Pharos University in Alexandria
Faculty of Engineering
Fifth Year Students
Computer Engineering Department
Spring 2021 – 2022
SHEET 02
S T O C H A S T I C P R O C E S S
P R O C E S S E S
Modeling and Simulation
Course Code: EC 384
Lecturer: Prof. Magdy A. Azim
TA:
Eng. Karim Soliman
&
M A R C O V
1. In a study of the movement between the city and the suburbs, it is noticed that every
year 40% of the city population moves to the suburbs and 30% of the suburb’s
population moves to the city. If 70% of the population live in the city and 30% live in the
suburbs, calculate the population in the city and in the suburbs after
a. 2 years
b. 4 years
2. A study on the population in EGYPT showed that 40% of them leave EGYPT to live
outside the country and 15% of the Egyptian people that live outside the country come
to live in EGYPT each year. Knowing that only 45% of the Egyptian live in EGYPT.
a. Find the percentage of those who leave EGYPT after 3 years.
b. Find the steady state values of the people distribution.
3. A car rental has three locations A , B , and C, a car rented from any location can be
returned to any of the three locations according to the following probabilities.
Rented To
Rented From
A
B
C
A
0.3
0.4
0.3
B
0.2
0.3
0.5
C
0.4
0.5
0.1
a. If a car is rented from location A returned then rented again, what is the
probability that is returned to C the second time ?
b. If the agency has a total of 1000 cars, what rates of space should they use for
locations A, B, and C?
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Pharos University in Alexandria
Faculty of Engineering
Fifth Year Students
Computer Engineering Department
Spring 2021 – 2022
Modeling and Simulation
Course Code: EC 384
Lecturer: Prof. Magdy A. Azim
TA:
Eng. Karim Soliman
4. From the following transition graph
a. Find the transition matrix of the following state diagram that describes the
weather state conditions.
b. Find the steady state probabilities.
5. Trees in a forest are assumed to fall into four age groups :
• Baby trees (age 0-15 years) , young trees (age 16-30 years) ,
• Middle trees (age 31-45 years) , and old trees (older than 45 years of age).
Assume the following:
- a certain percentage of trees in each group age dies
- Surviving trees enter into the next age group and old trees remain old.
- Lost trees are replaced by baby trees (total tree population doesn’t change )
If 10%, 20% , 30% and 40% are the loss rates in each age group respectively.
a. Find the steady state values for the age distribution in the forest.
b. If the total population is 50000 trees and suppose the forest is newly planted.
Find the age distribution in the forest after :
i. 15 years
ii. 30 years
iii. 90 years
iv. 15n years, n=1, 2, 3, 4, ……
6. Find the transition matrix for the following state diagrams
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