Analytical: Time and Arrival Rate Tellers Essay

Submitted By mawjirah
Words: 2485
Pages: 10

Solution to First City National Bank1
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 