(1) Markov Chains in Continuous Time. Two mechanics work in a workshop with a capacity of 3 cars. If there are 2 or more cars in the shop, each mechanic works individually and each completes a car repair in a timely manner exponential of average 4 hours. If there is only 1 car in the workshop, both mechanics work together completing the repair in a time an exponential time of average 3 hours. Cars arrive randomly according to a Poisson process at the rate of 1 every 2 hours when there is at least one mechanic unoccupied, but at the rate of 1 every 4 hours when both mechanics are busy. If there are 3 cars in the shop, no more cars can come. (a) Model the process as a CMTC, establishing the transition graph with the rates included, and the intensity matrix. Is it a birth-death process? (b) Determine the steady-state probabilities. (c) What percentage of the day will both mechanics be unemployed? (d) What percentage of the day will both mechanics be working on the same car? (e) What is the average number of cars in the shop? (f) How many cars are expected to arrive during an 8-hour workday?