A work center has three machines (MA, MB, and MC) and tries to run (if possible) two of them in parallel at the same time. The management prefers to run MA and MC. Thus, MB (if it is not under repair and MA and MC are both working) stays in standby mode. MA/MB/MC, while working, can fail according to an exponential distribution with mean 32 / 15 / 44 .When a machine fails its repair starts right away. MA/MB/MC has exponential repair times with mean 8 / 13 / 11. When MA and MC are repaired, they start working right away. If MA or MC is repaired and the other two machines are working, MB is put in standby mode. If MB is repaired, it starts working only if MA or MC is not working; if both MA and MC work, obviously MB is put in standby mode. If MA/MB/MC is working, the system earns 160 / 30 / 220per unit time independently of the status of the other machines. For instance, if MA has been working for the last 5 time units, the company has made 5 x 160 amount of money during this period. The times to failure and times to repair of machines are independent of each other. A machine, when under repair, and MB, when in standby mode, cannot fail.

a. ) Briefly explain and list the states of the underlying continuous-time Markov chain (CTMC) and sketch the transition rate diagram with rates on it: - If only MB is working: - Explain and provide the rate with which the CTMC enters the state with only MA under repair. - Compute with what probability this transition occurs. - If only MC is working: - Explain and provide the rate with which the CTMC enters the state with all machines under repair. - Compute with what probability this transition occurs.

b. Compute and write (in numbers, not with formulae) the Q matrix (you don’t need to give explanations) necessary to obtain the limiting distribution. Obtain via Python and write the limiting/stationary distribution. Write also the line number in your code that finds and prints the limiting distribution.

c) Theoretical results (based on the limiting/stationary distribution found using the Q matrix in part b). You can compute these yourselves on your PDF file without Python. No partial credit if the results are wrong. - (2 points) the stationary probability that only MC is working - (2 points) the stationary probability that MC is working - (2 points) the expected earnings per unit time