Various ways of selecting random numbers used in process simulations will be presented in this paper. Special attention will be given to complex phenomena not known enough to be precisely described. Modes of interaction are unknown; what is known are probabilities of interaction outcome. Such cases are found mostly in social and economic phenomena, such as population growth, economic predictions, decision-making risk analysis, etc.random numbers, simulation, process, distribution, model.

Dominika Crnjac Milic

Ljiljanka Kvesic

English

Research Papers in Economics

Dominika Crnjac Mili} • Ljiljanka Kvesi} 562
ROLE OF RANDOM NUMBERS IN SIMULATIONS 
OF ECONOMIC PROCESSES
Dominika Crnjac Milić, Ph.D.
 J.J. Strossmayer University in Osijek
Faculty of Electrical Engineering
Ljiljanka Kvesić, M.Sc.
University of Mostar, Bosnia and Hercegovina
Abstract
Various ways of selecting random numbers used in process simulations will be 
presented in this paper. Special attention will be given to complex phenomena not 
known enough to be precisely described. Modes of interaction are unknown; what 
is known are probabilities of interaction outcome. Such cases are found mostly in 
social and economic phenomena, such as population growth, economic predictions, 
decision making risk analysis, etc. 
Key words: random numbers, simulation, process, distribution, model.
1. Introduction
1.1 Types of simulation models
Classiﬁ  cations of simulation models resulted in four basic types of simulation 
models, which differ both in their approach to modeling and the type of the problem 
addressed, and in modeling and simulation techniques developed for such purposes. 
These are Monte Carlo simulation, continuous simulation, discrete event simula 
tion and mixed continuous/discrete simulation (Law and Kelton, 1982; Kreutzer, 
1986; Evans, 1988). All these types of simulation are dynamic simulations, exclud 
ing Monte Carlo simulation.
1.1.1. Monte Carlo simulation
Monte Carlo simulation1 (statistical simulation), as the name implies, is linked 
to random phenomena. It is interesting that it is characterized as one of the ﬁ  rst com 
puter programming applications. The method was developed in Los Alamos during 
1  Vlatko Čerić, (1993), Simulacijsko modeliranje, Školska knjiga, Zagreb, p. 36ROLE OF RANDOM NUMBERS IN SIMULATIONS OF ECONOMIC... 563
the Second World War II for the purpose of solving complex problems referring to 
creation of atomic bomb, such as calculation of dispersion of neutrons on the nucle 
us. However, the term itself is not used by full consent. Some authors call any type 
of software using random numbers Monte Carlo.  As in the majority of references in 
relation with simulation modeling, in this text this term will be used only for static 
types of simulations by which problems are solved by creating samples from random 
variable distributions. In such cases, problems might be of either deterministic or 
stochastic character.
The following types of applications of Monte Carlo simulations are differenti 
ated (Kleijnen, 1974):
(1) Deterministic problems whose solving is hard or expensive. 
A typical example of this type is calculation of values of certain integrals that 
cannot be solved analytically, i.e. whose subintegral function is such that a solution 
in form of a mathematical expression cannot be found.
(2) Complex phenomena not known enough.
The second class of problems solved by Monte Carlo simulation refers to phe 
nomena that are known insufﬁ  ciently to be precisely described. Instead of knowing 
modes of elements interaction, only probabilities of interaction outcome are known, 
which are in Monte Carlo simulation used for execution of a series of experiments 
giving samples of possible states of dependent variables. Statistical analysis of such 
samples provides a distribution of probabilities of dependent variables of interest. 
Most frequently social or economic phenomena, such as population growth, eco 
nomic predictions or risk analysis, are analyzed by this approach. 
(3) Statistical problems with no analytical solution. 
Statistical problems with no analytical solution are just one of broad classes of 
problems by solving of which Monte Carlo simulation is used. E.g. estimation of 
critical values or the power of testing new hypotheses belongs to this group. Genera 
