A cafe that serves cakes for takeaway has on average 9 customers arriving per hour. The cafe employs one staff who serves the cake to customers, and she takes on average 4 minutes to serve each customer. The cafe opens 10 hours per day and makes a profit of $10 for each cake 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.

1. If customers do not want to spend more than 6 minutes in the cafe (including waiting and service), how much is the potential profit that the cafe would lose per day?

2. The cafe management considers improving 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 cafe. Are there any other factors that need to be considered by the cafe management in making these decisions?

3. The cafe 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 time achieved by the automatic machine, how quickly would the staff have to serve each customer?