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Supplement E
Simulation
Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall
E- 01
What is Simulation?
Simulation
The act of reproducing
the behavior of a
system using a model
that describes the
processes of the
system.
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E- 02
Reasons for Using Simulation
• To analyze a problem when the relationship
between variables is nonlinear, or when the
situation involves too many variables or
constraints to handle with optimizing approaches.
• To conduct experiments without disrupting real
systems.
• To obtain operating characteristic estimates in
much less time (time compression).
• To sharpen managerial decision-making skills
through gaming.
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E -03
The Simulation Process
• Monte Carlo simulation
– A simulation process that uses random numbers to
generate simulation events
• Data collection
• Random-number assignment
– A random number is a number that has the same
probability of being selected as any other number (see
Appendix 2)
• Model formulation
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E - 04
Example E.1: Specialty Steel Products
The Specialty Steel Products Company produces items, such as
machine tools, gears, automobile parts, and other specialty
items, in small quantities to customer order.
Because the products are so diverse, demand is measured in
machine-hours.
Orders for products are translated into required machine-hours,
based on time standards for each operation.
Management is concerned about capacity in the lathe
department.
Assemble the data necessary to analyze the addition of one
more lathe machine and operator.
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E - 05
Example E.1 : Specialty Steel Products
Historical records indicate that lathe department demand
varies from week to week as follows:
Weekly Production Requirements
(hour)
200
250
300
350
400
450
500
550
600
Relative Frequency
0.05
0.06
0.17
0.05
0.30
0.15
0.06
0.14
0.02
Total 1.00
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E - 06
Example E.1 : Specialty Steel Products
To gather these data:
• Weeks with requirements of 175.00–224.99
hours were grouped in the 200-hour category.
• Weeks with 225.00–274.99 hours were grouped
in the 250-hour category, and so on.
The average weekly production requirements for the
lathe department are:
200(0.05) + 250(0.06) + 300(0.17) + ... + 600(0.02) = 400 hours
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E - 07
Example E.1 : Specialty Steel Products
Employees in the lathe department work 40 hours per week
on 10 machines. However, the number of machines actually
operating during any week may be less than 10. Machines
may need repair, or a worker may not show up for work.
Historical records indicate that actual machine-hours were
distributed as follows:
Regular Capacity (hr)
320 (8 machines)
360 (9 machines)
400 (10 machines)
Relative Frequency
0.30
0.40
0.30
The average number of operating machine-hours in a week is
320(0.30) + 360(0.40) + 400(0.30) = 360 hours
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E - 08
Example E.1 : Specialty Steel Products
The company has a policy of
completing each week’s workload
on schedule, using overtime and
subcontracting if necessary.
Resources and Costs
Maximum Overtime
100 hrs
Lathe Operators
$10/hr
Overtime Cost
$25/hr
Subcontracting Cost
$35/hr
To justify adding another machine and worker to the lathe
department, weekly savings in overtime and subcontracting costs
should be at least $650. Management estimates from prior
experience that with 11 machines the distribution of weekly capacity
machine-hours would be
Regular Capacity (hr) Relative Frequency
360 (9 machines)
0.30
400 (10 machines)
0.40
440 (11 machines)
0.30
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E - 09
Random-Number Assignment
• Random number
– A number that has the same probability of
being selected as any other number
• Events in a simulation can be generated in an
unbiased way if random numbers are
assigned to the events in the same
proportion as their probability of occurrence.
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E - 10
Random-Number Assignment
Weekly
Demand
(hour)
Probability
Random
Number
200
250
300
350
400
450
500
550
600
0.05
0.06
0.17
0.05
0.30
0.15
0.06
0.14
0.02
00-04
05-10
11-27
28-32
33-62
63-77
78-83
84-97
98-99
EVENT
Existing
Weekly
Capacity
(hr)
320
360
400
Probability
Random
Numbers
0.30
0.40
0.30
00-29
30-69
70-99
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E - 11
Model Formulation
• Decision variables
• Uncontrollable variables (random)
• Dependent variables
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E - 12
Example E.2 : Specialty Steel Products
Formulate a simulation model for Specialty
Steel Products that will estimate idle-time
hours, overtime hours, and subcontracting
hours for a specified number of lathes.
Design the simulation model to terminate
after 20 weeks of simulated lathe
department operations.
