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Taking randomness for granted: the complexities of applying random number streams in simulation modelling

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Abstract

Uncertainty, as a constant companion of our world, is one major reason why simulation modelling takes precedence over static calculations to achieve accurate predictions. Computational random number generators are able to algorithmically determine values on the basis of random distributions, which utilise seed values to calculate streams of random numbers. This deterministic approach to replicating seemingly non-deterministic numbers ensures stochastic models to be reproducible at any time-one of the major requirements of simulation models. However, there are some pitfalls in the application of random number streams in modelling and simulation, which may even mislead experienced developers. In addition to a general introduction of the history of random number generators, this article shares empirical considerations and means by which the utilisation of random number streams can be improved to deliver valid and reliable results.
Taking randomness for granted:
the complexities of applying random number streams
in simulation modelling
Maximilian Selmair
Tesla Manufacturing Brandenburg SE
15537 Gr¨
unheide (Mark), Germany
mselmair@tesla.com
Abstract—Uncertainty, as a constant companion of our world, is
one major reason why simulation modelling takes precedence over
static calculations to achieve accurate predictions. Computational
random number generators are able to algorithmically determine
values on the basis of random distributions, which utilise
seed values to calculate streams of random numbers. This
deterministic approach to replicating seemingly non-deterministic
numbers ensures stochastic models to be reproducible at any time
– one of the major requirements of simulation models. However,
there are some pitfalls in the application of random number
streams in modelling and simulation, which may even mislead
experienced developers. In addition to a general introduction of
the history of random number generators, this article shares
empirical considerations and means by which the utilisation of
random number streams can be improved to deliver valid and
reliable results.
KeywordsRandomness; Modelling and Simulation; Seed Values;
Random Number Generators
I. INT RO DU CTI ON A ND TH E HIS TO RIC AL PE RS PEC TI VE
The digital replication of any system or process involving
randomness requires a method to generate or obtain numbers
that are both random and reproducible. Practical examples for
random occurrences in the field of production and logistics are
service times, interarrival times and maintenance occurrences.
This article focuses on the history of random number generators
as well as the potential drawbacks of utilising deterministic
random number generators for simulation modelling. The first
paragraph introduces how random values can be generated
efficiently from a predetermined probability distribution in
order to provide data sets for simulation modelling.
The modest beginnings of generating random numbers date
back to over a century ago. (Hull and Dobell, 1962; Dudewicz
and Dalal, 1971; Morgan, 1984). The earliest samples were
not generated by computer, but literally carried out by hand:
flipping coins, throwing dice, dealing out game cards or the
lottery draws. Even today, most lotteries are still operated
in this manner to avoid fraud allegations. As early as the
20
th
century, gamblers were joined by statisticians in their
quest to explore random numbers and mechanised devices were
built to generate random numbers more efficiently. Particular
examples from the late 1930s are e. g. Kendall and Smith
(1938) – the use of a spinning disk to select values from a
turntable containing a hundred thousand random digits. Only
two years later, electric circuits based on randomly pulsating
vacuum tubes were used to deliver random digits at much
higher rates of up to 50 per second. Such a device, referred to
as a random number machine, was used by the British General
Post Office to pick the winners of the Premium Savings Bond
lottery (Thomson, 1959). Electronic devices were also used
by the Rand Corporation (2001) to produce a sequence of
a million random numbers. Some past examples of different
approaches were picking numbers randomly out of phone books
and using digits in an expansion of
π
, such as 0.1415926535,
0.8979323846, 0.2643383279, etc. (Tu and Fischbach, 2005).
Another notable example of retrieving random numbers was
proposed by Yoshizawa et al. (1999), who described a physics-
based random number generator that relied on the radioactive
decay of the nuclide Americium-241.
As computers and, later on, simulation modelling became
more relevant, computational random number generators began
to gain popularity. A first attempt in this direction was the
use of the previously mentioned Rand Corporation’s table
in a computer’s memory (Rand Corporation, 2001). This
solution depended on substantial memory requirements and
a vast amount of time to retrieve new values. Research in
the 1940s and 1950s developed towards more deterministic
strategies of generating random numbers. These values were
sequential, which means that each new number was determined
by one or sometimes several of its predecessors according to
a fixed algorithm. The first known deterministic generator was
proposed by von Neumann (1951). The well-known mid-square
method is demonstrated in the subsequent example.
A researcher may begin with a four-digit positive integer
Z0= 1234
and square it to obtain an integer with at least
eight digits. The central four digits of this eight-digit number
constitutes the next four-digit number,
Z1
. By placing a decimal
point on the left of
Z1
, the first random number between
0
and
1
(
U1
) is yielded. Following this procedure, an unlimited
sequence of deterministic ”random“ values can be created.
