© Chair and Institute of Industrial Engineering and Ergonomics, RWTH Aachen University

Simulation of Discrete Event Systems

Exercise to Unit 1

Introduction to Discrete Event Systems

Fall Winter 2016/2017

Dr.-Ing. Dipl.-Wirt.-Ing. Sven Tackenberg

Univ.-Prof. Dr.-Ing. Dipl.-Wirt.-Ing. Christopher M. Schlick

Chair and Institute of Industrial Engineering and Ergonomics

Instructor:

Jochen Nelles, M.Sc.

E-Mail: j.nelles@iaw.rwth-aachen.de

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1. Review of Set Theory

2. Review of Probability Theory

3. Exercises

Contents

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1. Review of Set Theory

1. Review of Set Theory

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A set is a collection of objects. The objects are called the elements of the set. Sets will be

denoted with capital letters and elements will be denoted with lower case letters. The empty

set does not contain elements and is denoted with the help of the symbol.

Example: A = {a, b, c} a A; d A

A set can be defined in two ways. First, we can simply enumerate its elements. Second, we can

specify the relevant properties characterizing the set elements (generation). In the second case

a lower case letter as a variable, e.g. the variable x, represents an arbitrary element of the set

and the colon “:” denotes “with the property”. Logical associations among properties are

denoted with the help of the binary operators “and“ (symbol ) as well as “or“ (symbol ) of

Boolean algebra. The unary “-” operator is used to denote a negation.

Enumeration example: A = {a, b, c}; B = {1, 2, 3, ...}

Generation example: A = {x: x is an English word}

B = {x: x is a prime number x is < 106}

C = {x: x is an English word x is an English sentence}

Review of set theory (I)

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Two sets are equal, if each element of the first set can be associated with an equal element of

the second set and the number of elements (set cardinality) of both sets are equal. Otherwise

they are not equal.

Example: A = {a, b, c}; B = {a, b, c}; C = {a, b, c, d} A = B; A C

A subset B of set A contains a sample of elements in A. The sample may represent the complete

set A and the binary operators or are used. A true subset is a subset with lower cardinality

than the reference set and the binary operators are are used. The empty set is a true

subset of any other set.

Example: A = {a, b, c}; B = {a, b}; C = {a, b, c} A B; A C; A B

A set of cardinality 1 is a single set. A set with a finite cardinality is a finite set. A set with an infinite

cardinality is an infinite set. The cardinality of a set is denoted with the help of the unary | |

operator.

Example: A = {a, b, c} A is a finite set; |A| = 3

Review of set theory (II)

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If A und B are two sets, a set C can be generated with the help of the following set operators:

- Complement:

C = Acomp = {x : x U x A}, where U is the universal set

- Conjunction:

C = A B = {x : x A x B}

- Intersection:

C = A B = {x : x A x B}

- Difference:

C = A \ B = {x : x A x B}

- Power:

2A = { X : X A}, that is the set of all subsets

- Cartesian product:

C = A B = { (x, y) : x A y B }

Review of set theory (III)

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Examples of set operators:

A = {a, b, c} 2A = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, A}

A = {a, b}; B = {c, d} C = A B = { (a, c), (a, d), (b, c), (b, d) }

A B is hatched A B is hatched

A \ B is hatched AC is hatched

Review of set theory (IV)

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Review of set theory (V)

Idempotency property 1a. A A = A 1b. A A = A

Associative property 2a. (A B) C =

A (B C)

2b. (A B) C =

A (B C)

Commutative property 3a. A B = B A 3b. A B = B A

Distributive property 4a. A (B C) =

(A B) (A C)

4b. A (B C) =

(A B) (A C)

Identity 5a. A Ø = A

6a. A U = U

5b. A U = A

6b. A Ø = Ø

Complement 7a. A AC = U

8a. (AC)C = A

7b. A AC = Ø

8b. UC = Ø, ØC = U

De Morgan law 9a. (A B)C = AC BC 9b. (A B)C = AC BC

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2. Review of Probability Theory

2. Review of Probability Theory

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In probability theory a random trial is a nondeterministic phenomenon where possible outcomes

of the trial are associated with a probability measure. The set S of all possible outcomes is

called the sample space.

A possible trial outcome as an element of the sample space S is called a sample. A random event

A is a set of samples or a subset of the sample space. An event with cardinality 1 is called an

elementary event.

The empty set and S are random events. The empty set is called the impossible event and

S is called the certain event.

Combined events of random trials can be simply defined bottom-up from elementary events when

using the previously introduced unary or binary set operators.

Example: Throwing a die S = {1, 2, 3, 4, 5, 6}

A = {a S: “even number of faces up”} = {2, 4, 6}

Review of probability theory (I)

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Let the sample space S be a finite set, e.g. S = {a1, a2, ..., an}. We can define a probability space,

if each ai S can be associated with a real number pi called probability of ai, so that the

following conditions are fulfilled:

- the pi are non negative

- the sum of all pi equals 1.

