Long-run share in maintenance

Question

A factory machine is idle, running, or in maintenance each hour, and next hour depends only on this one: - From idle: to idle 0.7, to running 0.3, to maintenance 0. - From running: to idle 0.2, to running 0.6, to maintenance 0.2. - From maintenance: to idle 0.5, to running 0.5, to maintenance 0. In the long run, what fraction of hours is the machine in maintenance?

Accepted Answer

We want the steady shares (\piI, \piR, \piM) that stop changing hour to hour. Each share equals its total inflow. Maintenance: it is only ever reached from running (0.2), and never stays: \piM = 0.2\,\piR. Idle: arrivals are stay-idle (0.7), from running (0.2), from maintenance (0.5): \piI = 0.7\,\piI + 0.2\,\piR + 0.5\,\piM \implies 0.3\,\piI = 0.2\,\piR + 0.5\,(0.2\,\piR) = 0.3\,\piR, so \piI = \piR. Now use that the shares add to 1: \piI + \piR + \piM = \piR + \piR + 0.2\,\piR = 2.2\,\piR = 1 \implies \piR = 	frac{5}{11}. Then \piM = 0.2 	imes 	frac{5}{11} = 	frac{1}{11}. So the machin…

Long-run share in maintenance

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