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Dr Jezdimir KNEZEVIC Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 8( Version 7), August 2014, pp.93-100
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Mirce Functionability Equation
Dr Jezdimir Knezevic
MIRCE Akademy, Woodbury Park, Exeter, EX5 1JJ, UK
Abstract
Scientific principles and concepts expressed through the laws, equations and formulas are the bedrock for the
prediction of the deign-in functionality performance of any engineering creation. However, there is no
equivalent when the in-service functionability performance predictions have to be made. Hence, Mirce
Mechanics has been created at the MIRCE Akademy to fulfil the roll. The main purpose of this paper is to
present the development and application of Mirce Functionability Equation which is the bedrock for the
prediction of the functionability performance of maintainable systems.
I. The Concept of Functionality
According to Einstein “Everything that the
human race has done and thought is concerned with
the satisfaction of felt needs”.
Human needs for transporting, communicating,
defending, entertaining and many other functions are
satisfied by ships, airplanes, tractors, computers,
radios and other systems. As they are functioning in
accordance to the laws of science, which are
independent of time, place and human impact, their
design-in performance, like speed, acceleration,
power, fuel consumption and many others, are
accurately predictable. [1]
II. The Science of Functionality
The theoretical foundations of designing systems
are laws of science that describe observable natural
phenomena, known to humans so far. Among them
laws of motion are the most significant from the life
cycle engineering and management point of view, in
respect to functionality of a system. Some of them
are very briefly addressed in this paper as the
scientific foundation for the development of the laws
of the motion of functionability. Hence:
Newton's laws of motion are three physical laws that
form the basis for classical mechanics. These laws
describe the relationship between the forces acting on
a body and the motion of that body. They were first
compiled by Sir Isaac Newton in his work
Philosophiæ Naturalis Principia Mathematica, first
published on July 5, 1687.Newton used them to
explain and investigate the motion of many physical
objects and systems, from the “apple” to planets.
Kepler's laws of planetary motion are three
astronomical laws that describe the motion of planets
around the Sun. From them it is possible to
accurately predict either what the position of the
planet is at a given time, or the time when the planet
is in a given position.
Maxwell's equations are a set of four partial
differential equations that relate the electric and
magnetic fields to their sources, charge density and
current density.
Navier–Stokes equations, describe the motion of
fluid substances. These equations arise from applying
Newton’s second law to the motion of fluid, together
with the assumption that the fluid stress is the sum of
a diffusing viscous term (proportional to the gradient
of velocity), plus a pressure term. The equations are
useful because they describe the physics of many
things from modelling the weather, ocean currents,
water flow in a pipe, air flow around a wing and
motion of stars inside a galaxy. In their full and
simplified forms help with the design of aircraft and
cars, the study of blood flow, the design of power
stations, the analysis of pollution, and many other
things.
Boltzmann transport equation, is used to study the
motion of physical quantities such as heat and charge
through fluid, and thus to derive transport properties
such as electrical conductivity, viscosity, and thermal
conductivity. Physicists today use the equation to
model gases in everything from nuclear power
stations to galaxies
Heisenberg's equation of motion was the first
complete and correct definition of quantum
mechanics, branch of physics that study the motion of
subatomic particles. It extended the Bohr model of
atom by describing how the quantum jumps occur, by
interpreting the physical properties of particles as
matrices that evolve in time. The Heisenberg
equation of motion, named after Werner Heisenberg
who formulate it in 1925.
Schrödinger equation describes how the quantum
state of a physical system changes in time. It is as
central to quantum mechanics as Newton's laws are
RESEARCH ARTICLE OPEN ACCESS
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to classical mechanics. In the standard interpretation
of quantum mechanics, the quantum state, also called
a wavefunction or state vector, is the most complete
description that can be given to a physical system.
The equation is named after Erwin Schrödinger, who
constructed it in 1926. Solutions to Schrödinger's
equation describe not only molecular, atomic and
subatomic systems, but also macroscopic systems,
possibly even the whole universe.
In summary, scientific principles and concepts
expressed through the laws, equations and formulas
are the bedrock of any engineering creation. They
have achieved that status by providing accurate
predictions for all engineering and management
concepts, scenarios and “dreams”.