Table I lists the first few examples.
At first sight, the mid-square method seems to provide a
seemingly suitable set of random numbers. However, a main
disadvantage of this method is the strong tendency of the
Communications of the ECMS, Volume 36, Issue 1,
Proceedings, ©ECMS Ibrahim A. Hameed, Agus Hasan,
Saleh Abdel-Afou Alaliyat (Editors) 2022
ISBN: 978-3-937436-77-7/978-3-937436-76-0(CD) ISSN 2522-2414
181
i ZiU
i Z2
i
0 1234 01522756
1 5227 0.5227 27321529
2 3215 0.3215 10336225
3 3362 0.3362 11303044
4 3030 0.3030 09180900
5 1809 0.1809 03272481
Law01323_ch07_393-425.indd Page 395 11/09/13 8:37 PM user /203/MH02090/Law01323_disk1of1/0073401323/Law01323_pagefiles
Table I: Sample calculation for a set of random numbers with
the mid-square method based on value 1234
generated values to converge to zero.
A more fundamental objection to the mid-square method is
that it does not yield ”random“ values at all, if one considers
random values to be unpredictable in nature. That is, if we know
one number, we have to acknowledge that it determines the
succeeding numbers as the method stipulates. In more recent
times, the first number in a sequence or stream of random
numbers is referred to as seed value. With a given
Z0
, the whole
sequence of
Zi
’s and
Ui
’s is determined. This characteristic
applies to all deterministic generators. These deterministic
random generators are often faulted to be generating pseudo-
random numbers. In the author’s opinion, pseudo-random is a
misnomer, indeed it is an oft-quoted remark by von Neumann
(1951), who declared that ”Anyone who considers arithmetical
methods of producing random digits is, of course, in a state
of sin. For, as has been pointed out several times, there is no
such thing as a random number – there are only methods to
produce random numbers, and a strict arithmetic procedure of
course is not such a method... We are here dealing with mere
”cooking recipes“ for making digits. . . “ (von Neumann, 1951).
Although this quote dates back to over 70 years ago, it does
not seem to have lost its relevance today. Furthermore, it is
rarely stated that von Neumann proceeds in the same paragraph
that these ”recipes“ ”probably can not be justified, but should
merely be judged by their results. Some statistical study of
the digits generated by a given recipe should be made, but
exhaustive tests are impractical. If the digits work well on
one problem, they seem usually to be successful with others of
the same type“ (von Neumann, 1951). Lehmer (1951) offered
a more practice-oriented definition: ”A random sequence is
a vague notion embodying the idea of a sequence in which
each term is unpredictable to the uninitiated and whose digits
pass a certain number of tests traditional with statisticians
and depending somewhat on the use to which the sequence is
to be put“ (Lehmer, 1951).
Since Lehmer’s proposal, further formal definitions and
empirical considerations about randomness were formulated in
the subsequent decades:
”Any one who considers arithmetical methods of produ-
cing random digits is, of course, in a state of sin. For, as
has been pointed out several times, there is no such thing
as a random number – there are only methods to produce
random numbers, and a strict arithmetic procedure of
course is not such a method“ von Neumann (1951).
”[A] sequence is random if it has every property that is
shared by all infinite sequences of independent samples of
random variables from the uniform distribution“ Franklin
(1963).
”[. . . ] random numbers should not be generated with a
method chosen at random. Some theory should be used“
Knuth (1968).
”The generation of random numbers is too important to
be left to chance“ Coveyou (1969).
”[In statistics] you have the fact that the concepts are not
very clean. The idea of probability, of randomness, is not
a clean mathematical idea. You cannot produce random
numbers mathematically. They can only be produced by
things like tossing dice or spinning a roulette wheel. With
a formula, any formula, the number you get would be
predictable and therefore not random. So as a statistician
you have to rely on some conception of a world where
things happen in some way at random, a conception which
mathematicians don’t have“ LeCam (1988).
”Sequences of random numbers also inevitably display
certain regularities. [. . . ] The trouble is, just as no real
die, coin, or roulette wheel is ever likely to be perfectly
fair, no numerical recipe produces truly random numbers.
The mere existence of a formula suggests some sort of
predictability or pattern“ Peterson (1998).
”The practical definitions of randomness – a sequence
is random by virtue of how many and which statistical
tests it satisfies and a sequence is random by virtue of
the length of the algorithm necessary to describe it [. . . ]“
Bennett (1999).
