A manufacturer has a machine that if it ran all day today has a proba- bility of 0.2 of breaking down sometime during the day tomorrow. When the machine breaks down, it goes offline for the remainder of the day and then a technician will spend the next day (after the breakdown) repairing it. A newly repaired machine only has a proability of 0.1 of breaking down sometime tomorrow. (a) Formulate the evolution of the status of the machine at the end of the day as a Markov Chain by identifying the three possible states at the end of the day and providing the transition probabilities between these states. (b) Determine the expected first passage times, µij , for all combinations of states i and j where i cannot equal to j (i.e., you don’t need to determine the recurrence times). You must provide the set of equations used to calculate these µij (micro with lowercase i and j). (c) Using your results from (b): (i) Identify the expected number of full days that the machine will remain operational before the next breakdown after a repair is completed. (ii) Suppose that the machine just finished 1 day of being fully operational. Determine the expected number of full days from now that the machine will remain operational before the next breakdown. (iii) Suppose that the machine has run 5 straight days - how does the expected number of days from now until the next breakdown differ from your answer in (ii)? Provide your reasoning behind your answer.

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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A manufacturer has a machine that if it ran all day today has a proba- bility of 0.2 of breaking down sometime during the day tomorrow. When the machine breaks down, it goes offline for the remainder of the day and then a technician will spend the next day (after the breakdown) repairing it. A newly repaired machine only has a proability of 0.1 of breaking down sometime tomorrow. (a) Formulate the evolution of the status of the machine at the end of the day as a Markov Chain by identifying the three possible states at the end of the day and providing the transition probabilities between these states. (b) Determine the expected first passage times, µij , for all combinations of states i and j where i cannot equal to j (i.e., you don’t need to determine the recurrence times). You must provide the set of equations used to calculate these µij (micro with lowercase i and j). (c) Using your results from (b): (i) Identify the expected number of full days that the machine will remain operational before the next breakdown after a repair is completed. (ii) Suppose that the machine just finished 1 day of being fully operational. Determine the expected number of full days from now that the machine will remain operational before the next breakdown. (iii) Suppose that the machine has run 5 straight days - how does the expected number of days from now until the next breakdown differ from your answer in (ii)? Provide your reasoning behind your answer.

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