 #1
Joe1998
 36
 2
 Homework Statement

A production line comprises three machines working independently of each other. For each machine, the probability of working through the day is p ∈ (0, 1) and may breakdown during a day with probability q = 1 − p, independently of its previous history. There is a single repairman who, if he has to, repairs exactly one machine overnight. If there is a machine not working by the end of the day, then the repairman works during the night. Let Xn denote the number of working machines at the end of day before the repairman begins any overnight repair.
a) Specify the state space of (Xn : n ≥ 0).
b) Determine the transition matrix in terms of p.
c) Given that 2 machines are idle tonight, what is the probability of one idle machine 3 nights later, if p = 0.8 (answer 6 decimal places).
 Relevant Equations
 Markov chain and probability transition matrix are mainly the two relevant equations required for this question.
I have tried solving this question, but I am not even sure where to start from and how to specify a state space for this question where it has a trick stating that a worker works overnight. However, for part a, is it the case that there are only 2 states to consider which are: {machine working, machine not working}
part b) I am very unsure, so I don't want to type some gibberish. I mean I am not even sure if the state space is indeed only 2 states or more, so that's why I am unsure about all parts because I am not sure about part (a) in the first place.
Same story with part c, I am not able to solve it unless I get the first two parts correct.
So I would really appreciate any help here, thank you very much.
part b) I am very unsure, so I don't want to type some gibberish. I mean I am not even sure if the state space is indeed only 2 states or more, so that's why I am unsure about all parts because I am not sure about part (a) in the first place.
Same story with part c, I am not able to solve it unless I get the first two parts correct.
So I would really appreciate any help here, thank you very much.
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