Hubble Gyroscope Maintenance 

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X proc = ContinuousMarkovProcess[ 1, {{-6 \[Lambda], 6 \[Lambda], 0, 0, 0, 0, 0, 0, 0}, {0, -5 \[Lambda], 5 \[Lambda], 0, 0, 0, 0, 0, 0}, {0, 0, -4 \[Lambda], 4 \[Lambda], 0, 0, 0, 0, 0}, {0, 0, 0, -3 \[Lambda], 3 \[Lambda], 0, 0, 0, 0}, {0, 0, 0, 0, -2 \[Lambda] - \[Mu], 2 \[Lambda], \[Mu], 0, 0}, {0, 0, 0, 0, 0, -\[Lambda] - \[Mu], 0, \[Mu], \[Lambda]}, {\[Eta], 0, 0, 0, 0, 0, -2 \[Lambda] - \[Eta], 2 \[Lambda], 0}, {\[Eta], 0, 0, 0, 0, 0, 0, -\[Lambda] - \[Eta], \[Lambda]}, {\[Nu], 0, 0, 0, 0, 0, 0, 0, -\[Nu]}}] /. {\[Lambda] -> 1/10, \[Mu] -> 100, \[Eta] -> 5};

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X Graph[MapThread[ Style, {{"6", "5", "4", "3", "2", "1", "Sleep2", "Sleep1", "Crash"}, {Green, Green, Green, Green, Orange, Orange, LightBlue, LightBlue, Red}}], proc /. {\[Nu] -> 0}, GraphLayout -> {"SpringElectricalEmbedding", "Rotation" -> Pi}, ImageSize -> Medium]

Model gyroscope failure and repair for the Hubble space telescope, where the telescope is put into sleep mode if fewer than three gyroscopes are operational. The gyroscope operating times, the time to reach sleep mode, and the time to prepare a shuttle mission and repair the gyroscopes are exponentially distributed with rates , , and , with units of inverse years. In the meantime, the other two gyroscopes may fail. If the last gyroscope fails, the telescope will crash. Visualize the process, with integers representing the number of operational gyroscopes.

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Find the probability that the telescope will crash in the next 10 years.

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X CDF[FirstPassageTimeDistribution[proc, 9], 10] // N

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Find the probability that sleep mode is not reached (no shuttle mission is required) in 10 years.

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X SurvivalFunction[ FirstPassageTimeDistribution[proc /. \[Nu] -> 1., {7, 8}], 10]

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X Plot[SurvivalFunction[ FirstPassageTimeDistribution[proc /. \[Nu] -> 1, {7, 8}], x], {x, 0, 20}, Filling -> Axis]

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