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.

(i) The expected number of machines waiting in the queue to be repaired at an arbitrary point in time is Lq = * Wq = 3 * (W - 1 / 5), where W is the expected time a machine spends in the system.

(ii) The expected amount of time that a machine must wait in the queue before the repairperson begins work on it is Wq = W - 1 / 5, where W is the expected time a machine spends in the system.

E) After some further analysis, the factory owners believe that the amount of time that the repairperson spends repairing a machine actually has a triangular distribution with minimum value hours, maximum value hours and mode hours. Given this new information, they believe that the system should be modelled as an M/G/1 queue rather than M/M/1. (Assume that they continue to use a single repairperson.) Recalculate the performance measures in (I and ii) for the new M/G/1 system. [Hint:look up the properties of the triangular distribution online.]