The author agrees with most researchers that deterministic
generators, if appropriately designed, can replicate numbers
which would have been generated by independent draws from
the
U(0,1)
distribution and would be also pass a series of
statistical tests. In summary, in the domain of simulation
modelling, deterministic random number generators are used
to replicate stochastic occurrences and measure their influence
on a system. These generators allow simulation modellers
to achieve reproducible scenarios with an unlimited set of
seemingly non-determined values. In simulation modelling, a
replication is defined as having the same set of parameters, but
different stochastic influences. Each replication is based on a
predetermined seed value (e. g. 1, 2, 3, .. .), defined as a specific
reproducible stream of random numbers. The purpose of this
article is to address the complexities of simulation modelling
with random number streams, which are associated with the
risk of generating skewed, unreliable and invalid results.
II. MODELLING PI TFALLS
In simulation modelling, the term iteration refers to different
combinations of parameter values. A single iteration is com-
posed of a series of replications, their number determined by
a sensitivity analysis, which differ in terms of their internal
seed of randomness. Models that do not contain any internal
stochastic are referred to as deterministic; however, these lie
outside the focal point of this article. In stochastic modelling,
182
Figure 1: Examined process flow modeled with AnyLogic
a set of replications is computed in order to ascertain the
influence of internal randomness on the final results. This
methodology is referred to as the Monte Carlo Experiment
(Shonkwiler and Mendivil, 2009).
In order to be able to compare two iterations with each other,
it is vital that both are subject to the the same random stochastic
behaviour. These can be e.
˙
g. machine maintenance occurrences,
interarrival times of supply or interfering influences. If such
random influences differ between iterations, the results, for
instance differences in a Key Performance Indicator (
KPI
) of
interest, may well be caused be the actual internal randomness.
As such, the risk of attributing a result to a certain combination
of parameters is considerable. Additionally, the adjustment of
parameters (e. g. a different rule set or policy) may not correlate
with any improvement of a
KPI
. The following conditions
have been identified as potential causes of deviation in random
behaviour in two or more iterations with the same seed of
randomness:
1)
When decisions in a model are stipulated by parameters,
and a particular decision causes consumers of randomness,
that is, every level of every variable, to require random
numbers in a different sequence, e. g. job shop machine
scheduling models
2)
When parameters affect the number of consumers of
randomness, e. g. Automated Guided Vehicle (
AGV
) fleet
with an initially randomly distributed battery level
3)
When parameters manipulate the model’s initialisation
sequence, e. g. data-driven models vs. applied random
distributions
The following sections are intended to illustrate how these
conditions can lead to different stochastic behaviour, even if the
same seed of randomness is utilised. An example is provided
to illustrate each item.
The first example presented is a basic process flow consisting
of a source, two queues with subsequent servers and a sink,
see Figure 1. The model’s parameters were set as follows:
Source, interarrival time:
30 seconds, exponentially distributed
Single stations 1 & 2
process time: 1.5 - 2.5 minutes, uniformly distributed
mean time between failures: 60 hours, exp. distributed
mean time to repair: 15 hours, based on the Erlang
distribution with n= 2
There are two policies that decide whether an agent, for
instance a customer, chooses the first or the second server.
Policy 1 is a random-only decision with a 50 : 50 chance.
With Policy 2, arriving agents will always choose the
shorter queue. If both queues are of the same length, a
random 50 : 50 chance is calculated.
If Policy 1 were to yield a higher throughput than the
”smarter“ Policy 2, it is deemed self-evident that the results
were skewed by the difference in stochastic behaviour, caused
by any one or combination of the above mentioned conditions.
To examine this scenario, a Monte Carlo simulation trial run
was performed on both iterations (Policy 1 and 2), each with
105
replications. The number of replications was determined
by the results of a sensitivity analysis carried out at an earlier
point. A different seed value was allocated to each replication,
which provides a stream of random numbers for the consumers
source, policy and both servers. Each simulated iteration covers
a duration of 24 actual hours.
The expected results were that the policy which considers
the length of the queues generated a greater throughput
than expected. In order to ascertain that this is the case in
every replication, a further analysis showed that in 14.8%
of replications, Policy 1 (random queue) lead to a higher
throughput than Policy 2 (shortest queue). In order to investigate
this further, the number of server maintenance occurrences
was analysed and the findings showed that, in some cases,
two iterations with the same random seed and stream were
correlated with different numbers of maintenance occurrences
per server. This seems to indicate that we are, in fact, comparing
not only two different policies, but also two different number
consumption patterns. Even though these differing patterns
183
are only obvious in the 14.8 % of the cases, it is suggested
that this methodological shortcoming affects all cases. It has a
certain influence on all the other scenarios, but in these cases
the impact is less substantial and not noticeable when one
only regards the number of maintenance occurrences. However,
an assessment of the down-time duration is likely to detect
stochastic differences in all replications. Focusing on those
14.8 % of cases where the throughput of Policy 1 (random
queue) is higher, it is notable that the number of maintenance
occurrences is lower than when applying Policy 2 with the
same seed value.
