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 bank’s process manager estimates the average rate of customer arrivals at 40 per hour. All arriving cars line up in a single queue 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 transaction-processing time to an average of 4 minutes per customer. Assume that interarrival and activity times are exponentially distributed.

a) 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) 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 Bank wants to meet this standard, should it lease the new high-speed equipment?

Answer 1

The average interarrival time between two successive arrivals is 1.5 minutes. When the store first opens, it takes an average of 4.5 minutes for three customers to arrive.

Step-by-step explanation:

a. The average interarrival time between two successive arrivals can be found by taking the reciprocal of the average arrival rate. In this case, the average rate of customer arrivals is 40 per hour, so the average interarrival time is 1/40 hour, or 1.5 minutes.

b. To find the average time it takes for three customers to arrive when the store first opens, we can add together the average interarrival times for the first three customers. Since the average interarrival time is 1.5 minutes, it will take an average of 4.5 minutes for three customers to arrive.

Answer 2

Economic analysis shows that the saving in customer's time may not cover the cost of the new equipment when valued at $20 per hour. However, the decision might depend on the actual value of reduced waiting time and whether the bank wants to meet a service standard of 8 minutes total time spent by the customer.

Step-by-step explanation:

Economic Justification for Leasing New Equipment

In the scenario presented, the average rate of customer arrivals is 40 per hour, which translates to one customer every 1.5 minutes (60 minutes divided by 40 customers). Currently, each teller completes a transaction in 5 minutes, so with 4 tellers, the bank processes customers at a rate of 48 per hour (4 tellers times 12 customers per teller per hour), which is above the arrival rate of 40 per hour. However, if customer waiting time is monetized at $20 per customer per hour, and the new equipment reduces each transaction time to 4 minutes, the new service rate with the equipment would be 60 customers per hour (4 tellers times 15 customers per teller per hour), which significantly exceeds the arrival rate, potentially reducing waiting time.

The cost of the new equipment is $30 per hour. If we assume that the reduced transaction time saves 1 minute per customer, for the 40 customers arriving per hour, this amounts to a total saving of 40 minutes of customer time. Monetized at $20 per hour, this equates to a saving of approximately $13.33 (40 minutes / 60 minutes x $20). This does not cover the cost of the new equipment. Therefore, on an economic basis, leasing the new equipment to save customer's time alone may not be justified unless the waiting time has a greater value than estimated.

In terms of the service standard of 8 minutes in and out of the drive-in facility, the current average service time is 5 minutes plus the average waiting time (which depends on queue length and service rate). Without the exact queue model and service rate calculation, we can't provide a conclusive answer, but if the current waiting time is significantly above 3 minutes on average, the new equipment might be necessary to meet the 8-minute standard.