Philadelphia University
Faculty of Information Technology
Lecturer: Dr. Nameer N. EL-Emam
Department of CS
Internal Examiner: Dr. Raad Alwan
Marking Scheme
Coordinator: Dr. Nameer N. EL-Emam
Module Name: Simulation and Modelling Second Exam Semester one of the academic year:
Module Number: 750472
2016-2017
Date: 18/12/2016 Time 50 Min.
Familar Part:
Objective: This part aims to show student capability to answer the problem solving questions.
Q1/ (12-marks):
Let us define the following set of observations (inter-arrival times X) for airplanes landing:
X
3.1
6.5
2.5
1.3
3.7
0.4
4.2
8.0
where the experimental mean of these observations is equal to (0.333). Answer the following:
a) ( 4 marks) Derive a mathematical formula to generate random numbers (RNs) from the
corresponding observations X and then apply this formula to calculate RNs.

1
X i   Ln (RN i )

(1)
  Xi  Ln (RN i )
(2)
Multiply Eq.(1) by ( -  ) then

Take the Exp. on both side of Eq. (2) then we find RNs

e Xi  RN i
Apply Eq. (3) to find RNs
(3)
Xi
3.1
6.5
2.5
1.3
3.7
0.4
4.2
8
RNi
0.35585569
0.11458367
0.43463443
0.64837244
0.29135582
0.87518499
0.24663149
0.06950198
b) (4 marks) Apply KS-test algorithm to check uniformity of random numbers which were
generated in section (a), where the KS benchmark table is defined below:
SIZE
(N)
1
2
3
4
5
6
7
8
9
10
11
LEVEL OF CONFIDENCE
98%
.900
.684
.565
.494
.446
.410
.381
.358
.339
.322
.307
97%
.925
.726
.597
.525
.474
.436
.405
.381
.360
.342
.326
.96%
.950
.776
.642
.564
.510
.470
.438
.411
.388
.368
.352
.95%
.975
.842
.708
.624
.565
.521
.486
.457
.432
.410
.391
.90%
.995
.929
.828
.733
.669
.618
.577
.543
.514
.490
.468
Class
Boundary
class
index
[0-0.2]
[0.21-0.4]
[0.41-0.6]
[0.61-0.8]
[0.81-1]
1
2
3
4
5
fi
2
3
1
1
1
ACC
2
5
6
7
8
Fn(ACCi)=
ACCi / n
Fi(index)=
index / s
ei
0.25
0.625
0.75
0.875
1
1/5 = 0.2
2/5 = 0.4
3 / 5 = 0.6
4 / 5 = 0.8
1
0.05
0.225
0.15
0.075
0
K-S Theoretically=max(ei)=0.225
K-S Experimentally = 0.457
Then RNs are not belong to uniform distribution since
K-S Theoretically> K-S Experimentally
c) (4 marks) Using Poisson distribution to find the number of airplanes arrived in the time
period 13 time unit.
n
 RN

i
 e
i 1
 T
n 1
  RN i
i 1
e   T  e 0.33313  e  4.329 =0.01318

3
4
i 1
i 1
 RN i  0.01318   RN i
Then we conclude that the number of airplanes arrived in time period 13 is equal to 3.
OR
Applying the following on observation directly:
n

X
i 1
i
T
n 1
X
i
i 1
Q2/ (8-marks):
a) (4 marks) Derive a uniform distribution formula to generate observations (By
converting a probability density function PDF to a cumulative distributed function CDF
and then using the inverse method).
b) (4 marks) Apply the uniform distribution formula to find the observations within the
interval [5, 15] of the following random numbers:
RN
0.7
0.1
0.4
0.2
0.5
RVi = 5 + (15 - 5) * RNi
12
6
9
7
10