Assume that a newborn baby has three general behaviors: crying, eating, and sleeping. The amount of time corresponding to each of his three behaviors is exponentially distributed. If he is crying, he will always switch to eating after an average of 10 minutes. When he is eating, he does so for an average of 20 minutes, and then switches to crying with probability 0.2 or to sleeping with probability 0.8. When he falls asleep, he does so for an average of 120 minutes. After he wakes up, he will switch to crying or eating with equal probability.
(a) Draw the CTMC rate diagram, clearly label your states and transition rates (qᵢⱼ's).
(b) Find the transition rates out of the states (vᵢ's) and transition probabilities (Pᵢⱼ's).
(c) Find the limiting probabilities (make sure to write out the balance equations).
(d) In the long run, what is the probability that the baby is not crying?
(e) The father is taking care of the baby while mum tries to sleep. She is awake 90% of the time the baby is crying, 30% of the time the baby is eating and none of the time the baby is sleeping. What fraction of the time is the mum able to sleep?