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The Problem of Stopping Software Tests and the Interpretation of
Software Programs
PAOLO ROCCHI
IBM,
via Luigi Stipa 150, 00148 Roma,
ITALY
also with
LUISS University,
via Romania 32, 00197 Roma,
ITALY
Abstract: - We conducted empirical research on the defects found by customers when using six large software
programs. The Wakeby (WAK) and Kumaraswamy (KUM) functions, recently discovered by statisticians,
proved to be the best in terms of fitting the six time-series of defects. Two analytical reports on this empirical
research have already been published; we do not repeat the details of the outcomes here, but instead illustrate
the consequences deriving from the published outcomes. The abstract and applied properties of WAK and
KUM provide new answers to the following managerial and theoretical problems.
- The first problem concerns managers who make the decision to stop software testing. The literature proposes
various directional criteria which leave margins for subjectivity and incertitude. WAK and KUM provide a
mathematical answer to the problem of terminating a software test.
- Most researchers have assigned software defects to the personal inabilities of developers, and the descriptions
of software programs usually neglect the objective mechanisms for errors. Based on WAK and KUM, this
paper suggests a territory model that illustrates the behavior of a software program with erroneous instructions.
This paper aims to raise a discussion of the original ideas generated by WAK and KUM. These ideas need
to be further verified, and in its current form, this work can therefore be considered a position paper.
Key-Words: - Software defects; Management of software test; Modelling software programs execution;
Wakeby and Kumaraswamy functions.
Received: March 17, 2024. Revised: August 21, 2024. Accepted: October 7, 2024. Available online: November 11, 2024.
1 Introduction
This paper reports on the third phase of a research
project. In this section, we briefly review the first
two phases for the convenience of the reader, and
more details can be found in [1], [2].
1.1 First Phase
We first investigated a large-scale product from
IBM that is used to monitor the performance and
availability of all of the hardware (e.g. disks,
memory, I/O, CPU, network) and software (e.g.
application programs, management programs, etc.)
resources of a large-scale computer installation.
IBM developed four versions of this product with
increasing complexity over time; each release
passed alpha and beta tests and was released on the
market, but had residual software defects that clients
detected during normal operation. Requests for
change (RFCs) by the clients covered a range from
severe errors, such as abnormal termination of the
program, to suggestions that vary or add simple
functions. Each RFC was sent by a customer to the
triage system in charge of handling software defects,
and teams of software specialists promptly fixed the
defects identified by the clients. It may be said that
both the clients and the software developers jointly
carried out an effective gamma test.
In a previous research work [1], we examined
various aspects of this process, such as the
discovery of software defects over time, the severity
of the impact of a defect on the system, the times
required to ‘open’ and ‘close’ a change (that are the
times to initiate the fixing process and to terminate
it), and other parameters. The time-series data on the
defects, which relate the number of detected
software defects to time, constitute the most
important outcome of this study, in fact histograms
exhibit very different shapes: concave, convex,
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oscillating and so on (Figure 1, Figure 2 and Figure
3). The variety of trends made it challenging to find
the best statistical function fitting the empirical data.
Fig. 1: Rapidly decreasing histogram
Source:[1]
Fig. 2: Increasing histogram
Source:[1]
Fig. 3: Bell histogram
Source:[1]
We used a program called EasyFit to try forty
candidate functions [3], and finally found that the
Wakeby (WAK) was best adapted to the four time-
series using the Kolmogorov-Smirnov test, which
was accepted at the 99% significance level. This
result shows the following features:
The literature presents the power functions
(e.g., Pareto, gamma, log-normal and other
traditional functions) as best fitting to time-series
data on software bugs [4], [5]; so WAK turns out to
be unusual and surprising from the statistical
viewpoint.
WAK is one of the most recent distributions
(Appendix A) defined for the purpose of modeling
hydrological phenomena, [6], [7]. It is also
employed in other research fields, such as sociology
[8], energy management [9], and reliability but has
been generally ignored in software engineering. To
be more precise, we explored various repositories
(ACM, IEEE, arXiv and others), and found only one
work employs WAK for the purpose of developing a
machine learning application, [10].
