A company owns three CNC machines. Although they go through a well-planned preventive maintenance program, each working machine breaks down five times per week, on the average. A repairperson is available to repair the machines when they break down. If a machine breaks down, it is served immediately if the repairperson is idle; otherwise, it waits for service. The repairperson can repair machines at an average rate of ten machines per week. The times that working machines take to breakdown and their repair times are continuous random variables which distributions possess the memoryless property. Thus, this system can be modeled as a continuous-time Markov chain.
a) Define the states of the Markov chain and draw the state transition diagram including the transition rates
b) Determine the steady-state probabilities.
c) What is the utilization of the repairman (i.e., fraction of time the repairman is busy)
d) What is the average number of working machines?
e) What is the average number of machines waiting to be repaired?