Solution to First City National Bank[1]
a)Considering the data supplied for arrival and service times, how would you calculate an average arrival rate and service rate?
We need to be consistent with our units here. Notice that the arrival data are in units of “customers per 30 minutes” and the service data are in units of “seconds per customer”. We can't use any of our queueing formulas until both the arrival rate and the service rate are in comparable units.
It would seem logical to convert both of these measures into “minutes per customer”, because the manager is apparently interested in a particular service target of 3 minutes waiting time per customer.
Example: On normal days between 8:30 and 9:00, we observed a total of 803 arrivals. There were 41 of these normal days, so the average number of arrivals during that half-hour interval was
To convert this into our desired units, we divide by 30:
Therefore, during the period from 8:30 to 9:00 on Normal days, we estimate that customers arrive at a rate of 0.653 per minute.
As for service times, we are told that the average customer service takes 45 seconds, which is equivalent to 0.02222 customers per second:
We multiply this times 60 to get the average number of customers served per minute:
Using Excel, we can quickly convert all of the raw data into these units of “customers per minute” and make a chart showing how customers tend to arrive at the bank over the course of a day:
Time of Day / Arrival Rate - Normal / Arrival Rate - Peak / Arrival Rate -Super Peak / Service Rate
(per Teller)
8:00-8:30 / 0.653 / 0.744 / 0.849 / 1.333
8:30-9:00 / 0.747 / 0.902 / 1.072 / 1.333
9:00-9:30 / 0.981 / 1.027 / 1.464 / 1.333
9:30-10:00 / 2.098 / 2.420 / 3.149 / 1.333
10:00-10:30 / 2.113 / 2.663 / 3.544 / 1.333
10:30-11:00 / 2.333 / 2.718 / 3.428 / 1.333
11:00-11:30 / 2.751 / 3.125 / 4.044 / 1.333
11:30-12:00 / 3.698 / 4.833 / 5.962 / 1.333
12:00-12:30 / 4.719 / 6.344 / 7.456 / 1.333
12:30-1:00 / 4.350 / 5.861 / 6.985 / 1.333
1:00-1:30 / 3.541 / 4.742 / 5.823 / 1.333
1:30-2:00 / 2.953 / 3.750 / 5.105 / 1.333
2:00-2:30 / 1.887 / 2.395 / 3.287 / 1.333
2:30-3:00 / 1.573 / 2.333 / 3.092 / 1.333
3:00-3:30 / 1.749 / 2.457 / 3.205 / 1.333
3:30-4:00 / 1.720 / 2.664 / 3.405 / 1.333
4:00-4:30 / 1.863 / 2.786 / 3.451 / 1.333
4:30-5:00 / 1.670 / 2.608 / 3.118 / 1.333
5:00-5:30 / 1.299 / 2.099 / 2.369 / 1.333
Average / 2.247 / 2.972 / 3.727 / 1.333
b)As Mr. Craig, what characteristics of this queuing system would you be most interested in observing?
According to the case, it would appear that there are two measures of interest:
- The average time spent in the waiting line by the customers (Wq). Mr. Craig seems to want this to be 3 minutes or less.
- The proportion of tellers' time actually spent helping customers (). He wants to have this be 80% to 90%.
c)What is the best number of tellers to use?
d)Calculate the waiting time for a customer (time in the queue before service) and determine which of the two line configurations you would recommend? Support your result with the appropriate quantitative queuing analysis.
Using Excel and/or HOM, we can study the effects of S (the number of tellers) on Wq (the amount of time an average customer spends waiting for service) and answer Parts c and d.
It is important not to forget several key assumptions we are making when we use the M/M/S queueing formulas:
- We assume that both the times between arrivals and the times between services are exponentially distributed. We can evaluate these assumptions either by using a statistical hypothesis test of goodness-of-fit (e.g. Chi-square test) or by a less-sophisticated visual inspection of the histograms provided in our case. In the case of arrivals the exponential assumption is reasonable, but in the case of service times the graph suggests that an exponential assumption might not be appropriate. We can go ahead and use the M/M/S formulas and see what implications they offer, but to do a really good job, we might want to do some sensitivity analysis: we might see how our conclusions vary when we try different service rate distributions in our model.
- We use (the arrival rate parameter) and (the service rate parameter) as though they are known and fixed quantities. In the case of this is clearly inappropriate from looking at the graph above. We will need to study different values for in order to arrive at some reasonable staffing decisions for this bank. If we use an average rate (say, 2.2 customers per minute — the observed average on normal days) we will be understaffed during the peak demand in the middle of the day and overstaffed during the slow periods in the early morning and late afternoon. As for the service rate, it apparently doesn't vary across different time periods, so a single estimate for is probably fine.
