Part I. (Modification of Problem 16. 4-1, page 761, of Hillier & Lieberman text, 8th edition)
(1) Given the following (one-step) transition matrices of a Markov chain, determine (1) the classes of the Markov chain and (2) whether they are recurrent, transient, or absorbing.
(2) Determine the period of each of the states in the Markov chain that has thefollowing (one-step) transition matrix (Modification of Problem 16.4-4, page 755, of Hillier & Lieberman text, 9th edition).
(3) Consider the Markov chain that has the following (one-step) transition matrix(Modification of Problem 16.4-5, page 755, of Hillier & Lieberman text, 9th edition).
(a) Determine the classes of this Markov chain and, for each class, determine whether itis recurrent or transient.
(b) For each of the classes identified in part (a), determine the period of the states in thatclass.
Part II. The admissions office has modeled the path of a student through ABCS as a Markovchain:
Each student's state is observed at the beginning of each fall semester. For example, if a studentis a junior at the beginning of the current fall semester, there is an 60% chance that he will be asenior at the beginning of the next fall semester, a 10% chance that he will still be a junior, and a30% chance that he will have quit. We assume that once a student quits, he never reenrolls.
(a) If a student enters ABCS as a freshman, how many years can he expect to spend as a studentat NMSU?
(b) What is the probability that a freshman graduates?
Part III. (Modification of Problem 16.7-2, page 758, of Hillier & Lieberman text, 9th edition)
A video cassette recorder manufacturer is so certain of its quality control that it is offering acomplete replacement warranty if a recorder fails within 3 years. Based upon compiled data, thecompany has noted the following:
- Only 3 percent of its recorders fail during the first year
- 8 percent of the recorders that survive the first year will fail during the second year
- 15 percent of the recorders that survive the second year will fail during the third year
The warranty does not cover replacement recorders.
(a) Formulate the evolution of the status of a recorder as a Markov chain whose states includetwo absorption states that involve needing to honor the warranty or having the recordersurvive the warranty period. Then construct the (one-step) transition matrix.
(b) What is the number of transient states in your Markov chain model?
(c) Find the probability that the manufacturer will have to honor the warranty.