The probability P(A) of an event A is defined as the sum of probabilities of the elements of A. A

compact notation for P({ai}) is often P(ai).

For the discrete probability distribution P being defined on the sample space the following axioms

according to Kolmogorov hold:

- For each event A: 0 P(A) 1

- P(S) = 1

- If A and B are mutually exclusive, then P(A B) = P(A) + P(B)

Example: Throwing a fair die S = {1, 2, ..., 6}; P(S) = 1

P({“even no. of faces up”}) = P({2, 4, 6}) = P({2}) + P({4}) + P({6}) = 1/2

Review of probability theory (II)

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An event E with P(E) > 0 is given. The probability, that another event A occurs under the

hypothesis of E being given is called a conditional probability of A given E. The functional

notation is P(A|E). The conditional probability is defined in the Bayes theorem:

( )( | )

( )

P A EP A E

P E

If the sample space is finite, the following equation holds:

( )( | )

( )

A EP A EP A E

P E E

Example: Throwing two fair dices

S = {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6} = {(1,1), (1,2), ..., (6, 6)}

E = {e S : “sum of faces up is 6“} = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

A = {a S : “2 is up in one die “}

A E = {(2, 4), (4, 2)} P(A | E) = 2/5 P(A)

Review of probability theory (III)

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For arbitrary events A1, A2, ..., An the Bayes theorem can be used recursively and we can derive

the chain rule:

1 2 1 2 1 3 1 2 1 2 1( ... ) ( ) ( | ) ( | )... ( | ... )n n nP A A A P A P A A P A A A P A A A A

An event A is independent from an event B, if the observation of B does not influence the

probability of A to be observed. If the following condition holds, the events A and B are called

(statistically) independent:

( | ) ( )P A B P A

Three events A, B, and C are called independent, if the event pairs are independent and the joint

probability can be completely factorized into unconditional probabilities:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

i P A B P A P B P A C P A P C P B C P B P C

ii P A B C P A P B P C

Review of probability theory (IV)

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Consider the sample space S of a random trial. Obviously, the elements of S do not need to be

numbers to apply the rules of probability theory. However, in engineering science the elements of

S are usually mapped onto numbers, e.g. the number of faces up for a fair die. This mapping is

called a random variable.

A random variable X on a probability space S is a function of S into the real numbers, so that the

preimage are random events. The domain of the random variable X is denoted RX: RX = X(S).

Example: Throwing two dices

According to the previous examples the sample space of the random trial is the cartesian product

of A = {1, 2, 3, 4, 5, 6} with itself:

S = A A = {(1, 1), (1, 2), ..., (6, 6)}

The random variable X represents the sum of faces up of both dices and is mapping each element

in S onto the sum of faces; hence, X is a random variable with the domain:

RX = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Review of probability theory (V)

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Given is a domain RX = {x1, x2, ..., xn} of a random variable X being defined on a finite sample

space S. X induces the following probabilities on RX:

pi = P(xi) : Sum of probabilities of all elements in S with the image xi.

The function which is mapping the point xi onto the probability pi is called the probability

distribution of the random variable X. The distribution can be specified easily as a table:

x1 x2 ... xn

p1 p2 ... pn → Sum must equal 1!

Example: Throwing two fair dices

S = {(1, 1), (1, 2), ..., (6, 6)} with |S| = 36

RX = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

The element (1, 1) has the image 2 and therefore P(2) = p1 = 1/36.

There are two elements in S with image 3: {1, 2}, {2, 1} and therefore P(3) = p2 = 2/36.

There are three elements in S with image 4: {1, 3}, {2, 2}, {3, 1} and therefore

P(4) = p3 = 3/36.

Et cetera...

Review of probability theory (VI)

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Given is a random trial being repeated and the repetitions are independent. The trial has binary

outcomes as events, which are called “success” and “failure”. We denote with p the probability of

a success being observed and with q = 1 - p the probability of failure. These repeated trials are

called Bernoulli trials.

According to the following proposition we can compute the probability of k successes being

observed in n repeated trials:

The function P(.) denotes the binomial probability distribution with parameters n and p.

Example: n times fair coin tosses

The probability after n tosses (n is even!) to observe n/2 times “head” is:

!,

! !

k n knP k n p p q

k n k

2 2

2 2

1 ! 1 1 ! 1 !,

2 2 2 2 2! ! ! ! 2

2 2 2 2

n nn n

n

n n n nP n

n n n nn

Review of probability theory (VII)

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2

n

List plot of probability to observe n/2 times “heads” after n tosses:

5 10 15 20

0.2

0.4

0.6

0.8

11

,2 2

nP n

Review of probability theory (VIII)

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Open Questions ???

Questions ?

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1. Proof the De Morgan laws.

2. Develop a state transition diagram of a coin-based soda machine with lemonade and water beverages.

Each beverage costs 3 euros and due to occupational safety and hygiene the slot with the plastic cup

must be closed when pouring in the beverage.

Short homework