7. (16 marks)Consider a machine whose condition at any time can be classified as being in one of the following three states:

State 1: Good operating order

State 2: Deteriorated operating order

State 3: In repair

The condition of the machine is observed at the end of each period in a sequence of periods. LetXndenote the condition of the machine at the end of periodnforn= 1,2,.... LetX0be the condition of the machine at the start. The sequence of machine conditionsXnis a Markov chain with transition probabilities

p11=0.9p21=0p31= 1

p12=0.1p22=0.9p32= 0

p13=0p23=0.1p33= 0

and the process starts in stateX0= 1.

a)FindP(X4=1).

b) Calculate the limiting distribution.

c) What is the long run rate of repairs per unit time.