Discrete Event “Hand” Simulation
of a GI/GI/1 Queue
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
The System
(Server)
Arriving
customers
7
6
5
Queue (FIFO)
•
Departing
customers
4
customer in Service
Simulation Objectives

Claimed objective: Estimate expected waiting time in
queue, expected queue length, and proportion of time the
server is busy.

Real objective: Introduce you to discrete event simulation
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Model Specifics
•
•
•
•
Initially (time 0) empty and idle
Base time units: minutes
Input data (assume given for now …), in minutes:
Customer Number
1
2
3
4
5
6
7
8
9
10
11
.
.
Arrival Time
0.00
1.73
3.08
3.79
4.41
18.69
19.39
34.91
38.06
39.82
40.82
.
.
Interarrival Time
1.73
1.35
0.71
0.62
14.28
0.70
15.52
3.15
1.76
1.00
.
.
.
Service Time
2.90
1.76
3.39
4.52
4.46
4.36
2.07
3.36
2.37
5.38
.
.
.
Stop when 20 minutes of (simulated) time have
passed
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Schematic System Representation
System
Clock
B(t)
(Server
state)
Lq(t)
Number of
completed waiting
times in queue
Total of
waiting times in queue
Arrival times of
custs. in queue
Area under
Lq(t)
Event calendar
Area under
B(t)
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by “Hand”:
t = 0.00, Initialize
System
Number of
completed waiting
times in queue
0
Clock
B(t)
Lq(t)
0.00
0
0
Arrival times of
Event calendar
custs. in queue
[1, 0.00,Arr]
<empty> [–, 20.00,
End]
Total of
waiting times in queue
Area under
Lq(t)
Area under
B(t)
0.00
0.00
0.00
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 0.00, Arrival of Customer 1
System
1
Number of
completed waiting
times in queue
1
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[2, 1.73,Arr]
<empty> [1, 2.90,Dep]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
0.00
1
0
0.00
0.00
0.00
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 1.73, Arrival of Customer 2
System
2
1
Number of
completed waiting
times in queue
1
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[1, 2.90,Dep]
(1.73) [3, 3.08,Arr]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
1.73
1
1
0.00
0.00
1.73
4
Lq(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 2.90, Departure of Customer 1
System
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[3, 3.08,Arr]
<empty> [2, 4.66,Dep]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
2.90
1
0
1.17
1.17
2.90
4
Lq(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 3.08, Arrival of Customer 3
System
3
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[4, 3.79,Arr]
(3.08) [2, 4.66,Dep]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
3.08
1
1
1.17
1.17
3.08
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 3.79, Arrival of Customer 4
System
4
3
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[5, 4.41,Arr]
(3.79, 3.08) [2, 4.66,Dep]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
3.79
1
2
1.17
1.88
3.79
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 4.41, Arrival of Customer 5
System
5
4
3
2
Number of
completed waiting
times in queue
2
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[2, 4.66,
Dep]
(4.41, 3.79, 3.08) [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
4.41
1
3
1.17
3.12
4.41
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 4.66, Departure of Customer 2
System
5
4
3
Number of
completed waiting
times in queue
3
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[3, 8.05,
Dep]
(4.41, 3.79) [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
4.66
1
2
2.75
3.87
4.66
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 8.05, Departure of Customer 3
System
5
4
Number of
completed waiting
times in queue
4
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[4, 12.57,
Dep]
(4.41) [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
Area under
Lq(t)
B(t)
8.05
1
1
7.01
10.65
8.05
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 12.57, Departure of Customer 4
System
5
Number of
completed waiting
times in queue
5
Clock
B(t)
Lq(t)
12.57
1
0
Arrival times of
custs. in queue
Total of
waiting times in queue
Area under
Lq(t)
15.17
15.17
Event calendar
[5, 17.03,
Dep]
() [6, 18.69,
Arr]
[–, 20.00,
End]
Area under
B(t)
12.57
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 17.03, Departure of Customer 5
System
Number of
completed waiting
times in queue
5
Clock
B(t)
Lq(t)
17.03
0
0
Arrival times of
custs. in queue
()
Event calendar
[6, 18.69,
Arr]
[–, 20.00,
End]
Total of
waiting times in queue
Area under
Lq(t)
Area under
B(t)
15.17
15.17
17.03
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 18.69, Arrival of Customer 6
System
6
Number of
completed waiting
times in queue
6
Clock
B(t)
Lq(t)
18.69
1
0
Arrival times of
custs. in queue
()
Total of
waiting times in queue
Area under
Lq(t)
Event calendar
[7, 19.39,
Arr]
[–, 20.00,
End]
[6, 23.05,
Dep]
Area under
B(t)
15.17
15.17
17.03
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 19.39, Arrival of Customer 7
System
7
6
Number of
completed waiting
times in queue
6
Clock
B(t)
Lq(t)
Total of
waiting times in queue
Arrival times of
Event calendar
custs. in queue
[–, 20.00,
End]
(19.39) [6, 23.05,
Dep]
[8, 34.91,
Arr]
Area under
Area under
Lq(t)
B(t)
19.39
1
1
15.17
15.17
17.73
4
3
Lq(t) graph
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Discrete Event Simulation by Hand:
t = 20.00, The End
System
7
6
Number of
completed waiting
times in queue
6
Clock
B(t)
L(t)
20.00
1
1
Arrival times of
Event calendar
custs. in queue
[6, 23.05,
Dep]
(19.39) [8, 34.91,
Arr]
Total of
waiting times in queue
Area under
Lq(t)
Area under
B(t)
15.17
15.78
18.34
4
Lq(t) graph
3
2
1
0
B(t) graph
0
5
10
15
20
0
5
10
15
20
2
1
0
Interarrival times
Time (Minutes)
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, ...
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts
Simulation by Hand:
Finishing Up
•
Average waiting time in queue:
Total of times in queue 15.17

 2.53 mins per cutomer
No. of times in queue
6
•
Time-average number in queue:
•
Area under Lq(t ) curve 15.78

 0.79 customers
Final clock value
20
Server Utilization :
Area under B(t ) curve 18.34

 0.92 (dimension less)
Final clock value
20
Simulation with Arena
Chapter 2 – Fundamental Simulation Concepts