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E - 13
Example E.2 : Specialty Steel Products
• Use the first two rows of random numbers in the
random number table for the demand events and the
third and fourth rows for the capacity events.
Because they are five-digit numbers, only use the first
two digits of each number for our random numbers.
• The choice of the rows in the random-number table
was arbitrary.
• The important point is that we must be consistent in
drawing random numbers and should not repeat the
use of numbers in any one simulation.
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E - 14
Example E.2 : Specialty Steel Products
To simulate a particular capacity level, we proceed as follows:
Step 1: Draw a random number from the first two rows of the
table. Start with the first number in the first row, then
go to the second number in the first row, and so on.
Step 2: Find the random-number interval for production
requirements associated with the random number.
Step 3: Record the production hours (PROD) required for the
current week.
Step 4: Draw another random number from row 3 or 4 of the
table. Start with the first number in row 3, then go to
the second number in row 3, and so on.
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E - 15
Example E.2 : Specialty Steel Products
Step 5: Find the random-number interval for capacity (CAP)
associated with the random number.
Step 6: Record the capacity hours available for the current
week.
Step 7: If CAP ≥ PROD, then IDLE HR = CAP – PROD.
Step 8: If CAP < PROD, then SHORT = PROD – CAP.
If SHORT ≤ 100, then OVERTIME HR = SHORT and
SUBCONTRACT HR = 0.
If SHORT > 100, then OVERTIME HR = 100 and
SUBCONTRACT HR = SHORT – 100.
Step 9: Repeat steps 1–8 until you have simulated 20 weeks.
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E - 16
Example E.2 : Specialty Steel Products
• We used a unique random-number sequence for weekly
production requirements for each capacity alternative
and another sequence for the existing weekly capacity
to make a direct comparison between the capacity
alternatives.
• Based on the 20-week simulations, we would expect
average weekly overtime hours (highlighted in orange)
to be reduced by 41.5 – 29.5 = 12 hours and
subcontracting hours (highlighted in blue) to be reduced
by 18 – 10 = 8 per week. (See Slide E-20)
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E - 17
Example E.2 : Specialty Steel Products
The average weekly savings would be:
Overtime:
(12 hours)($25/hours) = $300
Subcontracting: (8 hours)($35/hour) =
280
Total savings per week = $580
• This amount falls short of the minimum required savings
of $650 per week.
• The savings are estimated to be $1,851.50 – $1,159.50 =
$692 and exceed the minimum required savings for the
additional investment from a 1000 week simulation.
• This result emphasizes the importance of selecting the
proper run length for a simulation analysis.
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E - 18
Example E.2 : Specialty Steel Products
10 Machines
11 Machines
Week
Demand
Random
Number
Weekly
Production
(hr)
Capacity
Random
Number
Existing
Weekly
Capacity
(hr)
1
71
450
50
360
90
400
50
2
68
450
54
360
90
400
50
3
48
400
11
320
80
360
40
4
99
600
36
360
100
400
100
5
64
450
82
400
50
440
10
6
13
300
87
400
7
36
400
41
360
8
58
400
71
400
9
13
300
00
320
10
93
550
60
360
11
21
300
47
360
60
400
100
12
30
350
76
400
50
440
90
Idle
Hours
Overtime
Hours
Subcontract
Hours
140
100
Existing
Weekly
Capacity
(hr)
440
40
Overtime
Hours
Subcontract
Hours
100
140
400
20
100
Idle
Hours
90
440
40
360
60
400
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100
50
E - 19
Example E.2: Specialty Steel Products
Results from 20-week simulation
11 Machines
10 Machines
Week
Demand
Random
Number
Weekly
Production
(hr)
Capacity
Random
Number
Existing
Weekly
Capacity (hr)
Idle
Hours
Overti
me
Hours
Subcontract
Hours
20
Existing
Weekly
Capacity
(hr)
Idle
Hours
360
60
13
23
300
09
320
14
89
550
54
360
15
58
400
87
400
440
40
16
46
400
82
400
440
40
17
00
200
17
320
360
160
18
82
500
52
360
19
02
200
17
320
20
37
400
19
320
100
90
120
100
40
120
400
80
100
400
360
Overtime
Hours
Subcontract
Hours
50
100
160
360
Total
490
830
360
890
590
200
Weekly
average
24.5
41.5
18.0
44.5
29.5
10.0
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E- 20
Example E.2 : Specialty Steel Products
Results from 1,000-week simulation.
