What fraction of time is the heater running?

Question

A thermostat is either heating or idle each minute, and the next minute depends only on the current one: - If heating now, it goes idle next minute with probability 0.5 (and keeps heating with probability 0.5). - If idle now, it starts heating next minute with probability 0.2 (and stays idle with probability 0.8). In the long run, what fraction of minutes is the thermostat heating?

Accepted Answer

We want the steady shares (\piH, \piI) that stop changing minute to minute. Balance the flow out of "heating" against the flow in. Flow leaving heating: \piH 	imes 0.5. Flow entering heating: \piI 	imes 0.2. Balance: \piH 	imes 0.5 = \piI 	imes 0.2 \implies \piH = 0.4\,\piI. Since the shares add to 1: 0.4\piI + \piI = 1 \implies 1.4\,\piI = 1 \implies \piI = 	frac{5}{7}, \quad \piH = 	frac27. So the thermostat is heating two sevenths of the time, about 28.6\%. Two-state shortcut for the heating state: it is entered with probability b = 0.2 and left with probability a = 0.5, giving \frac…

What fraction of time is the heater running?

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