Why does the randomness differ between two iterations with
the same random seed? As mentioned above, the mid-square
method generates random numbers in such a manner that each
random number is based on the preceding one. Therefore, the
sequence of numbers is considered to be predetermined. In
the process flow described above, there are several consumers
of random numbers. Depending on the chosen policy, all the
consumers of random numbers (source, policy, servers) retrieve
a random number whenever the stochastic influence necessitates
it. For instance, Policy 1 (random queue) consumes a random
number every time an agent needs to decide whether it joins
Queue 1 or 2. In contrast, Policy 2 (shortest queue) only
consumes a random number when both queues are of the
same length, which only rarely occurs. Here, the sequence of
random number consumers differs substantially and this leads
to a different pattern of maintenance occurrences and process
times at all subsequent processes.
The question that arises from these deliberations is how
can researchers achieve the same stochastic influence for two
different parameter settings, in this case iterations with the same
random seed value. The solution appears to lie within the grasp
of the researcher, who can separate all random occurrences
within the model that are actually independent of each other. In
the presented scenario, this refers to the interarrival time of the
source, the random decision of the applied policy, the process
times of both servers and their maintenance occurrences. More
specifically, each random number consumer needs to retrieve
a new random number from its own stream. If this is the
case, the sequence of consumption will no longer influence
the generated random numbers for the other consumers.
Keeping random streams separate from each other is also
proposed to solve the issue addressed in condition 2) where
parameters affect the consumed random numbers during the
initialisation phase of a simulation model. A logistics scenario
considering the number of
AGV
s that are involved in a specific
material flow system lends itself to illustrate the case in point.
If the
AGV
initially retrieves a random number to establish its
initial battery level, the sequence is directly influenced by the
number of
AGV
s. Consequently, this leads to very different
random behaviour from the very beginning of the simulation.
This unwanted deviation can be prevented by either allocating
one specific stream of random numbers for each
AGV
or one
stream that only provides the random initial battery levels.
Point 3 directly relates to the topic of model initialisation.
Here, the issue arises when a model is capable of replicating
both, evaluating actual historical data or applied random
distributions. Both usually result in a different sequence in
the initialisation phase and therefore in a different sequence
of random numbers.
The conditions described apply intrinsically to simulation
modelling in general, regardless of the software used. De-
pending on the software utilised for modelling, the described
pitfalls must be tackled differently. FlexSim, for example,
offers a total of 100 random streams by default. They can
be allocated by only using digits from 1 to 100 as the
respective parameters in any distribution function. Beyond
that, FlexSim allows the user to create a new stream by using
the command
getstream(current)
, where
current
refers to the individual consumer of random numbers whom
a single stream is dedicated to commencing as of the first
retrieval. In contrast, there is an unlimited number of seed
values available in AnyLogic, which can be used to initialise
an unlimited number of Java objects by using, for instance,
Random r = new Random(1);
for a seed value of
1
.
These objects can be used as a parameter in any distribution
function instead of the default random generator of the
model. PlantSimulation, on the other hand, designates an
individual stream for each consumer automatically. Despite
the advanced settings and suggestions that PlantSimulation
offers, the developer requires a firm understanding of the
design of their simulation environment and the requirements
to maintain an empirically sound simulation when utilising
random numbers. In summary, while not explicitly naming all
common simulation tools, they all offer functions to maintain
control over random streams.
III. CONCLUSION
Based on this author’s previous reviews of the pertinent
literature in this domain, it is suggested that few simulation
studies tend to the exact assessment of random occurrences
from one replication to the another. Modellers frequently rely
on the software’s default random generator, as its main purpose
is to simply provide random numbers. Yet, depending on the
design of the simulation study, default settings may not suffice
when empirically reliable results require a more thorough
approach.
In this article, a brief historical review of random number
generators was provided and some substantial pitfalls in their
application to simulation modelling were highlighted. In this
particular context, it may appear as if some modellers do not
separate their random streams as their design may require
it. This oversight can lead to considerable differences in the
behaviour of random number consumers as well as the results
of the simulation, even if the same initial seed value is used.
As such, assuming the same stochastic behaviour may lead to
imprecise results and their comparison does not lead to reliable
conclusions. However, for many cases this difference of the
model’s stochastics may go unnoticed or perhaps be attributed
to the parameter settings. Finally, a number of suggestions were
made to remedy these procedural shortcomings of replicating
randomness.
184
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