1.2 Second Phase
We therefore wondered whether the fitting with
WAK a particular case or a strange coincidence
was. In the second phase, we examined two open-
source packages, Ubuntu and Android [2], whereas
the first phase of research had studied four
proprietary software packages. From the Ubuntu
bug repository, we identified all the defects
registered over six years. The errors noticed by
Android customers were reported over about five
years. An Anderson–Darling test (accepted at the
99% significance level) showed that the WAK
function fitted the defects reported in Ubuntu, while
a Kolmogorov–Smirnov test proved that the
Kumaraswamy distribution (KUM) was the best fit
for Android. The KUM distribution was devised to
explore rainfalls and floods from a statistical
perspective (Appendix B), and the results of the
second phase are therefore in line with the first
results.
This pair of research stages provides consistent
results. The functions WAK and KUM have similar
characteristics from the application stance,
moreover they suggest new insights about computer
science. In detail, they provide in-depth views about
stopping software tests, that is a practical problem
(Section 2), and software program modeling,
representing a theoretical problem (Section 3).
2 The Problem of Stopping Software
Tests
Error detection and correction require a great deal of
time and remain some of the costliest aspects of
software development.
2.1 State of the Art
Testing managers encounter significant difficulties,
because software tests indicate how a code works
only in certain controlled cases, and an erroneous
logical condition can remain hidden for a long time,
until it is executed. Program testing proves to be a
very effective way to demonstrate the presence of
software defects, but it is hopelessly inadequate for
demonstrating their absence. Consequently, the
more tests that are conducted, the more defects are
discovered, but this method proves to be expensive,
and determining a suitable number of tests is the
central issue to be resolved.
Researchers have put forward several ways to
follow. The authors of [11], [12] use software tools,
while the researchers in [13] set up a graph, and
those in [14] resort to artificial intelligence. A group
of researchers exploited the theory of reliability in
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[15], [16]. Testing managers are often inclined to
find a trade-off between costs and benefits [17]; on
the one hand, they aim to exploit the benefits of an
earlier market introduction, while on the other, they
prefer to defer the release of a product in order to
enhance its functionalities or to improve its quality.
Despite great efforts, solid criteria are lacking, [18].
The paper [19] concludes: “In current practice,
deciding when to close a crowd-testing task is
largely done by guesswork due to lack of decision
support”.
2.2 Separation Effect
WAK and KUM provide a precise answer to why
have researchers have missed the mark so far. These
two functions share a special mathematical property:
the right and left tails of the function are
independent of each other (Appendix A). This
property, called the "separation effect," implies that
the final course of the curve cannot be predicted
based on the initial data. In the practice of software
testing, this implies that it is not possible to
accurately determine when to stop a test based on
the faults found so far. WAK and KUM disprove in
mathematical terms those who theorize about
predictions, and prove that it is impossible to
forecast when 100% of the defects will be
discovered. The problem of stopping a software test
has no rigorous solution, meaning that the
difficulties encountered by test managers do not
come from professional inabilities but from the very
software defects, the number of which increases
over time or decreases, is asymmetric or constant,
etc. (Figure 1, Figure 2 and Figure 3).
2.3 Managerial Tactics
Although WAK and KUM demonstrate on a
theoretical level that the problem of stopping a
software test has no rigorous solution, we can adopt
the following pragmatic criteria.
There are two mechanisms underpinning the test
processes analyzed here: the discovery of defects
and their correction. The former raises the time
curve, while the latter lowers the curve. Together,
they create variable and unpredictable time
distributions of defects. Thus, the best that a testing
manager can do is to analyze the most recent results
and apply the following rules:
(1) If the number of the latest found defects is
steadily decreasing, then the validation process can
be stopped in conformity with the mathematical
projections and the available resources (economic,
organizational, technical etc.).