10 Machines
11 Machines
Idle hours
26.0
42.2
Overtime hours
48.3
34.2
Subcontract
hours
18.4
8.7
Cost
$1,851.50
$1,159.50
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E - 21
Application E.1: Famous Chamois
Famous Chamois is an automated car wash that advertises
that your car can be finished in just 15 minutes. The time
until the next car arrival is described by the following
distribution.
Minutes
Probability
Minutes
Probability
1
0.01
8
0.12
2
0.03
9
0.10
3
0.06
10
0.07
4
0.09
11
0.05
5
0.12
12
0.04
6
0.14
13
0.03
7
0.14
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1.00
E - 22
Application E.1 : Famous Chamois
Assign a range of random numbers to each event so that
the demand pattern can be simulated.
Minutes
Random
Numbers
Minutes
Random
Numbers
1
00–00
8
59-70
2
01–03
9
71-80
3
04–09
10
81-87
4
10–18
11
88-92
5
19–30
12
93-96
6
31–44
13
97-99
7
45–58
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E - 23
Application E.1
Simulate the operation for 3 hours, using the following random
numbers, assuming that the service time is constant at 6 minutes
(or :06) per car.
Random
Number
Time
to
Arrival
Arrival
Time
Number
in Drive
Service
Begins
Departure
Time
Minutes in
System
50
7
0:07
0
0:07
0:13
6
63
8
0:15
0
0:15
0:21
6
95
12
0:27
49
68
11
40
93
61
48
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E - 24
Application E.1
Simulate the operation for 3 hours, using the following random
numbers, assuming that the service time is constant at 6 minutes
(or :06) per car.
Random
Number
Time
to
Arrival
Arrival
Time
Number
in Drive
Service
Begins
Departure
Time
Minutes in
System
50
7
0:07
0
0:07
0:13
6
63
8
0:15
0
0:15
0:21
6
95
12
0:27
0
0:27
0:33
6
49
7
0:34
0
0:34
0:40
6
68
8
0:42
0
0:42
0:48
6
11
4
0:46
1
0:48
0:54
8
40
6
0:52
1
0:54
1:00 hr.
8
93
12
1:04
0
1:04
1:10
6
61
8
1:12
0
1:12
1:18
6
48
7
1:19
0
1:19
1:25
6
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E - 25
Application E.1
Random
Number
Time to
Arrival
Arrival Time
Number in
Drive
Service Begins
Departure
Time
Minutes in
System
82
10
1:29
0
1:29
1:35
6
09
3
1:32
1
1:35
1:41
9
08
3
1:35
1
1:41
1:47
12
72
9
1:44
1
1:47
1:53
9
98
13
1:57
0
1:57
2:03 hrs.
6
41
6
2:03
0
2:03
2:09
6
39
6
2:09
0
2:09
2:15
6
67
8
2:17
0
2:17
2:23
6
11
4
2:21
1
2:23
2:29
8
11
4
2:25
1
2:29
2:35
10
00
1
2:26
2
2:35
2:41
15
07
3
2:29
2
2:41
2:47
18
66
8
2:37
2
2:47
2:53
16
00
1
2:38
3
2:53
2:59
21
29
5
2:43
3
2.59
3:05 hrs.
22
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E - 26
Application E.1
Analysis
– The average time a car is in the system:
(234/25) = 9.36 minutes
– The percentage of cars that take more than 15
minutes: (4/25)100 = 16%
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E- 27
Computer Simulation
• Steady state
– The state that occurs when the simulation is
repeated over enough time that the average
results for performance measures remain
constant.
• Manual simulations can be excessively timeconsuming.
• Simple simulation models can be developed
using Excel.
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E - 28
Computer Simulation
• Random numbers can be generated
using the RAND function.
• Excel can translate random numbers
into values for the uncontrollable
variables using the VLOOKUP function.
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E - 29
Computer Simulation
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E- 30
Example E.3: BestCar
The BestCar automobile dealership sells new automobiles.
The BestCar store manager believes that the number of cars
sold weekly has the following probability distribution:
Weekly Sales (cars)
0
1
2
3
4
5
Relative Frequency (probability)
0.05
0.15
0.20
0.30
0.20
0.10
Total 1.00
The selling price per car is $20,000.
