At a small work center that runs without any time-breaks or holidays, the production is carried on with two distinct machines (lets call them Machine A and Machine B), and these two machines are working simultaneously if they are functional. These machines can fail from time to time and times to failures are exponentially distributed with a mean of 10 days and 15 days for Machine A and Machine B, respectively. There are two repairmen to fix these machines when they are broken. If only one machine is broken only a single repairman is working on that machine and if there are more than one broken machine, both repairmen are working on separate machines. Repair times are exponentially distributed and the repair rates are 2 per day for each machine. Whenever a machine is fixed, it starts production again. Whenever a machine fails, its repair starts right away (if there is an idle repairman). All times to failures and repair times are independent of each other. When Machine A and B are functional, they produce 15 and 10 electrical wires per hour, respectively and each wire is sold at 10 TL. Repair cost for each machine is 10,000 TL per day (if a machine is waiting to be repaired, the repair cost is not incurred for that machine). a) Construct an appropriate continuous-time Markov chain (CTMC) by stating what would be the states telling us. Then, compute the limiting probabilities for each state. b) Compute the expected profit per day by including profit obtained by the production of wires and cost of repairs. c) Write Monte Carlo Simulations to obtain the following values (a) proportion of times in which both machines are working, both machines are in repair, Machine A is working and Machine B is in repair, Machine A is in repair and Machine B is working.

At a small work center that runs without any time-breaks or holidays, the production is carried on with two distinct machines (lets call them Machine A and Machine B), and these two machines are working simultaneously if they are functional. These machines can fail from time to time and times to failures are exponentially distributed with a mean of 10 days and 15 days for Machine A and Machine B, respectively. There are two repairmen to fix these machines when they are broken. If only one machine is broken only a single repairman is working on that machine and if there are more than one broken machine, both repairmen are working on separate machines. Repair times are exponentially distributed and the repair rates are 2 per day for each machine. Whenever a machine is fixed, it starts production again. Whenever a machine fails, its repair starts right away (if there is an idle repairman). All times to failures and repair times are independent of each other. When Machine A and B are functional, they produce 15 and 10 electrical wires per hour, respectively and each wire is sold at 10 TL. Repair cost for each machine is 10,000 TL per day (if a machine is waiting to be repaired, the repair cost is not incurred for that machine). a) Construct an appropriate continuous-time Markov chain (CTMC) by stating what would be the states telling us. Then, compute the limiting probabilities for each state. b) Compute the expected profit per day by including profit obtained by the production of wires and cost of repairs. c) Write Monte Carlo Simulations to obtain the following values (a) proportion of times in which both machines are working, both machines are in repair, Machine A is working and Machine B is in repair, Machine A is in repair and Machine B is working. At a small work center that runs without any time-breaks or holidays, the production is carried on with two distinct machines (lets call them Machine A and Machine B), and these two machines are working simultaneously if they are functional. These machines can fail from time to time and times to failures are exponentially distributed with a mean of 10 days and 15 days for Machine A and Machine B, respectively. There are two repairmen to fix these machines when they are broken. If only one machine is broken only a single repairman is working on that machine and if there are more than one broken machine, both repairmen are working on separate machines. Repair times are exponentially distributed and the repair rates are 2 per day for each machine. Whenever a machine is fixed, it starts production again. Whenever a machine fails, its repair starts right away (if there is an idle repairman). All times to failures and repair times are independent of each other. When Machine A and B are functional, they produce 15 and 10 electrical wires per hour, respectively and each wire is sold at 10 TL. Repair cost for each machine is 10,000 TL per day (if a machine is waiting to be repaired, the repair cost is not incurred for that machine). a) Construct an appropriate continuous-time Markov chain (CTMC) by stating what would be the states telling us. Then, compute the limiting probabilities for each state. b) Compute the expected profit per day by including profit obtained by the production of wires and cost of repairs. c) Write Monte Carlo Simulations to obtain the following values (a) proportion of times in which both machines are working, both machines are in repair, Machine A is working and Machine B is in repair, Machine A is in repair and Machine B is working