III. Concept of Maintainable System
At the end of production or construction process,
when all consisting components are assembled
together and relationships between them established,
a new physical system is “born” with capability to
deliver all expected functionality characteristics. That
unique, infinitesimally short instant of time, is
denoted as t=0, to mark the beginning of the system
operational process. Thus, each system will have its
own “birth” time, which is very important from the
system life point of view. At that instant the system
is, for the very first time in its life, able to satisfy
users’ needs by delivering functionality (function,
performance and attributes). Hence, functionality
characteristics of the system are inherited from its
design process and cannot be changed during the
system life, apart from implementing some
modifications and redesigns.
For example, in 1969, engineers and managers of
the Boeing Corporation have deliver to the world first
wide body aircraft, named Boeing 747, series 100
with the known functionality characteristics.:
Passengers
3-class configuration
2-class configuration
1-class configuration
366
452
N/A
Cargo;
6,19 ft3 = 30 LD-1
containers
Engines
maximum thrust
Pratt & Whitney
JT9D-7A
46,500 lb (20,925 kg)
Rolls-Royce
RB211-524B2
50,100 lb (22,545 kg)
GE CF6-45A2
46,500 lb (20,925 kg)
Maximum Fuel
Capacity
48,445 U.S. gal (183,380 L)
Maximum Takeoff
Weight
735,000 lb (333,400 kg)
Maximum Range
6,100 statute miles (9,800
km)
Typical Cruise Speed
at 35,000 feet
Mach 0.84
555 mph (895 km/h)
Basic Dimensions
Wing Span
Overall Length
Tail Height
Interior Cabin Width
195 ft 8 in (59.6 m)
231 ft 10.2 in (70.6 m)
63 ft 5 in (19.3 m)
20 ft (6.1 m)
It is expected that each Boeing 747-100 series
aircraft have the same functionality, under identical
environmental conditions, because the laws of nature
are independent of time and the location in the
universe, However, experience teaches us that in-
service performance of these systems is dominated by
phenomena like fatigue, operator induced errors,
corrosion, creep, foreign object damage, a faulty
weld, bird strike, perished rubber, carburettor icing,
to name just a few. These phenomena generate
energy exchanges between systems and environment,
leading to the loss of the design-in function or
performance. Hence, maintaining the deign-in
performance beyond the delivery day requires actions
like troubleshooting, repairs, replacements,
modifications, diagnostics, “cannibalisations” and
similar to be performed.
In summary, any entity that satisfy human needs
by performing a measurable function whose design-
in functionality is maintained by humans is defined
as a maintainable system.
IV. The Concept of Functionability
Thus, the ability of being functional through
time, known as functionability, is an essential in-
service property of maintainable systems.
From functionability point of view, at any instant
of time a system can be in one of the following two
states:
Positive Functionability State, PFS, which is the
state of being functional
Negative Functionability State, NFS, which is
the state of not being functional
The motion of the system through functionability
states is governed by the occurrence of
functionability events, which are classified as:
Positive Functionability Events, PFE, which
cause the transition from NFS to PFS
Negative Functionability Events, NFE, which
cause the transition from PFS to NFS
Consequently, the life of a maintainable system
could be considered as motion of system through
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functionability states. The pattern generated by the
motion of functionability through functionability
states, in respect to the passage of time, forms the
functionability trajectory.
V. Functionability Questions
One of the major concerns of design engineers
and project managers are predictions of operation,
maintenance and support resources required for
maintaining systems in positive operational state
through their life. These include diagnostic
equipment, skilled and trained maintenance
personnel, maintenance facilities, spare parts,
inspection tools, technical data, storage facilities,
means of transportations and so forth. Often the cost
of these resources considerably exceeds the purchase
cost of system itself. Equally, the lack of
maintenance resources causes further delays in the
recovery of functionality. Hence, some balance
between investment in the resources and the time
delays incurred by their deficiency is required. To
make that trade off, engineers and managers, need to
find the answer to the following functionability
related questions:
• How many Negative Functionability Events are
going to occur?
• What types of Negative Functionality Events are
going to occur?
• What frequencies of Negative Functionability
Events are going to be?
• How the cause of Negative Functionability Event
will be detected?
• How long systems are going to be in Negative
Functionability State?
• How long systems are going to be in Positive
Functionability State?
