Oceania Hair Dressing has space available in the hair salon for five patrons including those having their hair cut. Currently the salon has two hairdressers and only one hairdresser cuts the hair of a patron at a time. However if more than three patrons are in the salon then one apprentice hairdresser vill cut hair but only whilst there are more than three patrons in the salon. Potential patrons, upon seeing the salon is full, will not enter the salon and instead go elsewhere to get their hair cut. Patrons arrive according to a Poisson process at the mean rate of 12|84n| per hour where n is the number of patrons in the salon. The average time for a hairdresser to cut a patron's hair is 15 minutes and is exponentially distributed. (a) Draw the state transition diagram and write down the steady-state (balance) equations for the model. Solve these equations to find the probabilities that there are n patrons in the salon. (4 marks) (b) On average, how many patrons are at the salon either having their hair cut or waiting to do so? (2 marks) (c) How long on average would a patron spend at the salon including having their hair cut? (3 marks) (d) What is the expected number of hairdressers (including the apprentice) that are busy cutting hair? (3 marks) (e) What is the probability that the apprentice hairdresser will not be busy cutting hair? (3 marks)