A machine works for the eight hour day and five day week. If the machine is working in the morning, then with probability q(0< q <1), it is broken at the end of day. At the end of each working day, the machine is inspected. If it is broken, a repair is scheduled for following days. Fixing the machine takes exactly 2 days. The owner of the machine wishes to know the proportion of days that machine is not working.
A) Provide the transition matrix of Markov chain whose states are the condition of the machine at the end of working day. You must have exactly three states- each explaining the number of days of repair that the machine needs.
B) Find out the communicating classes, and classify the states: provide the period of each state, and classify each state as transient or recurrent.
C) Find out the steady state probabilities.
D) In the steady state, what proportion of days is the machine not working?
E) Should the steady state proportions adequately satisfy the owner of machine not working? Why or why not?