Machines in a factory break down at an average rate of 3 per day. When a machine breaks down, it enters a queue to be repaired. There is a single repairperson on duty, who repairs machines at an average rate of 5 per day. Assume that the process of machines breaking down and being repaired can be modelled as an M/M/1 queue with an arrival rate of per day and a service rate of per day. Although the factory is only open for 8 hours per day, the state of the system at the beginning of any particular day is the same as the state at the end of the previous day, so that the system effectively evolves in a continuous, uninterrupted way over time.

D) The factory owners pay the repairperson 120 per day. During the time that a machine is broken (including the time spent in repair), the factory incurs costs at a rate of 95 per day due to loss of productivity. The owners are considering hiring an extra repairperson, who would be paid at the same rate as the current one (you may also assume that they can repair machines at the same rate). Do you think it would be profitable for them to hire the extra repairperson? Show your calculations and also state any additional assumptions that you make.