Problem 3: K-Caf

The neighborhood coffee shop, K-Caf, is opening a new store. Management expects that once the store is open, customers will arrive according to a Poisson process with a mean of 32 customers per hour. In designing the store, the management can choose between two different operating configurations using different kinds of espresso machines:

• Two "Classic" Machines
• One "2Xfaster" Machine

You have been tasked with analyzing these alternatives and determining the customer waiting time and queue size for each, both important metrics that impact customer satisfaction and the atmosphere in the store.

Consider first the Classic machine. One such machine can serve, on average, 20 customers per hour, and service times for this machine are exponentially distributed. The management knows that if they purchase this type of machine, they will require two identical machines to meet demand; any more machines would be impractical given the physical layout of the store. The customer orders will be placed in a single queue in front of both machines, from which orders will be routed to the first available machine.

1. What is the utilization of the Classic machines? Please enter your answer as a value between 0 and 1. If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

2. With the two Classic machines, what is the expected total time (in minutes) that a customer spends in the store (i.e., the total time in queue and in service)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

3. With the two Classic machines, what is the expected total number of customers in the store (i.e., in queue and in service)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

Problem 3: K-Caf Continued

Next consider the "2Xfaster" machine. As its name suggests, 2Xfaster can work twice as fast as the Classic machine, i.e., it can serve, on average, 40 customers per hour, although it only processes one drink order at a time. The service time for 2Xfaster is also exponentially distributed. A 2Xfaster machine is significantly larger than the Coffee Classic, so the physical limitations of the store would only allow for the purchase of one machine.

4. What is the utilization of the 2Xfaster machine? Please enter your answer as a value between 0 and 1. If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

Correct Answer: 0.8

5. With the 2Xfaster machine, what is the expected total time (in minutes) that a customer spends in the store (i.e., the total time in queue and in service)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

6. With the 2Xfaster machine, what is the expected total number of customers in the store (i.e., in queue and in service)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

Hint: Please ensure that all quantities are in the same units!