# Question: the neighborhood coffee shop operated by the koffee kompany is...

###### Question details

The neighborhood coffee shop, operated by the Koffee Kompany, 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 36 customers per hour. In designing the store, the manager can choose between two different operating configurations using different kinds of espresso machines:

• 2 Coffee Classic Machines

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

16. What is the average service time of a single Coffee Classic machine in minutes? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

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

18. With the Coffee 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.

Next consider the Coffee “2Xfaster” machine. As its name suggests, 2Xfaster can work twice as fast as the Coffee 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.

19. What is the utilization of the 2Xfaster machine? State your answer as a percentage with at least three decimal points, e.g., 50.000 (for 50% utilization).

20. What is the coefficient of variation of the customer interarrival time to the store? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

21. What is the coefficient of variation of the service time for the 2Xfaster machine? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

22. What is the average service time of the 2Xfaster machine in minutes? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

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

24. 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.

25. Management is worried about the cost of excessive customer waiting time, because each moment that customers wait increases the chance of customer abandonment and balking. After some analysis, the manager estimates that the hourly cost to the coffee shop for a customer’s total time in the store is $10 (that is, the waiting cost rate is $10 per customer per hour, which accounts for time in the queue and time being served). If you estimate that the average total time in the store for each customer is T minutes, what is the total waiting cost per hour?

a. $10 x (T/60)

b. $10 x 36 x (T/60)

c. $10 x 40 x (T/60)

d. $10 x (36/40) x (T/60)