Modelling and Simulation IS 331. Lec (6)

1. IS 331

Faculty of Information Technology
Fall 2020
Modelling and Simulation
IS 331
Lec (6)
By Dr. Alaa Zaghloul

2.

Single server Queuing system
modeling
Discrete event Simulation (DES)

3.

Representation of the system
schematically
queue
Server=idle
queue=0
e0 =0
server

4.

Representation of the system
schematically
First customer arrival time :
A1 =1min , T1=1 min
queue
server
Generate service time :
S1=0.4 min
C1=1.4 min
Server=BUSY
queue=0
e1 = T 1

5.

Representation of the system
schematically
second customer arrival time :
A2 =0.2 min , T2=1.2 min
T2<C1 e2=T2
queue
Server=busy
queue=1
e2 = T 2
server

6.

Representation of the system
schematically
third customer arrival time :
A3 =0.4 min , T3=1.6 min
T3>C1 e3=C1
queue
Server=busy
queue=1
e3 = C1
server
Generate service time
For second customer :
S2=0.2 min
C2=C1+S2=1.6 min
Therefore first customer departs after 1.4 min with no
Delay in the system (D=0)
Second customer now enter service :
Delay in system = C1-T2=1.4-1.2=0.2 min.

7.

8.

Customers
1
2
3
4
5
6
7
8
9
Inter-arrivals Ai
0.4
1.2
0.5
1.7
0.2
1.6
0.2
1.4
1.9
Service Si
2
0.7
0.2
1.1
3.7
0.6
-
-
-
Arrival Times
0.4
1.6
2.1
3.8
4
5.6
5.8
7.2
9.1
End Service Time
2.4
3.1
3.3
4.9
8.6
9.2
Delays
-
0.8
1
-
0.9
3
3.4

9.

Customers
1
2
3
4
5
6
7
8
9
Inter-arrivals Ai
0.4
1.2
0.5
1.7
0.2
1.6
0.2
1.4
1.9
Service Si
2
0.7
0.2
1.1
3.7
0.6
-
-
-
Arrival Times
0.4
1.6
2.1
3.8
4
5.6
5.8
7.2
9.1
End Service Time
2.4
3.1
3.3
4.9
8.6
9.2
Delays
-
0.8
1
-
0.9
3
3.4
Q(t)
432-
10||||||||||
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2
3
4
6
7
8
9
10
1
5
0.4
1.6

10.

Customers
1
2
3
4
5
6
7
8
9
Inter-arrivals Ai
0.4
1.2
0.5
1.7
0.2
1.6
0.2
1.4
1.9
Service Si
2
0.7
0.2
1.1
3.7
0.6
-
-
-
Arrival Times
0.4
1.6
2.1
3.8
4
5.6
5.8
7.2
9.1
End Service Time
2.4
3.1
3.3
4.9
8.6
9.2
Delays
-
0.8
1
-
0.9
3
3.4
B(t)
432-
10||||||||||
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| ||||||||| T
2
3
4
6
7
8
9
10
1
5
0.4
3.3
3.8
9.2

11.

Average delays in the queue.
Sum of all delays
0.8 + 1 + 0.9 + 3 + 3.4
No of customers
7
9.1 minute/customer

12.

Time-average number of customers in the queue
T0
T1
T2
T3
(1.6-0.4)+(3.3-3.1)+(4-3.8)+(5.6-4.9)
(2.1-1.6)+(3.1-2.4)+(4.9-4)+(5.8-5.6)
(2.4-2.1)+(7.2-5.8)+(9.1-8.6)
(8.6-7.2)+(9.2-9.1)
1.2+0.2+0.2+0.7
0.5+0.7+0.9+0.2
0.3+1.4+0.5
1.4+0.1
2.3 (0) + 2.3 (1) + 2.2 (2) + 1.5 (3)
9.2
2.3
2.3
2.2
1.5
1.21 Customer

13.

Utilization of the server
Server on Times
(3.3 - 0.4) + (9.2 – 3.8)
Total Time
9.2
2.9 + 5.4
9.2
0.9021

14.

15.

Customers
1
2
3
4
5
Inter-arrivals Ai
1.445
1.215
0.241
1.565
0.126
Service Si 1
0.071
0.59
0.624
0.351
0.713
Service Si 2
2.082
0.904
1.406
0.566
-
Arrival Times
1.445
2.66
2.901
4.466
4.592
End Service Time 1
1.516
3.25
3.874
4.817
5.53
End Service Time 2
3.598
4.502
5.908
6.474
Delays 1
-
-
0.349
-
0.225
Delays 2
-
0.348
0.628
1.091
0.944

16.

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19.

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22.

END