imes (	ext{cleared chance at } i+1). Write each share as a multiple of \pi0: \pi1 = \pi0 \cdot \frac{0.5}{0.5} = \pi0, \pi2 = \pi1 \cdot \frac{0.4}{0.6} = 	frac23\,\pi0, \pi3 = \pi2 \cdot \frac{0.3}{0.9} = 	frac23 \cdot 	frac13\,\pi0 = 	frac29\,\pi0. Normalize so the four shares add to 1: \pi0\,\big(1 + 1 + 	frac23 + 	frac29\big) = \pi0 \cdot 	frac{26}{9} = 1 \implies \pi0 = 	fra…

How often are there no alarms?

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