(2) If the temporal distribution has a counter-
intuitive trend because the curve remains constant or
is even going up instead of down, then debugging
should proceed until condition (1) is met. The test
manager may have to ask for more resources in
view of the trend in the number of defects
discovered at a given time.
As practical examples, the decreasing trend in
Figure 1 indicates that the testing process is
approaching its end, while the growing curve in
Figure 2 indicates that tests must go on. Figure 3
shows a rapidly increasing trend which later drops
down, indicating that the condition for stopping
testing is not far away from being met.
Fig. 4: The awash terrain T (top), and the territory
model (bottom)
3 Interpretation of Computer
Programs
For theorists, a computer program is a set of
instructions [20] that can be formalized using
propositional logic, set algebra and other
mathematical tools, [21]. Methodological studies of
software programming provide further accurate
descriptions of programs. Block diagrams, pseudo-
code, hierarchical diagrams, module diagrams,
Voronoi maps, sequence diagrams, class diagrams
and so on are used to illustrate special aspects of the
software and to aid experts to implement software
products, [22], [23], [24]. Both theoretical and
professional models share a common characteristic:
they describe a software program that functions
correctly, and do not account for the software
program that fails or is terminated abnormally.
WAK and KUM are original statistical distributions
that help us fill this gap.
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3.1 Territory Model
A methodological premise needs to be established.
Suppose that topics X and Y belong to two distinct
fields, and that X is clear whereas Y is new and
mysterious. If a mathematical model that applies to
X also governs Y, then scientists can exploit the
attributes of X to interpret Y. For instance,
physicists have used the concept of water flow (X)
to describe an electrical current (Y), which was
difficult to understand in the early stages.
Both the proprietary programs and the open-
source programs studied in the introduction are
system programs which ordinarily run without
interruption for a long period; hence, we can assume
that on a global basis, the defect distributions
discovered in [1] and [2] are not influenced by
occasional events, human interferences or random
factors. We can reasonably conclude that WAK and
KUM relate to the objective behavior of incorrect
software programs.
Fig. 5: Software instructions executed once (left)
and three times (right)
The Wakeby and Kumaraswamy functions were
developed to qualify a geographical territory
deluged by floods, overflows and inundations; we
can therefore think of software in terms of
hydrology/geology. This is suitable because the
execution of a program also involves eddies and
flows, like water. Hence, we can use a territory T
consisting of several plots flooded by water (Figure
4 top) to represent the software program P
comprising modules affected by errors. The diagram
representing both T and P is called Territory Model
(TM) (Figure 4, bottom). We draw analogies
between hydrology and software programming in
order to better understand defective software
programs.
3.2 Running Programs
Turing’s machine carries out software operations
one after the other, and the program execution can
be understood as a continuous flow that conforms to
the logic of the algorithm (Figure 5, left). When the
program is activated several times, the processor
creates a flow with several eddies due to loops,
jumps, goto commands, etc. (Figure 5, right).
Suppose the mountain territory T, awash with
rainfall A, includes areas Sa, Sb, Sc etc. which have
different shapes, extensions, orientations and
textures. The land consists of permeable and
impermeable soils, cultivated and inhabited zones,
etc. In a similar way, input data A are used to launch
the program P that is subdivided into sectors Sa, Sb,
Sc etc. which perform a variety of functions. These
sectors are the routines and modules in procedural
programming, and the methods in object-oriented
programming. Some professional diagrams visualize
these sectors in a manner very similar to the TM.
For instance, the Nassi–Shneiderman diagram
illustrates the sequence, selection and iteration of a
structured program using special geometric shapes,
[25]. The labels indicate the functions of each
section and make the logic of the algorithm explicit
(Figure 6).