With those caveats, we can study a quantitative model of the system under various sets of assumptions.
Scenario A: Each Teller has Own Queue
We can model this scenario using the M/M/1 formulas, simply dividing the arrival rate by the number of tellers. Using Excel, we can study this situation with a spreadsheet that incorporates the formulas Wq and as functions of and .
By experimenting with this spreadsheet, we can look for arrangements whereby both of the performance targets are met. Here are some results using the same spreadsheet as above:
Lambda/Teller / Mu / Rho / Wq1.0664 / 1.333 / 0.7998 / 2.9963
1.0665 / 1.333 / 0.7999 / 2.9977
1.0666 / 1.333 / 0.8000 / 2.9991
1.0667 / 1.333 / 0.8000 / 3.0005
1.0668 / 1.333 / 0.8001 / 3.0019
1.0669 / 1.333 / 0.8002 / 3.0033
We can see that the target of 3 minutes maximum waiting time per customer is satisfied only if the arrival rate per teller is fewer than 1.0667 customers per minute. We can also see that the target of at least 80% utilization is only satisfied for arrival rates greater than 1.0665 customers per minute. If the M/M/1 model is a good representation of the real system, it will be practically impossible for Mr. Craig to have it both ways. He will need to decide whether to emphasize short customer waiting time or high teller utilization.
To answer the question of how many tellers to hire, I assume that we want to stick to the 3 minute customer waiting time target. We know that to do this, we need to keep the average number of arrivals per minute to less than about 1.06, so we can calculate the minimum number of tellers needed at any time of day using the formula:
The arrival rate is different at different times of day (and different types of days). A table on page 7 shows the minimum staffing levels that are needed based on this analysis.
The same analysis can be performed in HOM. In the Queueing Techniques module, we set the parameters as shown here:
We can run this simulation and get results consistent with the Excel method. The output from HOM includes a probability distribution graph of the number of customers in the system, a table of "Base Case Parameters" (the same parameters we input above), and some tables of results (shown on the following page) for all of the usual measures of performance for a queueing system.
Notice that Excel is quicker to set up and get results, but that HOM is more powerful and provides sensitivity analysis that isn't easily done with Excel. The advantages of HOM will be even more compelling in the next scenario, in which we study the system as an M/M/S system with a single queue for all tellers. The formula for Wq gets very complicated with more than one server, and using Excel becomes prohibitively time-consuming.
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B60.2350 Decision ModelsProfessor Juran
HOM Output
Results of Calculations
Base Case / Sensitivity AnalysisNumber of Servers / 1 / 1 / 1 / 0 / 2
Service Rate, (100%=1.3333) / Base Case / 105% Base Cs / 95% Base Cs / Base Case / Base Case
Method (Model) Used / M/M/C / M/M/C / M/M/C / Not Used / M/M/C
Utilization / 0.0937502 / 0.0892859 / 0.0986845 / N/A / 0.0468751
Avg. No. Customers in System / 0.103449 / 0.0980395 / 0.109489 / N/A / 0.0939567
Std.Dev. of No.Cust in Syst. / 0.337861 / 0.328102 / 0.348536 / N/A / 0.307198
Avg. No. Customers in Queue / 0.00969833 / 0.00875355 / 0.0108049 / N/A / 0.000206449
Std.Dev. of No.Cust in Queue / 0.107754 / 0.101948 / 0.114255 / N/A / 0.015057
Average Time Spent in System / 0.827588 / 0.784316 / 0.875915 / N/A / 0.751653
