Larry is a student who cannot seem to make up his mind whether to live in the city or in the suburb. Every time he lives in the city, he moves to the suburb after one semester. Half of the time he...

Larry is a student who cannot seem to make up his mind whether to live in the city or in the suburb. Every time he lives in the city, he moves to the suburb after one semester. Half of the time he lives in the suburb, he moves back to the city after one semester. The other half of Larry’s suburban living results in his moving to a new apartment in the suburb where he lives for a time that is geometrically distributed with a mean of two semesters.

a. Model Larry’s living style by a two-state discrete-time semi-Markov process, giving the state-transition diagram.

b. Assume that Larry lives in the city at the beginning of the current semester. What is the probability that he will be living in the suburb k semesters from now?

c. What is the probability that the total duration of any uninterrupted stay in the suburb is k semesters? What are the mean and variance of the duration of one such stay in the suburb?