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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 έ, where k = 1 Pr{=k} = 0.1 2 3 4 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 X₁ = 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 έ, where k = 1 Pr{=k} = 0.1 2 3 4 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 X₁ = 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.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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 έ, where k = 1 Pr{=k} = 0.1 2 3 4 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 X₁ = 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 έ, where k = 1 Pr{=k} = 0.1 2 3 4 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 X₁ = 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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