Answer Happy

A job-shop has five machines. Machines breakdown at random following a Poisson process at the rate of 1 per day. Suppose there are three repairmen who can (individually) fix a broken machine in half a day on average. Suppose the shop manager receives an offer to replace the three repairmen by one “super” repairman who can fix a machine in a third of the time, but costs the equivalent of the three repairmen to hire. Answer the following problems. 1. Model the number of machines failed (and hence in repair) as a continuous time Markov chain, in both settings stated above. State clearly what assumptions you must make to justify using a Markov model. What is the state space? What is the transition rate matrix? 2. What queuing models do these two settings correspond to (if any)? 3. Can you justify hiring the super-repairman? Give a full mathematical reasoning with derivation of necessary expressions. (Hint: look at the steady state mean number of machines in repair.)