An oil company operates three oil well pumps in a remote area. Time is measured in monthly intervals. If a pump is operating at the beginning of the month, it will fail by the end of the month with probability p. The probability that it will not fail is 1 - p. Failed pumps remain failed until they are repaired. The three pumps fail independently. It is expensive to send a repair crew to the area, so the company waits until all three pumps are failed before sending a crew. On a visit, the crew repairs all failed pumps. The repair process takes one month.

Use the Stochastic Analysis Add-in to analyze this system if p = 0.03. The cost of sending a repair crew is $10,000. The revenue for production from a working pump is $1,000 per month.

Answer numerical questions with the Stochastic Analysis Add-in.

a. Construct the DTMC Matrix that describes this situation.

b. What is the steady-state monthly profit for the three pumps?

c. What is the steady-state probability distribution for the number of failed pumps at the end of the month?

d. What is the expected time between repair visits?

e. What is the probability distribution on the number of months between repair visits?

f. Change the original situation so that the crew is sent whenever two pumps are failed. How does this change the steady-state monthly profit?

g. Change the original situation so that it takes two months to repair the failed pumps. How does this change the steady-state monthly profit?