Fig. 6: Nassi–Shneiderman diagram for a program
handling a customer order
3.3 The Advantages of TM
In similarity to the terrain T which gives out the
regular creek B or undergoes a flood (Figure 4 left),
the software package P, executed several times
(Figure 5 right), provides the correct result or
otherwise runs into failure. We can use the Territory
Model to explicate the performances of P using the
parallel characteristics of the terrain T. The
following list pinpoints the symmetries:
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1. The terrain T covers a certain geographical
area ↔ P occupies a portion of the computer
memory.
2. The assortment of sectors characterizes the
geographical territory ↔ The number of modules
indirectly indicates the complexity of the algorithm.
3. The territorial sections vary significantly in
a large area whereas diversities are negligible if T is
small ↔ Subdivision into modules is necessary
when P is large and negligible in a small program.
4. Each geographic sector has a special
geomorphological property: inhabited, arid,
cultivated etc. that facilitates or hinders waterways
↔ Each software sector executes a specific function
that runs linearly or makes intricate the program
running.
5. The rain A falls on the various zones of T
↔ Input-data activate the functions of the software
sectors.
6. The borders establish the precise relations
amongst the territorial sectors ↔ Software sectors
Sa, Sb, Sc etc. interact through well-defined
interfaces for data exchange.
7. Heavy rains hit every spot on the ground
and can result in overflowing ↔ The repeated
executions of P check every part of the algorithm
and reveal software defects.
8. A regular area of T conveys water into B;
instead, obstructed and rocky sectors cause
overflows or inundations ↔ The software module
brings forth the correct output; otherwise, it has
defects or even causes an abnormal end.
9. The flatter the sector of T, the better the rain
is absorbed and flows ↔ The more linear the logic
of the software module, the less affected by errors.
10. Excess water can generate an abnormal
‘cascade effect’ on T ↔ A software error can
multiply the number of subsequent faults and
creates a ‘cascade effect’.
11. Inundation has very different behaviors in
the initial and final stages ↔ The temporal series of
bugs present apparent dissimilarities in the left and
right tails.
In conclusion, the Territory Model illustrates the
failure proneness of software programs. TM shows
how errors deviate the correct execution of the
algorithm and can even result in a multiplicative
effect (points 8, 10 and 11). TM proves that the
extension and complexity of the algorithm are the
objective root-causes of software failures (points 2,
3, 4 and 9).
Authors normally ascribe the origin of software
errors to human factors, [26], [27]. The cause-effect
relationships are identified as cognitive problems,
misunderstandings, mistakes and subjective
inabilities, [28], [29], [30]; instead, TM highlights
mechanisms which are independent of human
factors. TM assigns the origins of the software
defects to objective situations characterized by
extension and complexity of logic. This also match
with recent inquiries which seek for the ‘error prone
regions’ that are identified using genetic algorithms,
[31].
The concept of software engineering has been
the subject of vigorous debate over the years.
Researchers have raised doubts about software
engineering that seems pointless and not a real
engineering field, [32]. Troubled software projects
and the theories of computing which addressing
abstract issues support this criticism in a way. In
fact, there is a certain divide between the
professional practice and the theories that
sometimes are fragmentary and abstract, [33]. The
Territory Model, underpinned by WAK and KUM,
enhances our knowledge about the fault-proneness
of programs in scientific terms and supports the
engineering status of software technology.
4 Discussion and Conclusion
The Wakeby and Kumaraswamy functions best fit
six temporal distributions of software defects and
constitute the first unexpected finding of this
research project since they are new in the literature
of software debugging. Moreover, WAK and KUM
provide innovative answers to a pair of long-
discussed issues.
The tails of WAK and KUM are independent
each other this property implies that the history of a
software test process does not influence the final
phase. The two distributions demonstrate in
mathematical terms that the problem of stopping test
cannot be resolved in a rigorous manner and this
conclusion matches with universal experience. As a
consequence, a manager should close testing based
on the current trend of bug series, and this method
also corresponds to and improves the ITIL
guidelines about the so called ‘problem
management’, [34].
