2. Queueing models (6 marks) A bakery that serves delicious sandwiches for take away has on average 9 customers arriving per hour. The bakery employs one staff who serves the sandwiches to customers and he takes on average 4 minutes to serve each customer. The bakery opens 10 hours per day, and makes a profit of $10 for each sandwich sold. The staff has a pay rate of $30/hour. Assume a Poisson distribution on the arrival rate of customers and exponential distribution on the service time performed by the staff. a) If customers do not want to spend more than 6 minutes in the bakery (including waiting and service), how much is the potential profit that the bakery would lose per day? (1 mark) b) The bakery management considers to improve customer service by reducing customers' waiting time. In doing so, the management considers the following two options: Option 1: to employ another staff member to split the tasks equally with the existing staff in serving customers. Option 2: to employ another staff member (with a similar level of skills to the original staff) to open a second service counter and provide full service to customers. Compare the two options in terms of the improvement in the customers' waiting time and provide your recommendation to the bakery. Are there any other factors that need to be considered by the bakery management in making these decisions? (3 marks) c) The bakery management considers purchasing an automatic machine that can perform full service for customers with the same service time (4 minutes under normal distribution). If the existing staff need to match the customers' average waiting tin achieved by the automatic machine, how quickly would the staff have to serve each customer? (2 marks)