Std.Dev. of Time in System / 0.827588 / 0.784316 / 0.875915 / N/A / 0.750866
Average Time Spent in Queue / 0.0775866 / 0.0700284 / 0.0864392 / N/A / 0.00165159
Std.Dev. of Time in Queue / 0.349857 / 0.323952 / 0.379414 / N/A / 0.0360123
Fraction of Time Syst. Empty / 0.90625 / 0.910714 / 0.901316 / N/A / 0.910448
Fraction of Customers Lost / 0 / 0 / 0 / N/A / 0
Fraction of Time N Customers in the System
No. of Customers in System,N / ------/ ------/ ------/ ------/ ------0 / 0.90625 / 0.910714 / 0.901316 / N/A / 0.910448
1 / 0.0849611 / 0.081314 / 0.0889458 / N/A / 0.0853547
2 / 0.00796513 / 0.00726019 / 0.00877757 / N/A / 0.00400101
3 / 0.000746732 / 0.000648233 / 0.00086621 / N/A / 0.000187548
4 / 7.00063e-005 / 5.78781e-005 / 8.54815e-005 / N/A / 8.79133e-006
5 / 6.56311e-006 / 5.1677e-006 / 8.43569e-006 / N/A / 4.12094e-007
Results from M/M/1 Model
Normal / Peak / Super PeakTime of Day / Arrival Rate / Tellers Needed / Rounded / Arrival Rate / Tellers Needed / Rounded / Arrival Rate / Tellers Needed / Rounded
8:00-8:30 / 0.653 / 0.616 / 1 / 0.744 / 0.702 / 1 / 0.849 / 0.801 / 1
8:30-9:00 / 0.747 / 0.705 / 1 / 0.902 / 0.851 / 1 / 1.072 / 1.011 / 2
9:00-9:30 / 0.981 / 0.926 / 1 / 1.027 / 0.969 / 1 / 1.464 / 1.381 / 2
9:30-10:00 / 2.098 / 1.979 / 2 / 2.420 / 2.283 / 3 / 3.149 / 2.970 / 3
10:00-10:30 / 2.113 / 1.993 / 2 / 2.663 / 2.512 / 3 / 3.544 / 3.343 / 4
10:30-11:00 / 2.333 / 2.201 / 3 / 2.718 / 2.564 / 3 / 3.428 / 3.234 / 4
11:00-11:30 / 2.751 / 2.595 / 3 / 3.125 / 2.948 / 3 / 4.044 / 3.815 / 4
11:30-12:00 / 3.698 / 3.488 / 4 / 4.833 / 4.560 / 5 / 5.962 / 5.624 / 6
12:00-12:30 / 4.719 / 4.452 / 5 / 6.344 / 5.985 / 6 / 7.456 / 7.034 / 8
12:30-1:00 / 4.350 / 4.104 / 5 / 5.861 / 5.529 / 6 / 6.985 / 6.589 / 7
1:00-1:30 / 3.541 / 3.340 / 4 / 4.742 / 4.473 / 5 / 5.823 / 5.493 / 6
1:30-2:00 / 2.953 / 2.786 / 3 / 3.750 / 3.538 / 4 / 5.105 / 4.816 / 5
2:00-2:30 / 1.887 / 1.780 / 2 / 2.395 / 2.260 / 3 / 3.287 / 3.101 / 4
2:30-3:00 / 1.573 / 1.484 / 2 / 2.333 / 2.201 / 3 / 3.092 / 2.917 / 3
3:00-3:30 / 1.749 / 1.650 / 2 / 2.457 / 2.318 / 3 / 3.205 / 3.024 / 4
3:30-4:00 / 1.720 / 1.622 / 2 / 2.664 / 2.513 / 3 / 3.405 / 3.212 / 4
4:00-4:30 / 1.863 / 1.757 / 2 / 2.786 / 2.628 / 3 / 3.451 / 3.256 / 4
4:30-5:00 / 1.670 / 1.575 / 2 / 2.608 / 2.461 / 3 / 3.118 / 2.941 / 3
5:00-5:30 / 1.299 / 1.226 / 2 / 2.099 / 1.980 / 2 / 2.369 / 2.235 / 3
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B60.2350 Decision ModelsProfessor Juran
Scenario B: One Queue for All Tellers
In this case, we need to use a more complicated formula for Wq, and HOM turns out to be faster than Excel. Below are tables showing HOM results for different arrival rates () and different numbers of tellers (S).
Average Minutes in Queue (Wq)
S / Lambda0.5 / 1.0 / 1.5 / 2.0 / 2.5 / 3.0 / 3.5 / 4.0 / 4.5 / 5.0 / 5.5 / 6.0 / 6.5 / 7.0
1 / 0.450 / 2.905 / - / - / - / - / - / - / - / - / - / - / - / -
2 / 0.027 / 0.123 / 0.347 / 0.964 / 5.444 / - / - / - / - / - / - / - / - / -
3 / 0.002 / 0.015 / 0.048 / 0.118 / 0.258 / 0.568 / 1.547 / - / - / - / - / - / - / -
4 / 0.000 / 0.002 / 0.008 / 0.022 / 0.051 / 0.103 / 0.948 / 0.382 / 0.813 / 2.595 / - / - / - / -
5 / 0.000 / 0.000 / 0.001 / 0.004 / 0.011 / 0.025 / 0.048 / 0.089 / 0.157 / 0.277 / 0.517 / 1.144 / 5.629 / -
6 / 0.000 / 0.000 / 0.000 / 0.001 / 0.002 / 0.006 / 0.013 / 0.025 / 0.044 / 0.076 / 0.126 / 0.211 / 0.362 / 0.681
7 / 0.000 / 0.000 / 0.161 / 0.000 / 0.000 / 0.001 / 0.003 / 0.007 / 0.013 / 0.024 / 0.040 / 0.065 / 0.104 / 0.166
The bold face type indicates the smallest number of tellers that is sufficient to meet the target of Wq < 3 minutes.
