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4.2.4 A component of a computer has an active life, measured in discrete units, that is a random variable {, where k = 1 2 3 4 Pr{k} = 0.1 0.3 0.2 0.4 Suppose that one starts with a fresh component, and each component is replaced by a new component upon failure. Let X, be the remaining life of the component in service at the end of period n. When X, = 0, a new item is placed into service at the start of the next period. (a) Set up the transition probability matrix for {X}. (b) By showing that the chain is regular and solving for the limiting distribution, determine the long run probability that the item in service at the end of a period has no remaining life and therefore will be replaced. (c) Relate this to the mean life of a component.

4.2.4 A component of a computer has an active life, measured in discrete units, that is a random variable {, where k = 1 2 3 4 Pr{k} = 0.1 0.3 0.2 0.4 Suppose that one starts with a fresh component, and each component is replaced by a new component upon failure. Let X, be the remaining life of the component in service at the end of period n. When X, = 0, a new item is placed into service at the start of the next period. (a) Set up the transition probability matrix for {X}. (b) By showing that the chain is regular and solving for the limiting distribution, determine the long run probability that the item in service at the end of a period has no remaining life and therefore will be replaced. (c) Relate this to the mean life of a component.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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4.2.4 A component of a computer has an active life, measured in discrete units, that is a random variable {, where k = 1 2 3 4 Pr{k} = 0.1 0.3 0.2 0.4 Suppose that one starts with a fresh component, and each component is replaced by a new component upon failure. Let X, be the remaining life of the component in service at the end of period n. When X, = 0, a new item is placed into service at the start of the next period. (a) Set up the transition probability matrix for {X}. (b) By showing that the chain is regular and solving for the limiting distribution, determine the long run probability that the item in service at the end of a period has no remaining life and therefore will be replaced. (c) Relate this to the mean life of a component.
Transcribed Image Text:4.2.4 A component of a computer has an active life, measured in discrete units, that is a random variable {, where k = 1 2 3 4 Pr{k} = 0.1 0.3 0.2 0.4 Suppose that one starts with a fresh component, and each component is replaced by a new component upon failure. Let X, be the remaining life of the component in service at the end of period n. When X, = 0, a new item is placed into service at the start of the next period. (a) Set up the transition probability matrix for {X}. (b) By showing that the chain is regular and solving for the limiting distribution, determine the long run probability that the item in service at the end of a period has no remaining life and therefore will be replaced. (c) Relate this to the mean life of a component.
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