tion of random numbers and variables is also used in problem solving. 
In case of comparison of various regression methods, Monte Carlo simulation 
is used for generation of input data, which are then analyzed by means of various 
regression methods, providing estimations of regression parameters of these data. 
Since the input data are generated by some predetermined parameters, it is possible 
to compare the quality of various regression methods by accuracy of regression pa 
rameter estimations they give to known parameters by means of which the data are 
generated. Dominika Crnjac Mili} • Ljiljanka Kvesi} 564
2. Random numbers
Existence of random values in a simulation model requires mechanisms which 
can generate values of variables from various probability distributions2 during simu 
lation experiments.  A series of generated values of a random variable is a sample 
from the probability distribution describing that variable. We will describe random 
numbers which make the basis for random variable generation. We will present the 
way how to generate random numbers, describe a linear congruent random number 
generator and list fundamental methods of testing a random number generator.
2.1. Using random numbers in simulation experiments 
Simulation process models containing components behaving randomly require 
corresponding methods of generating random numbers (Law and Kelton, 1982; 
Banks and Carson, 1984). During a simulation experiment, e.g. generation of a 
great number of servicing time values, demand size or interarrival times belong 
ing to some probability distributions might be requested. Therefore it is necessary 
to have an effective and high quality way of generating values of random numbers 
and variables. Unfortunately, the term “generation of random variables” is thereby 
not precise enough, i.e. this term implies generation of numerical values of random 
variables from corresponding probability distributions of the variable in question. 
Using random numbers and variables in simulation models enables reproduction 
of irregular behavior of system elements without having the model with an excel 
lent detailed description of that behavior. Random numbers and variables describe 
irregular behavior even in a compressed form. 
The terms random number and random variable must be used very carefully, 
since it is not easy to say whether a certain series of numbers is random, although 
some series seems not to be random at all. This question cannot be answered cor 
rectly on the basis of knowing the way a series of numbers in question is formed. 
The only approach that might provide a satisfactory answer is to put that series of 
numbers under corresponding statistical tests which will prove whether it is a sample 
of a random number and to which extent. Quality of the random number generator 
is tested in the same way.
Precisely, the term “random number” implies a continuous random variable with 
the uniform distribution on the interval [0,1]. This distribution will be denoted by 
U(0,1). Although this is the simplest continuous distribution, it is extremely impor 
tant since random variables of all other probability distributions (normal, binomial, 
2  Željko Pauše, (1978), Vjerojatnost, informacija, stohastički proces, Školska Knjiga Zagreb, p. 248.ROLE OF RANDOM NUMBERS IN SIMULATIONS OF ECONOMIC... 565
etc.) can be obtained by transforming3 the random variable U(0,1) by means of in 
dependent identical distribution.
2.2. How can random numbers be generated?
In a long and interesting history of generating random numbers various methods 
of generation have been used. Let us mention a successful random number genera 
tor.  Very successful generators of random numbers are nowadays linear congruent 
generators (abbr. LKG). LKG is represented by a choice of three positive integers:
a, multiplicator; 
c, increment;
m, module; m > a,m > c
In order to obtain a desired series of random numbers x1,x2,...,xN, we generate 
a series of integers X1, X2,...,XN starting from the initial value X0 (called the seed). A 
series of numbers is generated by means of a recursive rule 
Xn+1 = (aXn + c)(mod m), n=0,1,2,...,N-1.
After that, xk is deﬁ  ned as Xk / m, k  = 1,2,...,N.
Every LKG produces a deterministic series, i.e. every number Xk, and therefore 
Xk is known, by which X0,a,c and m.
LKG always produces a periodic series with p<m.
