Consider Problem 6.11. Assume that the time she spends with each of her children is exponentially distributed with the same means as specified. Obtain the transition probability functions {φij (t)} of the process.
Problem 11
In her retirement days, a mother of three grown-up children splits her time living with her three children who live in three different states. It has been found that her choice of where to spend her time next can be modeled by a semi-Markov chain. Thus, if the children are labeled by ages as child 1, child 2, and child 3, the transition probabilities are as follows. Given that she is currently staying with child 1, the probability that she will stay with child 1 next is 0.3, the probability that she will stay with child 2 next is 0.2, and the probability that she will stay with child 3 next is 0.5. Similarly, given that she is currently staying with child 2, the probability that she will stay with child 1 next is 0.1, the probability that she will stay with child 2 next is 0.8, and the probability that she will stay with child 3 next is 0.1. Finally, given that she is currently staying with child 3, the probability that she will stay with child 1 next is 0.4, the probability that she will stay with child 2 next is 0.4, and the probability that she will stay with child 3 next is 0.2. The length of time that she spends with child 1 is geometrically distributed with mean 2 months, the length of time she spends with child 2 is geometrically distributed with mean 3 months, and the time she spends with child 3 is geometrically distributed with mean 1 month.
a. Obtain the transition probability functions of the process; that is, obtain the set of probabilities {φij
(n)}, where φij
(n) is the probability that the process is in state j at time n given that it entered state i at time zero.
b. What is the occupancy distribution of the process?