Average Utilization ()
S / Lambda0.5 / 1.0 / 1.5 / 2.0 / 2.5 / 3.0 / 3.5 / 4.0 / 4.5 / 5.0 / 5.5 / 6.0 / 6.5 / 7.0
1 / 0.375 / 0.750 / - / - / - / - / - / - / - / - / - / - / - / -
2 / 0.188 / 0.375 / 0.563 / 0.750 / 0.938 / - / - / - / - / - / - / - / - / -
3 / 0.125 / 0.250 / 0.375 / 0.500 / 0.625 / 0.750 / 0.875 / - / - / - / - / - / - / -
4 / 0.094 / 0.188 / 0.281 / 0.375 / 0.469 / 0.563 / 0.656 / 0.750 / 0.844 / 0.938 / - / - / - / -
5 / 0.075 / 0.150 / 0.225 / 0.300 / 0.375 / 0.450 / 0.525 / 0.600 / 0.675 / 0.750 / 0.825 / 0.900 / 0.975 / -
6 / 0.062 / 0.125 / 0.188 / 0.250 / 0.313 / 0.375 / 0.438 / 0.500 / 0.563 / 0.625 / 0.688 / 0.750 / 0.813 / 0.875
7 / 0.054 / 0.107 / 0.000 / 0.214 / 0.268 / 0.321 / 0.375 / 0.429 / 0.482 / 0.536 / 0.589 / 0.643 / 0.696 / 0.750
Notice that the average waiting times are shorter for the single queue system than for the system with each teller having his/her own waiting line. In general, holding the arrival rate per server and all other factors constant, a larger number of servers will result in shorter waiting times in the queue. This is illustrated in the chart below, in which the horizontal axis is in units of arrivals per minute per teller.
Mr. Craig could use these results on two levels. First, he can conclude that a single line for all tellers is preferable to individual lines for each teller. Second, he can make a simple table that shows the number of tellers necessary to meet the 3 minute waiting time target for various arrival rates:
Average Customers per Minute / Tellers Needed0.5 / 1
1.0 / 1
1.5 / 2
2.0 / 2
2.5 / 3
3.0 / 3
3.5 / 3
4.0 / 4
4.5 / 4
5.0 / 4
5.5 / 5
6.0 / 5
6.5 / 6
7.0 / 6
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B60.2350 Decision ModelsProfessor Juran
Results from M/M/S Model
Normal / Peak / Super PeakTime of Day / Arrival Rate / Tellers Needed / Arrival Rate / Tellers Needed / Arrival Rate / Tellers Needed
8:00-8:30 / 0.653 / 1 / 0.744 / 1 / 0.849 / 1
8:30-9:00 / 0.747 / 1 / 0.902 / 1 / 1.072 / 1
9:00-9:30 / 0.981 / 1 / 1.027 / 1 / 1.464 / 2
9:30-10:00 / 2.098 / 2 / 2.420 / 2 / 3.149 / 3
10:00-10:30 / 2.113 / 2 / 2.663 / 3 / 3.544 / 3
10:30-11:00 / 2.333 / 2 / 2.718 / 3 / 3.428 / 3
11:00-11:30 / 2.751 / 3 / 3.125 / 3 / 4.044 / 4
11:30-12:00 / 3.698 / 3 / 4.833 / 4 / 5.962 / 5
12:00-12:30 / 4.719 / 4 / 6.344 / 5 / 7.456 / 6
12:30-1:00 / 4.350 / 4 / 5.861 / 5 / 6.985 / 6
1:00-1:30 / 3.541 / 3 / 4.742 / 4 / 5.823 / 5
1:30-2:00 / 2.953 / 3 / 3.750 / 3 / 5.105 / 4
2:00-2:30 / 1.887 / 2 / 2.395 / 2 / 3.287 / 3
2:30-3:00 / 1.573 / 2 / 2.333 / 2 / 3.092 / 3
3:00-3:30 / 1.749 / 2 / 2.457 / 2 / 3.205 / 3
3:30-4:00 / 1.720 / 2 / 2.664 / 3 / 3.405 / 3
4:00-4:30 / 1.863 / 2 / 2.786 / 3 / 3.451 / 3
4:30-5:00 / 1.670 / 2 / 2.608 / 3 / 3.118 / 3
5:00-5:30 / 1.299 / 1 / 2.099 / 2 / 2.369 / 2
Note that the number of tellers needed under the M/M/S assumptions are always less than or equal to the number of tellers needed under the M/M/1 assumptions.
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B60.2350 Decision ModelsProfessor Juran
[1] Source: Schroeder, R.C., Operations Management (1993) McGraw-Hill, Inc. Solution by David Juran, 2000.