IE 306
Systems Simulation
Spring 2007
INSTRUCTOR: Wolfgang Hörmann, Dept of IE, hormannw@boun.edu.tr www.ie.boun.edu.tr/~hormannw/ie306
ASSISTANT: Onur Özgün, onur.ozgun@boun.edu.tr
COURSE SCHEDULE: Monday 15.00 -17.00 (ETA A2)
Tuesday 12.00 -13.00 (ETA A2)
Wednesday 9.00-11.00 (ETA A2)
PS 1 & 2 Tue. 27.2. and Tue 6.3. (attendance will be controlled)
COURSE DESCRIPTION: This course presents the fundamental concepts of discrete event systems simulation: Modeling,
input and output analysis and random variate generation. In addition a simulation software (ARENA) will be covered and used
for solving real world simulation problems. The aim of the course is thus to lead the students to understand and experience the
theoretical and practical basics of simulation.
TEXTBOOK: Banks, Carson, Nelson, Nicol: Discrete event system simulation
REFERENCE BOOKS: Kelton, Sadowski, Sadowski: Simulation with ARENA
Law, Kelton: Simulation Modeling and Analysis
Course Outline:
Introduction to simulation and some examples (Chapter 1)
Basic statistical methods for Modeling input distributions (Chapter 5+9)
Random number and random variate generation (Chapter 7+8)
Output Analysis (Chapter 11.1-11.4)
Spreadsheet Simulation
Principles of Simulation Implementations and Examples in C (Chapter 3)
Introduction and examples with ARENA
Verification and Validation (Chapter 10)
Steady state simulations (Chapter 11.5)
Comparing two and more systems (Chapter 12)
EVALUATION:
Midterm 35 % (probably in the end of March)
Assignments and Final Project: 25 %
Final 40 %
To get a DD it is necessary to reach at least 50% for the assignments and the final project and to have a weighted total
of 50%!!!!!!!!!!
ASSIGNMNETS IE306 – SPRING 2007 (All problems must be prepared at home. Those with (*) will be
handed in and will be controlled. The attendance will be controlled during the PS!!!!)
PS 1: Random variables & important distributions (due: 27. 2. at 12.00)
1*) The waiting time, in hours, between successive speeders spotted by a radar unit is assumed to follow a continuous random
variable with cumulative distribution
0
x≤0
F(x)=
x /(x+2) x > 0
a) Find the probability of waiting less than 12 minutes between successive speeders
b) Find the density of the above distribution.
2*) A continuous random variable has density
f(x) = 1- x/2
for 0 < x ≤ 2
0
else
a) Find the CDF F(x). b) Compute the probability P(0.2 < X ≤ 1.2). c) Find the expectation of X.
3*) Suppose that the probabilities are 0.3, 0.3, 0.2 and 0.2 respectively, that X = 0,1,2, or 3 power failures will strike a certain
subdivision in any given year.
a) Find the probability X < 2 of the random variable X representing the number of power failures striking this subdivision.
b) Find the mean of X. c) Find the variance of X.
4*) If X and Y are independent random variables with expectations μx = 1 and μy = 3 and variances 2x=4 and 2y=3,
a) find the expectation and variance of the random variable Z= 2X + 4Y + 3.
b) Find the expectation and variance of Z if X and Y are not independent but have correlation 0.6.
5*) Two random variates X, Y have covariance 2 and variances 2x=9 and 2y=4. Calculate their correlation.
6) Among the discrete distributions with nonzero probabilities only for 0 and 1 find the distribution with
a) the minimal expectation; b) expectation equal to 0.8.
c) the minimal variance (2 solutions) d) The maximal variance
7*) The number of hurricans hitting the coast of Florida annually has a Poisson distribution with mean 0.9.
a) What is the probability that 2 or more hurricanes will hit the coast in a year?
b) What is the variance of the number of storms hitting the coast?
8*) Assume that that a battery has an exponential time to failure distribution with a mean of 50 months.
a) What is the probability that it breaks down in the first year?
b) What is the variance of the time to failure?
9*) It is assumed that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
a) What proportion of society has an IQ below 110?
b) A person with a higher IQ than 99.5% of the population is called a genius. Which IQ is necessary to be called a genius?
10) An electronic subassemply has a time to failure modeled by a Weibull distribution with β = 0.5 and α = 2000 hours (ν=0).
a) What is the mean time to failure?
b) What fraction of these subassemblies will fail before 3000 hours?
11) A random variate X with beta distribution with alpha = 3 and beta = 5 should be transformed to the random variate Y such
that the domain of Y is the interval (1; 11);
a) Find the necessary linear transform. b) Find the expectation and variance of Y. c) Find the density of Y.
12*)a) The interarrival times of customers are assumed to be independent and to follow an exponential distribution with mean
value 30 seconds. What is the probability that the sum of two interarrival times is smaller than 60?
b) The weight of a certain kind of apples is known to have a normal distribution with expectation 160 g and standard deviation
25 g. What is the probability that ten randomly (and independently) selected apples have a weight of less than 1400 g ?
c) What is the probability that the sample mean of 1000 randomly selected apples is below 155g?
PS 2: quantiles, estimation, Poisson process, Chi-Square goodness of fit test (due: 6. 3. at 12.00)
13*) Find the 90 % quantile the 10% quantile and the median of the normal distribution with μ = 25 and σ = 3.
14) Find the 90 % quantile the 25% quantile and the median of the Weibull distribution with β = 0.5 , α = 2000 and ν = 0.
15*) Assume that the arrivals to a supermarket follow a Poisson process with rate λ = 120 per hour.
a) Find the expectation and the variance of the interarrival times.
b) Find the probability that in the next minute no customer is going to enter the supermarket.
c) Find the expectation and the variance of the number of customers arriving per minute.
16*) Find the moment estimate of the Binomial distribution for a sample (only containing integers) with
sample mean 8 and sample variance 6.
17) Find the moment estimate of the Beta distribution for a sample of size 200 with sample mean 0.4 and variance 0.01.
18*) For a sample of size 100 we observe the following frequencies: “< 1” 23; “ <2” 29 ; “< 3” 26; “< 4” 22;
Is it possible that this sample comes from a uniform distribution on (0,4)?
19*) To see the influence of the sample size on the chisquare test make the same test as above.
a) Assume a sample size of 1000 and all observed values multiplied by 10.
b) Assume a sample size of 10000 and all observed values multiplied by 100.
20*) A shopkeeper counts the number of customers per day who ask for a certain brand of dog-food.
In the last two months he observed
# of customers asking
0
1
2
3
4
5
6
7
8
9
10
11
# of days observed
12
10
7
4
2
2
0
1
1
0
0
1
Make a chi-square test to check if it is possible, that the number of customers asking per day follows a Poisson distribution.
HINT: To estimate the parameter of the Poisson distribution you can simply calculate the average number per day.
Do not forget to select the classes such that all Ei are >= 5.
MISSING PREREQUISITE!!!!!!!!!!! MISSING PREREQUISITE!!!!!!!!!!! MISSING PREREQUISITE!!!!!!!!!!!
For simulation probability is an important prerequisite. You cannot understand the simulation course for example without knowing what a random variate, its
density and its CDF are. For my consent you have to pass an entrance exam in the second week. The topics of the entrance exam are:
Basic probability, density and CDF of random variates, expectation and variance of random variates.
We will discuss and solve some sample questions in the first week of the course.
It is necessary that contact me latest till Tuesday 20.2. If you want to get my consent by taking that exam.