Q1. The Creekside Townhomes Apartment building in Dallas has a total of 240 rental units. It offers its new tenants an 18-month lease, which is renewed on a month-to-month basis if the tenant chooses to stay longer than 18 months. The leasing office reports that The Creekside Townhomes signs about 80 leases with new tenants in an average year, and that 60% of tenants who move in end up staying in the building longer than 18 months. On average, the building is 90% occupied. You may assume that a negligible number of tenants break their lease in the first 18 months.

a. (1%) What is the average number of tenants who are in their first 18 months at Creekside Townhomes?

b. (2%) What is the average length of stay (in months) for those tenants who stay longer than 18 months (counted from when they first sign their original lease)?

Q2. M.M. Sprout, a catalog mail-order retailer, has one customer service representative (CSR) to take orders at an 800 telephone number. If the CSR is busy, the next caller is put on hold. For simplicity, assume that any number of incoming calls can be put on hold and nobody hangs up in frustration over a long wait. Suppose that, on average, one call comes every 5 minutes and that it takes the CSR an average of 4 minutes to take an order. Both interarrival and activity times are exponentially distributed (i.e., they have coefficients of variation equal to 1). The CSR is paid $20 per hour, and the telephone company charges $5 per hour for the 800 line. The company estimates that each minute a customer is kept on hold costs it $2 in customer dissatisfaction and loss of future business. Estimate the following:

a. (1%) The average time that a customer will be on hold.

b. (1%) The average number of customers on line.

c. (2%) The total hourly cost of service and waiting

Q3. 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 bank’s process manager estimates the average rate of customer arrivals at 40 per hour. All arriving cars line up in a single file and are served at one of 4 windows on a first-come/first-served basis. Each teller currently requires an average of 5 minutes to complete a transaction. The bank is considering the possibility of leasing high-speed information-retrieval and communication equipment that would cost $30 per hour. The new equipment would, however, serve the entire facility and reduce each teller’s transactionprocessing time to an average of 4 minutes per customer. Assume that interarrival and activity times are exponentially distributed.

a. (2%) If our manager estimates the cost of a customer’s waiting time in queue (in terms of future business lost to the competition) to be $20 per customer per hour, can she justify leasing the new equipment on an economic basis?

b. (1%) Although the waiting-cost figure of $20 per customer per hour appears questionable, a casual study of the competition indicates that a customer should be in and out of a drive-in facility within an average of 8 minutes (including waiting). If First Local wants to meet this standard, should it lease the new high-speed equipment?

Q4. (5%) A small independent shop that sells three types of products (bakery, dairy and charcuterie) is trying to optimize the service provided to customers in light of the restrictions imposed by lockdown. Under normal operating conditions, customers come in through one door, choose their products from all three types and pay at the two tills next to the other door from which they exit. With the lockdown, and given the space inside the store, a maximum of 9 customers are allowed in at a time. Other customers form a single line outside of the shop. Data collected by the shop suggests that, at the busiest times (weekends and holidays) customers arrive at the shop at an average rate of 42 per hour and that the inter-arrival times have a coefficient of variation of 2.5 (CVIAT=2.5). Once inside the store, each customer takes an average of 10 minutes to select and pay for their products. This time is normally distributed, and the standard deviation is 4 minutes. Assuming every customer that joins the queue to enter the shop will not leave it until they’ve entered the store and made their purchase, on average, how many customers are queueing outside of the shop at any time? (Please explain how you apply the queueing model in this business context.)