Question

First Local Bank would like to improve customer service at its drive-in facility by reducing waiting and transaction times. On the basis of a pilot study, the bankâ€™s process manager estimates the average rate of customer arrivals is 15 cars per hour. All arriving cars line up in front of a window and are served on a first-come, first-served basis. Each transaction requires on average 3 minutes to complete. Assume that the arrival of cars follows a Poisson process, and that the transaction times are exponentially distributed.

The bank now has one window that serves the drive-in customers. It costs $20 an hour to staff this window. The customer relations (CR) manager estimates that the cost (loss of goodwill and future business, etc.) due to a customerâ€™s waiting in the queue has a linear form, i.e., the bank incurs a cost of 10 dollars per hour per customer in queue. On the other hand, as soon as a customer goes into service, the bank does not incur any goodwill loss.

- What is the average time (in minutes) a customer spends in queue?

- What is the probability that a customer needs to wait before being served?

- What is the average number of customers in the system (including the ones waiting in line and the ones receiving service at the window)?

- What is the probability that there are 3 customers in the system (including the ones waiting in line and the ones receiving service at the window)?

- What is the utilization of the customer service representative at the window?

- The bank is considering the possibility of leasing high-speed information- retrieval and communication equipment that costs $20 an hour. The equipment is operated by the original staff employee. As a result, the average transaction time will be reduced by one minute. To minimize the total cost of staffing, equipment, and customer waiting, should the bank lease the communications equipment? That is, how does the total cost after leasing compare to that before?

- Another way to improve customer service is to open a second window instead of leasing high-speed information-retrieval and communication equipment. It costs $20 an hour to staff the second window, and both windows have the same average service time, 3 minutes. The customers will form
*two separate lines*, each in front of one window. Upon arrival, a customer will choose one of the windows to line up with equal probability (i.e., 50% chance), and jockeying between queues is not allowed. The service discipline is first-come, first-served. Based on the total cost of staffing and customer waiting, is this alternative better than status quo? What is the total cost now?

The manager realized that letting customers form two separate lines is not an efficient queue layout. Instead, the customers, upon arrival, are asked to form one single line in front of the two windows. (There are still two windows open).

- Comparing to the two separate lines layout, what is the difference, if any, in the average number of customers waiting in line when using the single line layout? Explain the difference (or indifference) with intuition.

- Comparing to the two separate lines layout, what is the difference, if any, in the utilization of the service windows when using the single line layout? Explain the difference (or indifference) with intuition.

- Now, suppose the customers
*form a single queue*, and the service discipline is first-come, first-served. Based on the total cost of staffing and customer waiting, is this alternative better than the status quo? What is the total cost now?