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(solution) OPERATIONS MANAGEMENT  DSC 410 FALL 2016  WAITING LINES 
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OPERATIONS MANAGEMENT ? DSC 410
FALL 2016 ? WAITING LINES ? HW #1
DUE MONDAY OCT 10th, 2016
NAME:
1. De Morgan?s Bank is the only bank in a small town in Georgia. On a
typical Friday, an average of 13.6 customers arrives per hour at the
bank to transact business. There is one single teller at the bank, and
the average time required to transact business is 4 minutes. It is
assumed that service times are exponentially distributed. Although
this is the only bank in town, some people in the town have begun
using the bank in a neighboring town about 20 miles away. A single
line is used and the customer at the front of the line goes to the first
available bank teller. It is estimated that waiting time cost per
customer per hour is $25. A bank teller is paid $12 per hour
Arrival rate = 13.6 customers per hour
Service time = 4 minutes = 0.0666666 hours
Cost per server = $12 per hour
Waiting cost = $25 per hour per customer
This problem was run using computer software. Time units were
hours. The problem was run twice, first under the assumption of one
teller, and secondly under the assumption of 2 tellers. Below is the
computer output:
De Morgan's Bank (1 Teller)
QUEUE 1 : M / M / C ?[TIME UNITS = HOURS]
Q U E U E S TAT I S TI C S
Number of identical servers . . . . . . . . .
Mean arrival rate . . . . . . . . . . . . . .
Mean service rate per server . . . . . . . . 1
13.6000
15.0000 Mean server utilization (%) . . . . . . . . .
Expected number of customers in queue . . . .
Expected number of customers in system . . .
Probability that a customer must wait . . . .
Expected time in the queue . . . . . . . . .
Expected time in the system . . . . . . . . . 90.6667
8.8076
9.7143
0.9067
0.6476
0.7143 1 De Morgan's Bank (1 Teller)
QUEUE 1 : M / M / C [TIME UNITS = HOURS]
COST ANALYSIS PER UNIT TIME
Current System
Optimal System *
Number of servers 
1

2

Cost per server
 12.0000
 12.0000

Cost of service

12.0000
24.0000
Mean number in system  9.7143
 1.1412

Waiting cost/customer  25.0000
 25.0000

Cost of waiting

242.8600
28.5300
TOTAL COST
254.8600
52.5300
* Optimization is over number of servers
De Morgan's Bank (1 Teller)
QUEUE 1 : M / M / C
PROBABILITY DISTRIBUTION OF NUMBER IN SYSTEM
Number Prob 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
+++++++++++
0 0.0933*****

1 0.0846****+
2 0.0767****
3 0.0696***+
4 0.0631***+
5 0.0572***
6 0.0518***
7 0.0470**+
8 0.0426**+
9 0.0386**
10 0.0350**
11 0.0318**
12 0.0288*+
13 0.0261*+
14 0.0237*+
15 0.0215*+ 2 De Morgan's Bank (2 Tellers)
QUEUE 2 : M / M / C [TIME UNITS = HOURS]
Q U E U E S TAT I S TI C S
Number of identical servers . . . . . . . . .
Mean arrival rate . . . . . . . . . . . . . .
Mean service rate per server . . . . . . . . 2
13.6000
15.0000 Mean server utilization (%) . . . . . . . . .
Expected number of customers in queue . . . .
Expected number of customers in system . . .
Probability that a customer must wait . . . .
Expected time in the queue . . . . . . . . .
Expected time in the system . . . . . . . . . 45.3333
0.2345
1.1412
0.2828
0.0172
0.0839 De Morgan's Bank (2 Tellers)
QUEUE 2 : M / M / C [TIME UNITS = HOURS]
COST ANALYSIS PER UNIT TIME
Current System
Optimal System *
Number of servers

2

2

Cost per server
 12.0000
 12.0000

Cost of service

24.0000
24.0000
Mean number in system  1.1412
 1.1412

Waiting cost/customer
 25.0000
 25.0000

Cost of waiting

28.5300
28.5300
TOTAL COST
52.5300
52.5300
* Optimization is over number of servers
De Morgan's Bank (2 Tellers)
QUEUE 2 : M / M / C
PROBABILITY DISTRIBUTION OF NUMBER IN SYSTEM
Number Prob 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
+++++++++++
0 0.3761*******************

1 0.3410*****************+
2 0.1546******** 
3 0.0701**** 
4 0.0318**
5 0.0144*
6 0.0065+
OVER 0.0055+ 3 +++++++++++ Use the above computer output to respond to the following
questions.
(a) On the average, how many minutes does a customer spend in the
line before getting served if one bank teller is on duty?
ANSWER
(b) On the average, how many minutes does a customer spend in the
line before getting served if two bank tellers are on duty?
ANSWER
(c) On the average, how many minutes does a customer spend in the
bank both waiting to be served and being served if one teller is on
duty?
ANSWER
(d)On the average, how many minutes does a customer spend in the
bank both waiting to be served and being served if two tellers are
on duty?
ANSWER
(e) On the average, how long is the waiting line if one teller is on
duty?
ANSWER (f) On the average, how long is the waiting line if two tellers are on
duty?
ANSWER 4 (g) On the average, how many customers are in the bank if one teller
is on duty?
ANSWER
(h) On the average, hoe many customers are in the bank if two tellers
are on duty?
ANSWER
(i) What percentage of the time is a teller busy if one teller is on
duty?
ANSWER
(j) What percentage of the time is a teller busy if two tellers are on
duty?
ANSWER
(k)If the bank opens for service 8 hours on Fridays, what is the
expected cost for both waiting and service for the day if one teller
is used?
ANSWER
(l) If the bank opens for service 8 hours on Fridays, what is the
expected cost for both waiting and service for the day if two tellers
are used?
ANSWER
(m) What measures of performance are cut into half when two
tellers are on duty as opposed to one teller on duty?
ANSWER 5 (n) How come other measures of performance are not cut into half
when we double the number of tellers?
ANSWER (o) What is your recommendation and why?? Should the bank
schedule one teller on duty or two tellers on duty?
ANSWER 6
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