Summary Simulation Modelling and Analysis | Ghent University | 2025/26

Simulation Modelling Analysis
Summary : :




H1 : Introduction

1
Introduction

1 1 .
The nature
of simulation

entities
people , machines that
*
System =
a collection
of ex :

act-interact together toward the accomplishment
Oflogical end
some

Discolt system
changes instantaneously at separate times
m :


~
Continuous system state variables change continuously in time
:




State
A
of the
system =
collection variables nuched toascribe
of
the system at a certain point in time
~
state =



f system , uncertainty performance measure
,




2
probability theory
2 .
1 General concepts

* Trial experiment-realization of probability experiment
mex :
rolling a die
Outcome the result
of a trial
~
=




A
Sample space - set
of all possible outcomes
~
ex .
die : e =
21 ,
2,3 ,
4 ,
5, 6]
H1 :
Intro D . 1

, PeSA)
* Price probability :



Conditional Pe(A BJ PeSAnB]
*
probability : =




Pr [B]
* Operations Joe independent events : ·
PeCAUB] =
Pe[A] + Pe[B]
·
Pr[AlB] =
Pe[A]
·
Pe[AnB] =
Pr[A] + Pr[B]


2 2 .
Random variables -
distribution
functions
# Random variable X : a
function w Xw from - to R
,
such
that Ax = [X =
x] = SWE -Xw =

xY
A
Probability mass
function :



PxX =
PeG(x x]] = =
PeSw ! XW =
x] =ID
m 0 = Px + 1 1
for all X


n[PxX = 1




Families variables
of
L
A random :
X =
0
·
Discrete : -Bernoulli : xx =
X =
1

otherwise
Binomial
Ge n 0, 1 9
,
=
-




, ...




-

Prisson :
pxx
=
otherwise
·
Continuous : -



Exponential -


Normal -




Erlang
* Cumulative distribution function Fx X
of X :



FX x =
Pe[X = x) =
Pe(\wXw =
x]]
0
fo
~
< Fx x = 1 -
a < x + 0



Mondecreasing function if b = Fa Fb
~ : a <




H1 :
Intro D . 2

,2 3 .
Joint - conditional distributions

A
Joint cumulative distribution a
of variables :



#x X =
Fx X1 , Xz , ...,
xn =
Pr[X = x] =
Pe[X1 = X, X(]
...,




An variables are
mutually independent if
# x =
Pe[M = X1
, ...,
XnXn) =xi Xi




2 4 . Expectations

*
Average-mean : xipi
* Median m PoEX my PrEX-mY: < =




* Mode m :
PrEX M3cPedX = =
X} goo all X



* Moments-set of values that characterise a Random variable X
·
1st moment mean E[X] SeXwdPw
: = =
Jaf
Continuous
~
: E(X) =

)XOXOX Discrete : E[X] =ipi
~
SUM : E[X + Y] =
E[X] + ECY]

<
·
qua moment :
variance of =
E[X]- E[X]


Continuous
~
: 02 =X-EX d Discrete :
Gi-E]pi
~
Sum :
val[X + Y] =
vac[X] + vm(y] + 2cv(X , Y]
with cov(X , Y) =
E[XY] -



ECX] E[Y]·




H1 :
Intro D 3 .

, . 5
2 Discrete distribution
functions
* Discreet uniform distribution
Probability mass
function
cumulativedistribution functiona
· ·




FxX
P xx1pxxi
(
=




0 . 4-

, 2-
0

1 2 3 4 534 12345 > X



·
1st moment : E[X] = in =
n+
1q
qua moment van (X) E(X) E(X) 1 u2 1
12n
· : = -
= N+ +1 -




n+ =




212


A Bernoulli distribution
Probability mass function
*unti distributiofunction
·
a
1
Px
Y
Px0 =
1 -


p
PX1 =
p
0 . 6-

0
,4 -
-




> X > X
a 1 J 1




·
1st moment : E[X) =
0 1 p + 1 -p
.

p
-
=




qna moment
· : Var (X] P pa =
p1 p -

= - =

p qa
. =
1 -




p



= phpahOther s
A Binomial distribution
Probability distribution
function mass : bein p
·

,


Cumulative
·


function : Bt ; N , p = -
·
1st moment : E[X] = :EE[Xi] =
ap
·
gnamoment : Van[X] = Var[Xi] =

up9




H1 :
Intro D 4 .