Unlike the functionality questions to which
existing laws of science readily provide the answers,
the above raised functionability questions stayed
unanswered. Existing equations of motion, some of
which are briefly presented at the beginning of this
paper, are not able to even the address the above
questions, not because they are incorrect, but because
they are not created to address these phenomena.
In summary, without ability to provide accurate
answers to functionability questions design
engineering and project management are not in the
position make the trade off between the cost of
resources required to maintain systems in positive
functionability states and the consequential losses
while they are in negative functionability states.
VI. Concept of Mirce Mechanics
The development of science started when people
began to study phenomena not merely observing
them. People developed instruments and learned to
trust their readings, rather than to rely on their own
perceptions. They recorded the results of their
measurements in the form of numbers. Supplied with
these numbers they began to seek relationships
between them and to write down in the form of
formulas. Then the formulas became the only things
they came to trust when they began to predict things
they could not physically experience.
Consequently, to address functionability
questions the author established the MIRCE
Akademy in 1999. Staff, Fellows, Members and
students of the Akademy study in-service behaviour
of maintainable systems to:
Physically observe the emerging trajectory of the
motion of functionability through the life of
maintainable systems and to measure emerging
in-service performance
Scientifically understand mechanisms that cause
the motion of a functionability through the life
of maintainable systems, within the physical
scale from 10-10 to 1010 metre [2,3,,4,5,6,7]
Mathematically define the scheme for the
prediction of in-service performance of a given
design-in system, for a given in-service
conditions and rules.
A science based body of knowledge, formulated
through axioms, formulas, methods, rules and
algorithms for predicting the in-service performance
of the future systems, resulting from their motion
through the functionability states in respect to time
constitutes Mirce Mechanics.
The ability to simultaneously predict the design-
in functionality performance and in-service
functionability performance of the future systems is
of fundamental importance for the engineers,
managers, investors, regulators and other specialists
who are responsible for the satisfaction of the
“human felt needs”, in reliable, economical and safe
manner, for the future transportation, communication,
defence, energy, entertainment and many other
functions delivered by maintainable systems.
VII. The Concept of Motion in Mirce
Mechanics
Motion is one of the most complex concepts of
science. The images it creates in our minds are
diverse as the “jiggling” of atoms and molecules to
the movement of planets, and beyond.
Since the earliest years of science the only idea
of motion imagined was that of mechanical motion,
so there is a tendency to view all other kinds of
motion in terms of the concept of trajectory. As the
science progressed, this naturally became impossible,
for instance when the attempt was made to conceive
the electrical motion. It could be possible, of course,
to think in the case of a high-voltage transmission
line that wire is the “trajectory” of the electric
signals. However, such a mental picture would have
no practical purpose, as the electromagnetic waves
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could not have been viewed as a liquid flowing
through the wires.
Consequently, the question by which the motion
of functionability through the life of maintainable
systems is to be described must contain only those
quantities that can be measured physically. Research
performed shown that it could only be seen as the
change in the functionability state of a system
through time. Hence, a life of any maintainable
system could be viewed as a sequence of occurrences
of positive and negative functionability events that
“move” systems through functionability states.
In summary, in Mirce Mechanics the motion of
functionability is perceived as the change in the
functionability state of a system in relation to
functionability state variables, with respect to the
passage of time. Functionability state variables are
measures of functionality performance of a system
that uniquely determine the functionability states of a
system.
VIII. Mechanisms of Motion
As statistics does not study the cause of
statistical behaviour, to understand that motion of
functionability it was necessary to scientifically
analyse the mechanisms that generate functionability
events.
To understand the mechanisms that generate
negative functionability events analysis of over tens
of thousands of components, modules and assemblies
of systems in defence, aerospace, transportation,
motorsport, nuclear, communication and other
industries, had been studied at the MIRCE Akademy.
As it has a profound impact on all aspects of the in-
service life on any maintainable system several
research studies have been performed by the Master
and Doctoral students of the MIRCE Akademy with
aim to understand the physical mechanisms that
caused their occurrences.
All physical phenomena that cause the motion of
a system from the positive to negative functionability
states are known as negative functionability events.
