A machine can be in one of four states: ‘running smoothly’ (state 1), ‘running but needs adjustment’ (state 2), ‘temporarily broken’ (state 3), and ‘destroyed’ (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine to explode. If the machine is running but needing adjustment there is a 20% chance that an engineer will repair it so it is running smoothly the next day and otherwise it will remain in the same state for the next day. Taking Xi to be the state of the machine on the morning of day i for i ∈ N we get a Markov chain which models the state of the machine.Calculate the expectation of the number of days of smooth running in the lifetime of the machine. 1.A machine can be in one of four states: 'running smoothly' (state1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and'destroyed' (state 4). Eacl. morning the state of the machine is recorded. Supposethat the state of the machine voorrow morning depends only on the state of themachine this morning subject to the following rules.• If the machine is running smoothly, there is 1% chance that by the next morningit will have exploded (this will destroy the machine), there is also a 9% chancethat some part of the machine will break leading to it being temporarily broken.If neither of these things happen then the next morning there is an equalprobability of it running smoothly or running but needing adjustment.If the machine is temporarily broken in the morning then an engineer willattempt to repair the machine that day, there is an equal chance that theysucceed and the machine is running smoothly by the next day or they fail andcause the machine to explode.• If the machine is running but needing adjustment there is a 20% chance thatan engineer will repair it so it is running smoothly the next day and otherwiseit will remain in the same state for the next day.Taking X₂ to be the state of the machine on the morning of day i for i E N we get aMarkov chain which models the state of the machine. Calculate the expectation of the number of days of smooth running in thelifetime of the machine.

Take to be the state of the machine in the morning of day when .The states of the given Markov…

A machine can be in one of four states: ‘running smoothly’ (state 1), ‘running but needs adjustment’ (state 2), ‘temporarily broken’ (state 3), and ‘destroyed’ (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine to explode. If the machine is running but needing adjustment there is a 20% chance that an engineer will repair it so it is running smoothly the next day and otherwise it will remain in the same state for the next day. Taking Xi to be the state of the machine on the morning of day i for i ∈ N we get a Markov chain which models the state of the machine. Calculate the expectation of the number of days of smooth running in the lifetime of the machine.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 28EQ

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A machine can be in one of four states: ‘running smoothly’ (state 1), ‘running but needs adjustment’ (state 2), ‘temporarily broken’ (state 3), and ‘destroyed’ (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules.

If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment.

If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine to explode.

If the machine is running but needing adjustment there is a 20% chance that an engineer will repair it so it is running smoothly the next day and otherwise it will remain in the same state for the next day. Taking Xi to be the state of the machine on the morning of day i for i ∈ N we get a Markov chain which models the state of the machine.

Calculate the expectation of the number of days of smooth running in the lifetime of the machine.

1.
A machine can be in one of four states: 'running smoothly' (state
1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and
'destroyed' (state 4). Eacl. morning the state of the machine is recorded. Suppose
that the state of the machine voorrow morning depends only on the state of the
machine this morning subject to the following rules.
• If the machine is running smoothly, there is 1% chance that by the next morning
it will have exploded (this will destroy the machine), there is also a 9% chance
that some part of the machine will break leading to it being temporarily broken.
If neither of these things happen then the next morning there is an equal
probability of it running smoothly or running but needing adjustment.
If the machine is temporarily broken in the morning then an engineer will
attempt to repair the machine that day, there is an equal chance that they
succeed and the machine is running smoothly by the next day or they fail and
cause the machine to explode.
• If the machine is running but needing adjustment there is a 20% chance that
an engineer will repair it so it is running smoothly the next day and otherwise
it will remain in the same state for the next day.
Taking X₂ to be the state of the machine on the morning of day i for i E N we get a
Markov chain which models the state of the machine.

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