In order to obtain a high period p, module m is usually selected close to the 
greatest integer on the computer. It is usually m=231 1. In order to ensure a high qual 
ity of pseudorandom numbers, multiplicator a has to be selected carefully. One of 
the possibilities when m=231 1 is a=16807. For the purpose of generating a series X1, 
X2,...,Xn, increment c is set to be 0. By selecting c=0, “randomicity” of a generated 
series (if X0≠0) is not reduced more signiﬁ  cantly. The represented LKG 
is an example of a frequently used generator of pseudorandom numbers. Period p of 
this LKG is 231 2.
3  Vladimir Vranić, (1961), O graničnim teoremima teorije vjerojatnosti, Matematička biblioteka, Beo 
grad , pp. 151 165.Dominika Crnjac Mili} • Ljiljanka Kvesi} 566
Product a and Xn in LKG goes over the maximum value of the 32 bit integer if 
Xn>231/a. Thus, direct implementation of LKG with m=231 1 in an advanced language 
is impossible. Let us mention the following algorithm which gives us the opportunity 
to avoid that problem. Module is represented in the form m=aq+r, where r<q, and 
then aXn(mod m) is calculated by using the identity
where [Xn/q] is the integer of Xn/q. This expression does not include numbers greater 
than m.
There are special algorithms for random number generation from important 
distributions. For example, Box Muller algorithm and polar algorithm are used for 
generation of random numbers from a standard normal distribution. 
Numbers which are said to be strictly random can be generated. In 1955 RAND 
corporation published a table with a million random numbers obtained by such de 
vice. Input into the computer’s memory, these obviously pure random numbers seem 
to be suitable for usage as well as pseudorandom numbers generated by means of 
LKG. However, the table might e too short. Scientiﬁ  c Monte Carlo simulations re 
quire several billions of random numbers. 
On the other hand, a combination of LKGs might produce a series of pseudo 
random numbers which is, in spite of having a ﬁ  nite period, for all practical needs as 
good as an inﬁ  nite series.
A great advantage of LKGs is reproducibility of random numbers. If the same 
seed is speciﬁ  ed at the beginning of two series, then it will produce the same series 
of pseudorandom numbers. 
Monte Carlo simulation is mostly used for determining expectation E(X) of a 
random variable X referring to a speciﬁ  c stochastic model. Model simulation results 
in output data x1, realization of a random variable X. The second simulation gives 
a new output x2. Simulation is continued until we accumulate a sufﬁ  ciently great 
number n of outputs x1,x2,...,xn. Arithmetic mean of these output data   
is used as an estimator for  .
The second set of Monte Carlo simulations can be carried out with either the 
same or an independent set of random numbers. The same set of random numbers is 
e.g. used for:
1.Sensitivity analysis
2. Comparative simulations ROLE OF RANDOM NUMBERS IN SIMULATIONS OF ECONOMIC... 567
1. Sensitivity analysis
If the distribution of a random variable X depends on the model parameter, say 
a, then the expectation of X is a parameter function a, i.e. E(X)=A(a). One of the 
most important parts of simulation model analysis is sensitivity analysis.
Sensitivity analysis is used for determining whether simulation results, in our 
case expectation A(a), signiﬁ  cantly differ when the value of parameter a is changed. 
Two sets of simulations are carried out for two different values, say a1 and a2, of pa 
rameter a. It is important to use a joint set of random numbers for both simulations, 
since otherwise the effect of parameter change might be mixed up with the change 
of random numbers.
2. Comparative simulations
Two sets of simulations can be conducted in two different situations of the given 
model, or for two different model conﬁ  gurations. In such simulations a joint set of 
random numbers is to be used. 
Let us explain the usage of the joint set of random numbers. Comparative simu 
lation usually includes calculation of the difference 
Notice that by increasing the number of simulations n, variance of estimators   
(a1),   (a2) and   decreases. However, the number of simulations n is always ﬁ  nite, 
and thus the estimator 
has a ﬁ  nite variance
Clearly, the greater the covariance of X(a1) and X(a2), the less the variance of 