WAK and KUM establish a logical parallel
between hydrology and the software, especially
between an inundated territory and executed
software programs. To make this concrete, we
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propose the Territory Model, which expounds
analytical analogies and improves our understanding
of incorrect software. TM highlights how the risks
of software failures increase in proportion to the
complexity and extent of the software package. This
reading key, focusing on objective features,
provides insights rather distant from the current
subjective views of programming errors and
supports the engineering status of software
technology.
The original ideas that WAK and KUM spawn
should be further verified. It would be useful to
examine various time-series of defects, but this
inquiry takes much time. The present report is a
position paper which aims to quickly inform the
scientific community of the results, while further
initiatives will start in the future.
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DOI: 10.37394/232018.2025.13.5
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Volume 13, 2025
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Appendix A
Climate change cause abnormal rainfalls and
unusual water floods. Extreme values and apparent
skewness characterize hydrologic phenomena, so
statisticians have made considerable efforts to set up
suitable mathematical tools to address such special
events, say new indices, accurate distribution
functions, a theory of extremes etc. Distribution
functions, such as GEV (Generalized Extreme
Value), TCEV (Two Components Extreme Value)
and WAK (Wakeby), characterized by three or more
parameters, are able, better than traditional
distributions, to reproduce the statistical behaviors
of floods.
Wakeby is a five parameters function, more than
most of the common distributions; and was
originally introduced in 1978 by John C. Houghton
who named it after Wakeby pond on Cape Cod,
[35]. Houghton parameterized WAK by the
following quantile function
( ) 1 (1 ) 1 (1 ) .x F F F
(A.1)
Where x(F) is the probability of occurrence of x,
that can be regarded as a function to calculate
percentile. For example, the value x(0.5) would
correspond to the 50th percentile. The value ξ is the
location parameter; α and γ are the scale parameters;
β and δ are the shape parameters of the left end-tail
and of the right-end tail respectively. In fact, the
Wakeby can be thought of in two parts, the right-
hand tail
(1 )F
and the left-hand tail
(1 )F
.
WAK mimics the final trend of data that is
independent of the initial trend (and vice versa) by
means of δ. This ‘separation effect’ ordinarily
draws the attention of experts in hydrology, [36].
When δ > 0, WAK has a heavy upper tail and can
model occasional high outliers.
The parameters of WAK are constrained by the
following conditions:
(a) If α = 0, then β = 0.
(b) If γ = 0, then δ = 0.
(c) γ ≥ 0, (α + γ) ≥ 0.
(d) Either (β + δ) > 0 or (β + γ) = δ = 0.
The moments of all orders exist, provided δ ≤ 0
[36]. If δ is positive, E[Xr] exists for 0 ≤ r < (δ−1).
The probability moment can be defined this way
αr = E{X[1 − F(X)] r} (A.2)
And has this simple form for r ≥ 0
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αr = (r + 1)−1{ξ + α (r + β + 1)−1 + γ (r − δ + 1)−1}
(A.3)
In particular, if r = 0 we get from (A.2)
α0 = ξ + α (β + 1)−1 + γ (−δ + 1)−1 = E[X] (A.4)
All this allows for a wider variety of shapes and
the distribution is well suited to simulation of
intricate physical phenomena. The Wakeby
distribution exhibits more stability under small
perturbations when compared to the Beta distribution
and other more common distributions. In summary,
WAK is highly general; it can describe complex
events whose final behaviours dislike the initial
trends; it is robust against outliers, and it has a closed
functional form for determining percentiles.
Here the outcomes obtained in the first research
phase, [1]. Table A1 includes the number of defects
discovered by users, the time interval covered by the
gamma test of each release, the histograms that
relate the number of defects to time and the
dimension of the bars. Table A2 shows the
parameters that fit with the four releases of the IBM
software package. Columns D and P indicate the
statistic and the p-values.
Table A3 exhibits the parameters depicting the
right and left tails of time-series which demonstrate
far different trends. Table A4 shows the values of
the parameters fitting with the empirical data of
Ubuntu, [2].
