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Problem # 1 (30 points) A steel shop has three plasma machines to shear steel sheets and produce various part types automatically using information from a CAD/CAM database. Although the machines are relatively new, each machine breaks down at an average rate of 2 times per week. Two repairmen is available to repair the machines that break down. When a machine breaks down, and the other two are working, it is served immediately by one repairman. If a second machine breaks down (when one machine is down and the other one is working), the idle repairman serves the machine. When the third machine breaks down, and the other two are also down, it waits for service. Each repairman can repair plasma machines at an average rate of 5 machines per week. Machine breakdown times and machine repair times are continuous random variables which distributions possess the memoryless property. Thus, the system can be modeled as a continuous-time Markov chain. (a) (10 points) Define the states of the Markov chain and draw the state transition diagram including the transition rates (b) (15 points) Compute the steady-state probabilities. (c) (5 points) Determine the average number of working machines