Exercises :
Simulation Modelling Analysis :
H3 : Random number generation
2
a =
13 ; C =
5 ; N =
64
a Full period 3 criteria :
= C
relatively paine to N 64 Can't be divided
by 5
:
2 a mod p 1
if p is panie of N 13 Mod 2 1
=
, ↑
: =
3 a mod 4
if ↳ is factor of N 13 Mod 4 1
1 :
= =
,
& x453 =
2 No =
X64 =
1120 =
(192 =... x448 =
7
R Xn xn + 1 =
13Xn + 5 Mod 64
X448
=
No 8 7 32
X449
I
X1 1 3237
X450 I
XS 2 3738
X251 =
13 .
3 38 51
X452
=
X4 4 51 28 => X453 =
28
X453 =
Xj 5 28 => U453 = 2864 =
0 , 4375
H3 :
RNG D .
1
,3
1st recurrence relation : 21 , n
=
1221 ,
n 1
-
+ 8921 ,
n -
3 Mod 127
N z1 , -1 [1 , 1-2 [1 , 1-3 12. I , -1 + 8921 , -3 in Un =
J 57 100 45 4 689 117 0 , 92126
1 117 57 100 10304 17 0
, 13386
& 17 117 57 5277 70 0
, 55118
370 17 117 11253 77 0 , 60630
47778 17 2437 24 0 , 18998
5 24 77 70 6518 410 , 32283
andrecurrence relation : 22 , n
=
1422 , n-2 + 9822 ,
n-3 Mod 31
N 22 , n 1
-
22 , N-2 22, M -
3 14-22 , n -
2 + 2822 , 1 - 3 22, n Un =
2
J 24 1 17 490 250 , 80605
195 24 1 364 230 , 74194
I 23 25 24 1022 300 , 96774
330 25 25 1022 300 , 96774
4 30 30 23 1064 10 0 , 32258
5 10 30 30 1260 20 0 , 64516
N
Combining
z1in
these
Is , N
:
In
21 , n
=
-22 , n
Zin-2s,n most
an Un =
F
127
-
& 117 g5 92 92 0 , 72441
6 M11210
T
17 23 -
,
95276
70 30 40 40 0 , 51497
3 77 30 47 47 8 , 37008
4 24 10 14 14 0 , 11024
g 41 20 21 21 0
, 16535
H3 :
RNG D L .
,4
a Chie ↑04 0, ,
,8
0
, 11
1
Expected #Obs . N .
Xj
: 222222222h
"
Observed #obs .
Nj
:
h 223132h -L
=>
X Nj-Nj = 0, =16 ,
NXj
Accept distributed
Numbers are
uniformly
=> Ho :
Xi D=
↓
Kolmogorov-Smirnov Ex Fu Fy D Fy 1
-
-
=
, 85120
0 0 , 85 O 0
, 05
·
Empirical distribution : 0, 06 220 0 , 06 0 , 04 0
, 01
FX = #Xi = X
, 153280 . 15
0 O , 05
0
Th
, 18
0 4200 , 180 , 02 0
, 03
Uniform distribution
· : : i : : :
20
FX =
Xi , 96
0 20 0 , 96 0 , 04 0
, 01
MAX :
0, 1 0
, 05
=> D Max D+
=
,
D= 0 , 1 <
D1-00 , 20
=
0 , 294
Accept distributed
Numbers are
uniformly
=> Ho :
H3 :
RNG D.
, 5 Ex :
empirical diste .
Ey : Poisson diste .
with 1 =
6, 4
Kolmogorov-Smirnov
# cust 8 12 34 5
- -
-
6 78 9
-
-
10 1 -
12 13
-
14 15-
# Obs 18244068371618 8
Cum #obs . . 18 42 82 150 187 203 213 221 Accept Ho
FX 182214292182921150gq11879912032912139211 I
FY 0
, 012 19 0 , 38. . . . . . . . . . . . . . . .
Fx -
Fy 0 , 0690 , 8710 , 013--- ......
=> D =
0, 071 < 0 , 0915 =
Does ; 221
xt -X
G
-
F :
r =
1 -
2
F" :
x = -
(n 1 -
2
At
-
= In e
Xt
Obs .
r X Arrival times With Xt =
14 min
A 0
, 73 -
4 In 0 , 75
.
=
11 26 8 : 01 () 1 =
4
& , 36
0
-
4 In 0 , 36
.
=
4 , 09 8 :
05 At
.
3 Q , 45
-
4 In 0 , 45
.
=
3 , 19 8 :
09
4 0
, 02
-
4 In 0 , 02 15 , 65
-
=
8 : 24
5 0
, 94
-
4 In0 , 94
.
=
0 , 25 8 : 24
H3 :
RNG D 4 .
