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which implies the inequality (3515) This completes the proof Exponential convergence of the n-step transition probabilities does not hold in general for an in nite-state Markov chain Strong recurrence conditions should be imposed to establish exponential convergence in in nite-state Markov chains

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31 A production machine has two crucial parts which are subject to failures The two parts are identical The machine works as long as one of the two parts is functioning A repair is done when both parts have failed A repair takes one day and after each repair the system is as good as new An inspection at the beginning of each day reveals the exact condition of each part If at the beginning of a day both parts are in good condition, then at the end of the day both parts are still in good condition with probability 050, one of them is broken down with probability 025 and both are broken down with probability 025 If at the beginning of the day only one part is in good condition, this part is still in good condition at the end of the day with probability 050 De ne a Markov chain to describe the functioning of the machine and specify the one-step transition probabilities 32 To improve the reliability of a production system, two identical production machines are connected in parallel For the production process only one of the machines is used; the other machine is standby At the end of the day the used machine is inspected Regardless how long the machine has already been in uninterrupted use, the probability that an inspection 1 reveals the necessity for revision is 10 A revision takes exactly two days During the revision the other machine takes over the production if that machine is available The production process must be stopped when both machines are in revision Assuming that there are two repairmen, de ne an appropriate Markov chain to describe the functioning of the production system and specify the one-step transition probabilities of the Markov chain 33 Containers are temporarily stored at a stockyard with ample capacity At the beginning of each day precisely one container arrives at the stockyard Each container stays a certain amount of time at the stockyard before it is removed The residency times of the containers are independent of each other Specify for each of the following two cases the state variable(s) and the one-step transition probabilities of a Markov chain that can be used to analyse the number of containers present at the stockyard at the end of each day (a) The residency time of a container is exponentially distributed with a mean of 1/ days (b) The residency time of a container has an exponential distribution whose mean is 1/ 1 days with probability p and is 1/ 2 days with probability 1 p 34 Two teams, A and B, meet each other in a series of games until either of the teams has won three games in a row Each game results in a win for either of the teams (no draw is possible) The outcomes of the games are independent of each other De ne an appropriate Markov chain to determine the probability distribution of the length of the match when the two teams are equally strong 35 Consider Exercise 34 again, but assume now that team A wins a given game with a probability larger than 1 2 (a) Use Markov chain analysis to determine the probability distribution of the length of the match Explain how to calculate the probability that team A wins the match (b) Explain how to modify the Markov chain analysis when a draw between the teams is possible with positive probability 36 You play the following game A fair coin is ipped until heads appears three times in a row You get $12 each time this happens, but you have to pay $1 for each ip of the coin Use Markov chain analysis to nd out whether this game is fair 37 Consider the following variant of the coupon-collecting problem A fair die is thrown until each of the six possible outcomes 1, 2, , 6 has appeared Use a Markov chain with seven states to calculate the probability distribution of the number of throws needed 38 The gambler Joe Dalton has $100 and his goal is to double this amount Therefore he plays a gambling game in which he loses his stake with probability 060, but wins two or

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