You are assigned the task of deciding how many tellers to employ for the 11:30-12:00 slot on a normal day. Assume that the cost of using a teller for an hour is $40. It is clear that customers are not happy to wait in order to quantify economically the pain of waiting, you have been asked to value a customers time at the rate of $300 per hour that is, if a customer waits for 6 minutes (0.1 hours) in the queue (not including the service time), this waiting time is equivalent to the customer incurring a cost of $30. Also, assume that the single line teller arrangement of Exhibit 1 is used. Find the optimal number of tellers to employ for this half an hour slot your objective is to minimize the total cost of hiring some number of tellers plus the total expected waiting time costs for the customers. What is the utilization when this number of tellers is used? Hints: 1. The spreadsheet requires 3 inputs, an arrival rate, a service rate and the number of servers. The first two inputs should be gathered from the data. The number of servers is our decision which you can vary. 2. The service rate and arrival rate should both be expressed in the same units, for example customers per hour or customers per half-hour. If you choose to express them in customers per hour, then the expected waiting time that you see on the spreadsheet is the expected waiting time in hours. If you choose to express them in customers per half-hour, then the expected waiting time that you see on the spreadsheet is the expected waiting time in half-hours (for example, if the spreadsheet says the expected waiting time is 2.2, then, it means that the expected waiting time is 1.1 hours in this case). 3. The average service time is given in the case. The service rate is the reciprocal of this. 4. The average arrival rate given in Exhibit 4 is the average arrival rate in customers per half-hour.

M/M/s 1 Arrival rate Service rate Number of servers 2 1 Utilization 50.00% Lq, expected queue length Ls, expected number in system Wq, expected time in queue Ws, expected total time in system 0.5000 1.0000 0.5000 1.0000 First City National Bank In March 1987, David Craig, vice president of operations for First City National Bank of Philadelphia was considering a change in teller operations. Currently, the bank's tellers were arranged in pods to handle customer transactions. There were four pods containing three teller stations each. One pod was used primarily for savings accounts since some savings transactions took longer than other types of deposits or withdrawals. The major problem with the pod system was that one pod might be crowded while another was vacant. The distance between pods was such that the customers were unwilling to move from one to another. Mr. Craig was considering two alternatives to the pod system. The first was a single-line teller arrangement as shown in Exhibit 1. Using this plan, all customers would wait in a single line until a teller became available. The person at the head of the line would then move to the open teller. Mr. Craig thought that 10 tellers would be required to handle the bank's usual business. However, he could not be sure of the exact number without further study. Alternative 1-Single Line EXHIBIT 1 Teller arrangements. . . Alternative 2-Multiple Lines Source: Schroeder, R.G., Operations Management, 1993, McGraw-Hill, Inc. Exhibit 1 also shows the second alternative teller arrangement. Using this more conventional plan, the customers would form separate lines in front of each of the teller windows. Thus for 10 tellers, a total of 10 different lines could be formed. In evaluating these alternatives, several issues were of utmost importance. First, Mr. Craig was concerned with both customer waiting time and teller efficiency. On the basis of past experience, Mr. Craig felt that more than 3 minutes of waiting time would be unacceptable to most customers. He also felt that teller utilization should be as high as possible, perhaps in the 80 to 90 percent range. Since demand varied during the day, the number of tellers provided would have to vary to meet the customer-service and teller- utilization goals. 70 60 Day 4/20/91: Friday Time Period: 11:45-12:45 Total Arrivals: 431 customers Average Time Between Arrivals: 8.4 Ser 50 40 Number of Observations 30 20 10 0 1 3 5 7 9 11 21 23 25 27 29 31 13 15 17 19 Interarrival time Exhibit 2 - Histogram of Interarrival times The statistical distribution of service time and arrival time is shown in Exhibits 2 and 3. The service time averages 45 seconds per customer and does not vary by time of day. On the other hand, the average time between arrivals does vary with the time of day. For example, between 11:45 and 12:45 on one particular day sampled, 431 customers arrived at the bank, with an average of 8.4 seconds between customers. 70 60 50 40 Number of Observations Avg. = 45.5 seconds s= 37.8 seconds n= 308 observations 30 20 10