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System Clock B ( t ) Q ( t ) Arrival times of custs. in queue
Event calendar
Number of completed waiting times in queue
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
Setup
0
1
2
3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
Q ( t )
Arrival times of custs. in queue
Event calendar [1, 0.00, Arr] [–, 20.00, End]
Number of completed waiting times in queue 0
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 0.00, Initialize
0
1
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4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
Q ( t )
Arrival times of custs. in queue (1.73)
Event calendar [1, 2.90, Dep] [3, 3.08, Arr] [–, 20.00, End]
Number of completed waiting times in queue 1
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 1.73, Arrival of Part 2
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1
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3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
Q ( t )
Arrival times of custs. in queue
Event calendar [3, 3.08, Arr] [2, 4.66, Dep] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 2.90, Departure of Part 1
0
1
2
3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
Q ( t )
Arrival times of custs. in queue (3.79, 3.08)
Event calendar [5, 4.41, Arr] [2, 4.66, Dep] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 3.79, Arrival of Part 4
0
1
2
3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
1
Q ( t )
3
Arrival times of custs. in queue (4.41, 3.79, 3.08)
Event calendar [2, 4.66, Dep] [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 2
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 4.41, Arrival of Part 5
0
1
2
3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
1
Q ( t )
1
Arrival times of custs. in queue (4.41)
Event calendar [4, 12.57, Dep] [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 4
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 8.05, Departure of Part 3
0
1
2
3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
1
Q ( t )
0
Arrival times of custs. in queue ()
Event calendar [5, 17.03, Dep] [6, 18.69, Arr] [–, 20.00, End]
Number of completed waiting times in queue 5
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 12.57, Departure of Part 4
0
1
2
3
4
0 5 10 15 20
0
1
2
0 5 10 15 20
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System Clock
B ( t )
1
Q ( t )
0
Arrival times of custs. in queue ()
Event calendar [7, 19.39, Arr] [–, 20.00, End] [6, 23.05, Dep]
Number of completed waiting times in queue 6
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 18.69, Arrival of Part 6
0
1
2
3
4
0 5 10 15 20
0
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0 5 10 15 20
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System Clock
B ( t )
Q ( t )
Arrival times of custs. in queue (19.39)
Event calendar [–, 20.00, End] [6, 23.05, Dep] [8, 34.91, Arr]
Number of completed waiting times in queue 6
Total of waiting times in queue
Area under Q ( t )
Area under B ( t )
Q ( t ) graph
B ( t ) graph
Time (Minutes)
Interarrival times 1.73, 1.35, 0.71, 0.62, 14.28, 0.70, 15.52, 3.15, 1.76, 1.00, ...
Service times 2.90, 1.76, 3.39, 4.52, 4.46, 4.36, 2.07, 3.36, 2.37, 5.38, ...
t = 19.39, Arrival of Part 7
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Finishing Up
• Average waiting time in queue:
• Time-average number in queue:
• Utilization of drill press:
2 53 minutes per part
No.of timesinqueue
Total of times inqueue
0 79 part
Finalclock value
Area under curve
Q t
0 92 (dimension less)
Finalclock value
Area under curve
B t
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Complete Record of the Hand
Simulation
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Simulation Dynamics: The Process-
Interaction World View
• Identify characteristic entities in the system
• Multiple copies of entities co-exist, interact,
compete
• “Code” is non-procedural
• Tell a “story” about what happens to a “typical”
entity
• May have many types of entities, “fake” entities
for things like machine breakdowns
• Usually requires special simulation software
Underneath, still executed as event-scheduling
• The view normally taken by Arena
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Randomness in Simulation
• The above was just one “replication” — a sample
of size one (not worth much)
• Made a total of five replications:
• Confidence intervals for expected values:
In general,
For expected total production,
X ± tn − 1 , 1 − α / 2 s / n
Note
substantial
variability
across
replications
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