Design a simulation model that determines the probability distribution
and mean of the weekly sales.
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E- 31
Example E.3 : BestCar
• The first step in creating this spreadsheet is to
input the probability distribution, including the
cumulative probabilities associated with it.
• These inputs values are highlighted in yellow in
cells B6:B11 of the spreadsheet, with
corresponding demands in D6:D11 on Slide E-34.
• The cumulative values provide a basis to associate
random numbers to the corresponding demand,
using the VLOOKUP() function.
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E- 32
Example E.3: BestCar
• Excel’s logic identifies for each week’s random number (in
column H) which demand it corresponds to in the Lookup
array defined by $C$6:$D$11 on Slide E-34.
• Once it finds the probability range (defined by column C) in
which the random number fits, it posts the car demand (in
column D) for this range back into the week’s sales (in
column I).
• Finally, the results table is created at the lower left portion
of the spreadsheet to summarize the simulation output.
• Percentage and cumulative columns next to the frequency
column show the frequencies in percentage and cumulative
percentage terms.
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E- 33
Example E.3 : BestCar
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E - 34
Simulation with Two
Uncontrollable Variables
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E- 35
Simulation with More
Advanced Software
• Even more computer power comes from commercial,
prewritten simulation software.
• Simulation programming can be done in generalpurpose programming languages such as VISUAL
BASIC, FORTRAN, or C++.
• Special simulation languages, such as GPSS,
SIMSCRIPT, and SLAM, are also available.
• Simulation is also possible with powerful PC-based
packages, such as SimQuick, Extend, SIMPROCESS,
ProModel, and Witness.
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E- 36
SimQuick Software
• Easy-to-use package that is simply an Excel
spreadsheet with some macros.
• Models can be created for a variety of simple
processes.
• A first step with SimQuick is to draw a flowchart
of the process using SimQuick’s building blocks.
• Information describing each building block is
entered into SimQuick tables.
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E- 37
SimQuick Software
Entrance
Arrivals
Buffer
Sec. Line 1
Workst.
Insp. 1
Buffer
Sec. Line 2
Workst.
Add. Insp. 1
Workst.
Insp. 2
Dec. Pt.
DP
Workst.
Add. Insp. 2
Flowchart of
Passenger
Security
Process
Buffer
Done
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E- 38
SimQuick Software
Element
Types
Element
Names
Statistics
Overall
Means
Entrance(s)
Door
Objects entering process
237.23
Buffer(s)
Line 1
Simulation
Results of
Passenger
Security
Process
Line 2
Done
Mean Inventory
5.97
Mean cycle time
3.12
Mean Inventory
0.10
Mean cycle time
0.53
Final Inventory
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224.57
E- 39
Solved Problem 1
• A manager is considering production of several products in an
automated facility.
• The manager would purchase a combination of two robots.
• The two robots are capable of doing all the required operations.
• Every batch of work will contain 10 units.
• A waiting line of several batches will be maintained in front of
Mel.
• When Mel completes its portion of the work, the batch will then
be transferred directly to Danny.
Waiting line
Mel
Danny
Each robot incurs a setup before it can begin processing a batch. Each
unit in the batch has equal run time. The distributions of the setup
times and run times for Mel and Danny are identical. But because
Mel and Danny will be performing different operations, simulation of
each batch requires four random numbers from the table.
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E - 40
Solved Problem 1
Setup Time
(min)
Probability
Run Time per Unit
(sec)
Probability
1
0.10
5
0.10
2
0.20
6
0.20
3
0.40
7
0.30
4
0.20
8
0.25
5
0.10
9
0.15
• Estimate how many units will be produced in an hour.
• Then simulate 60 minutes of operation for Mel and
Danny.
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E - 41
Solved Problem 1
Except for the time required for Mel to set up and run the
first batch, we assume that the two robots run
simultaneously.
The expected average setup time per batch is:
[(0.1  1 min) + (0.2  2 min)(0.4  3 min)(0.2  4 min)
+ (0.1  5 min)]
= 3 minutes or 180 seconds per batch
The expected average run time per batch (of 10 units) is:
[(0.1  5 sec) + (0.2  6 sec) + (0.3  7 sec) +
(0.25  8 sec) + (0.15  9 sec)]
= 7.15 seconds/units  10 units/batch = 71.5 seconds per
batch
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E - 42
Solved Problem 1
Mel
Batch
No.