Table A1. Data about the time-series
Release
Number of defects
Time interval (days)
Figure
Histogram Segment
(days)
1
838
1471
1
73.5
2
322
1074
3
71.6
3
495
975
N.A.
75.0
4
593
939
2
85.36
Table A2. Parameters that optimize the Wakeby distribution for each release
Release
D
P
1
0.01986
0.88908
261.35
0.82955
121.3
0.10885
39082.0
2
0.02816
0.95414
803.23
4.4198
228.77
-0.21498
39460.0
3
0.0205
0.98272
742.2
5.5649
333.95
-0.44585
39749.0
4
0.04052
0.27695
6.7581E+
8
16826.0
676.17
-1.0297
0
Table A3. Parameters that detail the tails of each distribution
Release
Tail
D
P
1
Left
0.02256
0.91524
-97.495
3.8297
330.08
-0.85276
39085.0
1
Right
0.07706
0.97820
1.5900E+9
39906.0
226.47
-0.18278
0
2
Left
0.05626
0.65205
1008.8
12.49
304.76
-1.11
39427.0
2
Right
0.06502
0.90994
1.0533E+10
2.6363E+5
122.36
-0.00949
0
3
Left
0.05435
0.62191
396.72
6.8342
242.63
-1.1144
39733.0
3
Right
0.04618
0.62882
6.9669E+9
1.7384E+5
240.32
-0.42067
0
4
Left
0.09203
0.68535
7030.3
27.326
269.83
-1.5908
39813.0
4
Right
0.0579
0.2075
407.44
1.1267
0
0
40452.0
Table A4. Parameters that optimize the Wakeby distribution for Ubuntu
R
D
P
Ubuntu
4
0.09182
0.17134
1021.8
0.3368
0.0
0.0
38330.0
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Volume 13, 2025
Appendix B
Poondi Kumaraswamy, an engineer and hydrologist
from India, developed a two-parametric function for
double bounded random processes, [37]. Following
the approach of [38], the cdf of Kumaraswamy is as
follows for any baseline cumulative distribution
function G(x):
( ) 1 [1 ( ) ] , 0, 0.
ab
F x G x a b= - - > >
(B.1)
and the corresponding probability density function
(pdf) is
11
( ) ( ) ( ) [1 ( ) ] .
a a b
f x ab g x G x G x
--
=-
(B.2)
Where g(x) = dG(x)/dx is the baseline pdf; a and b
are shape parameters whose role is partly to
introduce skewness and to vary tail weights. The pdf
can be unimodal, increasing, decreasing or constant,
depending on G(x) and the two parameters that is
why this distribution is applicable to many natural
phenomena, [39].
KUM is used to analyze rainfall time series,
regional flood frequencies, wind extremes, river
flow modeling, atmospheric temperatures etc. KUM
is also used in computing literature, for example,
about Shannon information [40], data mining [41],
and especially in the design of neural networks [42],
[43], [44], [45].
The simplest cdf and pdf of Kumaraswamy are
the followings:
( 1)
( ) 1 (1 ) 0, 0; 0 1.F x x x
hg hg
-
= - - > > < <
(B.3)
( 1) ( 1)
( ) (1 ) .f x x x
h m g
hg --
=-
(B.4)
Where η and γ are the shape parameters. EasyFit
utilizes the following pdf version:
( )
2
11
12
( 1)
( 1) 1
, where .
()
()
zz xa
z
b a b a
fx
a
aa
aa
=
-
---
--
=
(B.5)
Table B1 exhibits the values fitting the
Kumaraswamy distribution with the timeseries of
Android defects [2].
Table B1. Parameters of KUM
Sample Size
Statistic
P-Value
Rank
1,016
0.05482
0.00428
1
0.2
0.1
0.05
0.02
0.01
Critical Value
0.03366
0.03837
0.0426
0.04762
0.05111
a
b
1.0
430.38
1021.8
0.3368
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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