Simulation Modelling Analysis :
H3 : Random number generation
2
a =
13 ; C =
5 ; N =
64
a Full period 3 criteria :
= C
relatively paine to N 64 Can't be divided
by 5
:
2 a mod p 1
if p is panie of N 13 Mod 2 1
=
, ↑
: =
3 a mod 4
if ↳ is factor of N 13 Mod 4 1
1 :
= =
,
& x453 =
2 No =
X64 =
1120 =
(192 =... x448 =
7
R Xn xn + 1 =
13Xn + 5 Mod 64
X448
=
No 8 7 32
X449
I
X1 1 3237
X450 I
XS 2 3738
X251 =
13 .
3 38 51
X452
=
X4 4 51 28 => X453 =
28
X453 =
Xj 5 28 => U453 = 2864 =
0 , 4375
H3 :
RNG D .
1
,3
1st recurrence relation : 21 , n
=
1221 ,
n 1
-
+ 8921 ,
n -
3 Mod 127
N z1 , -1 [1 , 1-2 [1 , 1-3 12. I , -1 + 8921 , -3 in Un =
J 57 100 45 4 689 117 0 , 92126
1 117 57 100 10304 17 0
, 13386
& 17 117 57 5277 70 0
, 55118
370 17 117 11253 77 0 , 60630
47778 17 2437 24 0 , 18998
5 24 77 70 6518 410 , 32283
andrecurrence relation : 22 , n
=
1422 , n-2 + 9822 ,
n-3 Mod 31
N 22 , n 1
-
22 , N-2 22, M -
3 14-22 , n -
2 + 2822 , 1 - 3 22, n Un =
2
J 24 1 17 490 250 , 80605
195 24 1 364 230 , 74194
I 23 25 24 1022 300 , 96774
330 25 25 1022 300 , 96774
4 30 30 23 1064 10 0 , 32258
5 10 30 30 1260 20 0 , 64516
N
Combining
z1in
these
Is , N
:
In
21 , n
=
-22 , n
Zin-2s,n most
an Un =
F
127
-
& 117 g5 92 92 0 , 72441
6 M11210
T
17 23 -
,
95276
70 30 40 40 0 , 51497
3 77 30 47 47 8 , 37008
4 24 10 14 14 0 , 11024
g 41 20 21 21 0
, 16535
H3 :
RNG D L .
,4
a Chie ↑04 0, ,
,8
0
, 11
1
Expected #Obs . N .
Xj
: 222222222h
"
Observed #obs .
Nj
:
h 223132h -L
=>
X Nj-Nj = 0, =16 ,
NXj
Accept distributed
Numbers are
uniformly
=> Ho :
Xi D=
↓
Kolmogorov-Smirnov Ex Fu Fy D Fy 1
-
-
=
, 85120
0 0 , 85 O 0
, 05
·
Empirical distribution : 0, 06 220 0 , 06 0 , 04 0
, 01
FX = #Xi = X
, 153280 . 15
0 O , 05
0
Th
, 18
0 4200 , 180 , 02 0
, 03
Uniform distribution
· : : i : : :
20
FX =
Xi , 96
0 20 0 , 96 0 , 04 0
, 01
MAX :
0, 1 0
, 05
=> D Max D+
=
,
D= 0 , 1 <
D1-00 , 20
=
0 , 294
Accept distributed
Numbers are
uniformly
=> Ho :
H3 :
RNG D.
, 5 Ex :
empirical diste .
Ey : Poisson diste .
with 1 =
6, 4
Kolmogorov-Smirnov
# cust 8 12 34 5
- -
-
6 78 9
-
-
10 1 -
12 13
-
14 15-
# Obs 18244068371618 8
Cum #obs . . 18 42 82 150 187 203 213 221 Accept Ho
FX 182214292182921150gq11879912032912139211 I
FY 0
, 012 19 0 , 38. . . . . . . . . . . . . . . .
Fx -
Fy 0 , 0690 , 8710 , 013--- ......
=> D =
0, 071 < 0 , 0915 =
Does ; 221
xt -X
G
-
F :
r =
1 -
2
F" :
x = -
(n 1 -
2
At
-
= In e
Xt
Obs .
r X Arrival times With Xt =
14 min
A 0
, 73 -
4 In 0 , 75
.
=
11 26 8 : 01 () 1 =
4
& , 36
0
-
4 In 0 , 36
.
=
4 , 09 8 :
05 At
.
3 Q , 45
-
4 In 0 , 45
.
=
3 , 19 8 :
09
4 0
, 02
-
4 In 0 , 02 15 , 65
-
=
8 : 24
5 0
, 94
-
4 In0 , 94
.
=
0 , 25 8 : 24
H3 :
RNG D 4 .
