A hospital is interested in tracking the movements of heart patients through dierent units. The four units of interest are emergency admissions, surgery, intensive care, and recovery. Emergency patients (that is, people who have suffered heart and were brought into the hospital through emergency admissions) are always assigned to the intensive care unit for a period of observation following the emergency treatment. Once a doctor determines that it is safe to do so, the patient is moved to a recovery unit. Of course, a relapse requires a return to intensive care. The surgical unit receives patients of two kinds: those who are scheduled for heart operations through non-emergency procedures (planned operations), and those who come from intensive care (semi-emergency operations). Only one patient in a hundred dies in surgery; the rest are always moved in the intensive care after surgery. Of the patients in intensive care, whether or not they have previously had surgery, about 10% will undergo surgery; the others are moved into recovery after some period of intensive care.
Once into recovery, most patients (95%) eventually either recover suffciently to be discharged, but about 2 percent have to be returned to intensive care. The others die in recovery.
(a) Draw a (weighted) directed graph to set up an appropriate Markov chain. Give the corresponding probability transition matrix.
(b) Classify the states.
(c) What is the expected number of operations required once a heart surgery patient enters the hospital?
(d) What is the probability that a patient will undergo surgery at least once?
(e) What is the expected number of transfers that a patient will go through during a hospital stay, if he or she is admitted as an emergency?