A machine is used to produce precision tools. The machine can be considered in one of three possible working conditions: good, fair, poor. If the machine is in good condition today, then 40% of the time it will be good tomorrow and 30% of the time in fair condition. If the machine is in fair condition today then it will be in fair condition tomorrow 10% of the time and poor 20% of the time. If the machine is in poor condition today then it will be poor tomorrow 40% of the time and good 60% of the time. A machine in good condition will produce about 156 tools/day, one in fair condition will produce about 69 and one in poor condition only produces 40. a. Define states and construct the one-step transition probability matrix P.
b. Assume that long-run probabilities (IT vector) are given. In terms of these probabilities, write a mathematical expression for the expected production of tools per day.
c. Assume that long-run probabilities (īt vector) are given. In terms of these probabilities, what is the expected daily loss in production due to the machine not always operating in the good condition?