Solutions
Simulation
1
20.1-2) The weather can be considered a stochastic system, because it evolves in
a probabilistic manner from one day to the next. Suppose for a certain
location that this probabilistic evolution satisfies the following:
The probability of rain tomorrow is 0.6 if it is raining today. The
probability of its being clear tomorrow is 0.8 if it is clear today.
a. Use the following uniform random numbers to simulate the evolution
of the
weather for 10 days, beginning the day after a clear day.
b. Repeat using excel and the rand() function to perform the simulation.
Day
1
2
3
4
5
6
7
8
9
10
Uniform
0.6996
0.9617
0.6117
0.3948
0.7769
0.5750
0.6271
0.2017
0.7660
0.9918
P{rain tom orrow| rain today}  0.6
P{cleartom orrow| rain today}  0.4
P{clear tom orrow| clear today}  0.8
P{rain tom orrow| clear today}  0.2
Use CDF above
CDF
Wx Today
Rain
Clear
Wx Tomorrow
Rain
Clear
0.0 - 0.6
0.6 - 1.0
0.0 - 0.2
0.2 - 1.0
Day
0
1
2
3
4
5
6
7
8
9
10
Uniform
0.6996
0.9617
0.6117
0.3948
0.7769
0.5750
0.6271
0.2017
0.7660
0.9918
Logical If
Clear
Rain
Clear
Clear
Clear
Clear
Clear
Clear
Clear
Clear
Rain
Solutions
Simulation
1
20.1-2) Alternate Solution
Assume CDF for Rain, Clear is shown as below.
Day
1
2
3
4
5
6
7
8
9
10
Probability Tomorrow
Weather CDF
1.0
0.8
0.6
Rain
0.4
Clear
0.2
0.0
Rain
Clear
Weather Today
Uniform
0.6996
0.9617
0.6117
0.3948
0.7769
0.5750
0.6271
0.2017
0.7660
0.9918
CDF built on condition
CDF
Wx Today
Rain
Clear
Wx Tomorrow
Rain
Clear
0.0 - 0.6
0.6 - 1.0
0.8 - 1.0
0.0 - .08
Day
0
1
2
3
4
5
6
7
8
9
10
Uniform
0.6996
0.9617
0.6117
0.3948
0.7769
0.5750
0.6271
0.2017
0.7660
0.9918
Logical If
Clear
Clear
Clear
Clear
Clear
Clear
Clear
Clear
Clear
Clear
Rain
Solutions
Simulation
2
20.1-4) The William Graham Entertainment Co. will be opening a new box office
where customers can come to make ticket purchases in advance for the
many entertainment events being held in the area. Simulation is being
used to analyze whether to have one or two clerks on duty at the box
office. While simulating the beginning of a day at the box office, the first
customer arrives 5 minutes after it opens and the interarrival times for the
next four customers are 3 minutes, 1 minute, and 4 minutes. The service
times are 8 minutes, 6 minutes, 2 minutes, 4 minutes, and 7 minutes.
a. Plot the no. of customers at the box office over time.
b. Estimate L, Lq, W, and Wq for this queueing system.
c. Repeat for two clerks.
Solutions
Simulation
3
20.1-4) solution
M/M/1 Queue
i
1
2
3
4
5
Ui
------
Ai
5.0
3.0
9.0
1.0
4.0
Ti
5.0
8.0
17.0
18.0
22.0
Ui
------
Si
8.0
6.0
2.0
4.0
7.0
Part No.
0
1
2
3
4
5
Time of
Arrival
-5.0
8.0
17.0
18.0
22.0
Start
Service
-5.0
13.0
19.0
21.0
25.0
Depart
Time
0
13.0
19.0
21.0
25.0
32.0
Avg =
Time in
Queue
-0.0
5.0
2.0
3.0
3.0
2.600
Time in
System
-8.0
11.0
4.0
7.0
10.0
8.000
Event
Time
0
5.0
8.0
13.0
17.0
18.0
19.0
21.0
22.0
25.0
32.0
Part No.
-1
2
1
3
4
2
3
5
4
5
Event
Type
start
arrive
arrive
depart
arrive
arrive
depart
depart
arrive
depart
depart
Q(t)
0
0
1
0
1
2
1
0
1
0
0
S(t)
0
1
2
1
2
3
2
1
2
1
0
0.41
1.25
B(t)
0
1
1
1
1
1
1
1
1
1
0
Sum =
TimeAvg
a.) L = 1.25, Lq = 0.41,
Busy
Time
0
0
3.0
5.0
4.0
1.0
1.0
2.0
1.0
3.0
7.0
27.0
0.844
20.1-4) solution
Number in Queue
Q(t)
3
0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
25.0
30.0
35.0
25.0
30.0
35.0
Time
Number in System
S(t)
3
0
0.0
5.0
10.0
15.0
20.0
Time
Busy/Idle
B(t)
2
0
0.0
5.0
10.0
15.0
20.0
Time
Solutions
Simulation
5
20.3-1) Use the mixed congruential method to generate the following sequence of
random numbers.
a. 10 one-digit integer numbers such that Xn+1 = (Xn + 3) modulo 10 and Xo =
2.
m=
a=
c=
xo =
n
0
1
2
3
4
5
6
7
8
9
10
10
1
3
2
xn
2
5
8
1
4
7
0
3
6
9
2
axn + c
5
8
11
4
7
10
3
6
9
12
5
(axn + c)/m
0
0
1
0
0
1
0
0
0
1
0
xn+1
5
8
1
4
7
0
3
6
9
2
5
b. 8 integers (0-7) such that Xn+1 = (5Xn + 1) modulo 8 and Xo = 1.
m=
a=
c=
xo =
n
0
1
2
3
4
5
6
7
8
9
10
8
5
1
1
xn
1
6
7
4
5
2
3
0
1
6
7
axn + c
6
31
36
21
26
11
16
1
6
31
36
(axn + c)/m
0
3
4
2
3
1
2
0
0
3
4
xn+1
6
7
4
5
2
3
0
1
6
7
4
Solutions
Simulation
6
20.3-1) Use the mixed congruential method to generate the following sequence of
random numbers.
c. 5 two-digit integer numbers such that Xn+1 = (61Xn + 27) modulo 100 and
Xo=10.
m=
a=
c=
xo =
n
0
1
2
3
4
5
6
7
8
9
10
100
61
27
10
xn
10
37
84
51
38
45
72
19
86
73
80
axn + c
637
2284
5151
3138
2345
2772
4419
1186
5273
4480
4907
(axn + c)/m
6.37
22.84
51.51
31.38
23.45
27.72
44.19
11.86
52.73
44.8
49.07
xn+1
37
84
51
38
45
72
19
86
73
80
7
Solutions
Simulation
7
20.4-11) Consider the following CDF.
1.0
0.8
.06
0.4
0.2
7
U1 = .2655
X1 = 9.2
U2 = .3472
X2 = 9.5
U3 = .0248
X3 = 7.0
U4 = .9205
X4 = 12.2
U5 = .6130
X5 = 10.5
9
11
13