2. Queueing problems A small take away cafe that serves hot beverages opens 12 hours per day, and makes a profit of $3 for each cup. Daily observations show that the average arrival rate of customers is 8 per hour. The cafe employs one barista who serves hot beverages to customers and he takes on average 5 minutes to serve each customer (including the time to take the orders and the payment as well as making the hot drinks). Assume a Poisson distribution on the arrival rate of customers and exponential distribution on the service time performed by the barista. Answer the following questions: a) Due to a limited space and social distancing regulation, there are a maximum of 2 customers can be accommodated inside the caf, and the rest has to wait outside. What is the probability of customers have to wait outside the bakery? b) To improve the service, the management wants to reduce the probability of customers waiting outside the cafe down to 15%. How would that decision impact on the baristas service time? and customers waiting timeUse trial and error to find the answer, and show the two closest figures above and below your final answer at 2 decimal points. c) Supposed customers do not want to wait in a queue more than 7 minutes, how much the average potential profit that the cafe would lose per day due to customers walk away from the queue? d) The management considers to add another barista (with a similar level of skills) in the cafe to serve more customers. How would this plan affect the average waiting time of the customers? (1 mark). What is your view on this plan? e) If the barista is replaced by an automatic machine with the same service time (5 minutes per serve with normal distribution), how would this new system affect the average waiting time of customers? f) If the barista wants to match the customers average waiting time achieved by the automatic machine, how quick he has to serve each customer? (1 mark) What is the probability of customers have to wait outside the bakery?