Start
Time
Danny
Random
No.
Setup
Random
No.
Process
Cumulative
Time
Start
Time
Random
No.
Setup
Random
No.
Process
Cumulative
Time
1
0.00
71
4
50
7
5:10
5:10
21
2
94
9
8:40
2
5.10
50
3
63
8
9:30
9:30
47
3
83
8
13:50
3
9.30
31
3
73
8
13:50
13:50
04
1
17
6
15:50
4
13.50
96
5
9
9
20:20
20:20
21
2
82
8
23:40
5
20.20
25
2
92
9
23:50
23:50
32
3
53
7
28:00
6
23.50
00
1
15
6
25:50
28:00
66
3
57
7
32:10
7
28.00
00
1
99
9
30:30
32:10
55
3
11
6
36:10
8
32.10
10
2
61
8
35:30
36:10
31
3
35
7
40:20
9
36.10
09
1
73
8
38:30
40:20
24
2
70
8
43:40
10
40.20
79
4
95
9
45:50
45:50
66
3
61
8
50:10
11
45.50
01
1
41
7
48:00
50:10
88
4
23
6
55:10
12
50.10
57
3
45
7
54:20
55:10
21
2
61
8
58:30
13
55.10
26
2
46
7
58:20
58:30
97
5
31
7
64:40
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E- 43
Solved Problem 1
• The total of average setup and run times per batch is 251.5
seconds.
• Even though the robots used the same probability
distributions and therefore have perfectly balanced
production capacities, Mel and Danny did not produce the
expected capacity of 14 batches because Danny was
sometimes idle while waiting for Mel (see batch 2) and Mel
was sometimes idle while waiting for Danny (see batch 6).
• The simulation shows the need to place between the two
robots sufficient space to store several batches to absorb the
variations in process times.
• Subsequent simulations could be run to show how many
batches are needed.
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E- 44
Solved Problem 2
• Customers enter a small bank, get into a single line, are
served by a teller, and finally leave the bank.
• Currently, this bank has one teller working from 9 A.M.
to 11 A.M.
• Management is concerned that the wait in line seems
to be too long.
• Therefore, it is considering two process improvement
ideas: adding an additional teller during these hours or
installing a new automated check-reading machine that
can help the single teller serve customers more quickly.
• Use SimQuick to model these two processes.
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E- 45
Solved Problem 2
Entrance
Entrance
Door
Door
Buffer
Buffer
Line
Line
Workstation
Workstation
Teller
Teller
Buffer Buffer
Served
ServedCustomers
Customers
Flowchart for a One-Teller Bank
Workstation
Workstation
Teller
1 1
Teller
Entrance
Entrance
Door
Door
Buffer
Buffer
Line
Line
Buffer Buffer
Served
ServedCustomers
Customers
Workstation
Workstation
Teller
2 2
Teller
Flowchart for a Two-Teller Bank
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E- 46
Solved Problem 2
• Three key pieces of information need to be entered: when
people arrive at the door, how long the teller takes to serve a
customer, and the maximum length of the line.
• Each of the three models is run 30 times, simulating the hours
from 9 A.M. to 11 A.M.
Element
Types
Element
Names
Entrance(s)
Door
Service Level
0.90
Buffer(s)
Line
Mean Inventory
4.47
Mean cycle time
11.04
Statistics
Overall
Means
Simulation Results of Bank
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Solved Problem 2
• The numbers shown are averages across the 30 simulations.
• The service level for Door tells us that 90 percent of the simulated
customers who arrived at the bank were able to get into Line.
• The mean inventory for Line tells us that 4.47 simulated customers
were standing in line.
• The mean cycle time tells us that simulated customers waited an
average of 11.04 minutes in line.
• When we run the model with two tellers, we find that the service
level increases to 100 percent, the mean inventory in Line
decreases to 0.37 customer, and the mean cycle time drops to
0.71 minute.
• When we run the one-teller model with the faster check-reading
machine we find that the service level is 97 percent, the mean
inventory in Line is 2.89 customers, and the mean cycle time is
6.21 minutes.
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