The operating status of a machine in a manufacturing facility can be modeled using a Markov chain. The states are 1 for operational, 2 for unknown malfunction, 3 for awaiting service, and 4 for awaiting parts. The status is updated every hour. The transition matrix is:
P = [[0.7, 0.3, 0, 0],
[0, 0, 0.4, 0.6],
[0.5, 0.5, 0, 0],
[0, 0, 0.2, 0.8]]
(a) If a machine is operational, how many hours can we expect it to stay operational?
E[N1] = MEAN SOJOURN TIME
(b) What percentage of time can we expect the machine to be awaiting parts in the long run?
STEADY STATE PROBABILITY, π4
(c) If the machine has an unknown malfunction, what is the average number of hours until it starts working again?
MEAN FIRST PASSAGE TIME, M21
(d) If the machine has an unknown malfunction, what is the average amount of time until the next unknown malfunction?
MEAN RECURRENCE TIME, M22 = 1/π2
Answers:
Part a. Expect to stay operational for 3.333333 hours.
Part b. Expect the machine to wait for parts 45.00% of the time.
Part c. Expect 10.000000 hours until the machine is working again.
Part d. The average time between unknown malfunctions is 6.666667
Need work shown. Can be handwritten or using Excel.