estimator  . It is reasonable to assume that the joint set of random numbers produces 
a positive correlation for |a1 a2| being small.Dominika Crnjac Mili} • Ljiljanka Kvesi} 568
2.3. How many simulations does it take for a certain purpose?
Monte Carlo simulation is usually used when the expectation A=E(X) of a ran 
dom variable X referring to a certain stochastic model is to be determined. Simula 
tion of the model results in the output data x1, realization of a random variable X. The 
second independent simulation gives the second output data x2. Simulations continue 
until we have the total of n output data x1,x2,...,xn. Arithmetic mean of these output 
data   is used as an estimator for A.
In typical numerical simulations the tolerance level ε is predetermined. Due to 
the stochastic nature of Monte Carlo simulations, prior to simulation execution, it is 
also necessary to specify the reliability level 1 a. The problem consists of the follow 
ing: determine the value of n such that the difference between   and A is less than 
ε (maximum value of the allowed error) with the probability greater than or equal to 
1 a. There are two common ways of measuring the difference: absolute error   
and relative error  .
It the absolute error is used as a measure, then the following argument can 
be used for the necessary number of simulations. Since simulations are indepen 
dent, the Central Limit Theorem4 says that for great values of n, the distribution of 
 is approximately normal with expectation zero and variance σ2. 
That is, 
,
where N(a,b) is a normal random variable with expectation a and variance b. 
Law of Great Numbers5 says that the sample variance 
is close to the unknown variance σ2 for sufﬁ  ciently great values of n. From the previous 
4  Željko Pauše, (1978), Vjerojatnost, informacija, stohastički proces, Školska knjiga, Zagreb, p. 147
5  Nikola Sarapa, (1987), Teorija vjerojatnosti, Školska knjiga, Zagreb, p. 647ROLE OF RANDOM NUMBERS IN SIMULATIONS OF ECONOMIC... 569
In tables indicating normal distributions we can ﬁ  nd the value za  such that 
Then
or
The following procedure might be used if we want to see whether the desired 
accuracy is reached or not. 
1. Generate n0=30 output data x1,x2,...,x30.
2. If
,
then the desired accuracy is reached.
Otherwise, generate output data xk until
knowing that the Central Limit Theorem and the Law of Great Numbers give 
correct results only for great values of n. As mentioned previously, estimator   is 
close to the unknown variance σ2, therefore it is not altered signiﬁ  cantly with n for 
. Hence, the number of simulations necessary for the desired accuracy should 
be close to 
A similar argument might be applied if the relative error   is used as 
an accuracy measure. Dominika Crnjac Mili} • Ljiljanka Kvesi} 570
 Generate  1.  n0=30 output data x1,x2,...,x30..
If 2. 
then the desired accuracy is reached.
Otherwise, generate output data xk until
Conclusion
In this paper we analyze ways of selecting random numbers and their usage in 
simulations, especially in Monte Carlo simulation. Application of independent sets 
of random numbers in sensitivity analysis, numerical calculations and comparative 
simulation is shown. The answer to the question how a simulation should be con 
ducted for certain purposes is given.
References:
Vlatko Čerić (1993) Simulacijsko modeliranje, Školska knjiga, Zagreb.
A. M. Law, W. D. Kelton (1982) Simulation Modeling and Analysis, Me Grew Hill, 
New York. 
Željko Pauše (1978) Vjerojatnost, informacija, stohastički proces, Školska knjiga, 
Zagreb.
Nikola Sarapa (1987), Teorija vjerojatnosti, Školska knjiga, Zagreb.
Vladimir Vranić (1961) O graničnim teoremima teorije vjerojatnost, Matematička 
biblioteka, Beograd, pp.151 165
Vladimir Vranić (1970) Vjerojatnost i statistika, Tehnička knjiga, Zagreb.
B. P. Zeigler (1976) Theory of Modelling and Simulation, Wiley, New York.

O graničnim teoremima teorije vjerojatnost,

Simulacijsko modeliranje, Školska knjiga,

Teorija vjerojatnosti, Školska knjiga,

Theory of Modelling and Simulation,

Vjerojatnost, informacija, stohastički proces, Školska knjiga,

Role of Random Numbers in Simulations of Economic Processes

http://www.efos.hr/repec/osi/journl/PDF/InterdisciplinaryManagementResearchIV/IMR4a33.pdf

Role of Random Numbers in Simulations of Economic Processes

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