Mechanisms that generate negative functionability
events belong to the following three categories:
Overstress mechanisms, where acting stresses
generated by mechanical, electrical, thermal,
radiation, chemical and other type of energy
exceed that strength of components and systems
subjected, resulting from phenomena like foreign
object damage (birds, hail, rain, snow),
lightening, abuse by operators, maintenance
errors and similar
Wearout mechanisms, where cumulative
damage, generated by mechanical, electrical,
thermal, radiation, chemical and other type of
energy, is accumulated through processes like,
corrosion, fatigue, creep, wear and similar.
Human actions, where the transition from
positive to negative state results from direct
decision taken by humans. Most frequently these
actions are performed as a part of scheduled
maintenance tasks performed to check the state
of a system, to preventively replace
predetermined components or to install modified
components.
All physical phenomena that cause the motion of
a system from the negative to positive functionability
states are known as positive functionability events.
Mechanisms that generate positive events belong to
the following categories [8]:
Servicing: replenishment of consumable fluids,
cleaning, washing, painting, etc.
Lubrication: installing or replenishing lubricant.
Inspection: Examination of an item against a
defined physical standard.
General visual inspection: performed to detect
obvious unsatisfactory conditions.
It may require the removal of panels and access
doors, work stands, ladders, and may be required
to gain access.
Detailed visual inspection: consists of intensive
visual search for evidence of any irregularity.
Inspection aids, like mirrors, special lighting,
hand lens, boroscopes, etc. are usually required.
Surface cleaning may be required, as well as
elaborate access procedure.
Special visual inspection: an intensive
examination of specific area using special
inspection equipment such as radiography,
thermography, dye penetrant, eddies current,
high power magnification or other NDT.
Elaborate access and detailed disassembly may
be required.
Check: a qualitative or quantitative assessment
of function.
Examination: a quantitative assessment of
one/more functions on an item to determine
whether it performs within acceptable limits.
Operational: a qualitative assessment to
determine whether an item is fulfilling its
intended function. It does not require quantitative
tolerances.
Restoration: perform to return an item to a
specific standard. This may involve cleaning,
repair, replacement or overhaul.
Discard: removal of an item from service.
All of the above listed mechanisms of the motion
of systems through positive and negative
functionability states are observable physical
processes or recognisable human actions. [9]
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IX. Mathematics of Motion in Mirce
Mechanics
Results of experiments and observations
performed thus far unquestionably lead to conclusion
that the deterministic regularity found in the
continuous motion of functionality, such as speed,
acceleration and similar, studied by classical
mechanics, cannot be found in respect to the motion
of functionability through time. What can be found is
discrete motion with statistical variability, as shown
in Figure 1.
Thus, functionability trajectories, generated by
similar individual systems, under similar
circumstances vary among them self, to the degree
that no two trajectories are identical. Therefore, the
proven formulas of Newtonian mechanics that govern
the motion of macroscopic bodies through time
cannot be used for predicting the motion of
functionability through time, as far as the
functionability trajectory is concerned
The relative frequency histogram of the motion
of functionability through the life of sample size of
497 systems at specific instances of time is obtained
by using well known statistical expression:
Number fo systems in PFS @ t
'( ) Total Number of Systems Orserved
yt
1.
0
50
100
150
200
250
300
350
400
450
040 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800 840 880 920 960 1000
Figure 1: Relative Frequency Histogram of the
Motion of Functionability through the life of 497
Systems at specific instances of time
Clearly, functionability histograms can be
produced only after the data have been generated,
which means after the events. However, the objective
of Mirce Mechanics is to develop equation that will
be able to predict the data that are going to be
observed, in the similar manner as the predictions
made by Newton’s, Maxwell’s, Schrödinger’s
become confirmed by the future events.
Mirce Mechanics Formulas, developed at the
MIRCE Akademy, by D Knezevic, are mathematical
expressions of the physically observed processes of
the motion of systems through functionability states
and they define and predict physically measurable
properties of system functionability in the
probabilistic terms.
The laws of probability are just as rigorous as
other mathematical laws. However, they do have
certain unusual features and clearly delineated
domain of application. For example, it can be readily
verify that in the case of a large number of systems
failure phenomena will occur in a specific number of
the cases, and the law is more accurate the more
systems are observed. However, this accurate
knowledge will be of no help in predicting the
occurrence of functionability events in each
individual case. This is what distinguishes the laws
of probability: the concept of probability is valid only
for an individual event and it is possible to work out a
number that corresponds to it. However, it can only
be measured when identical tests are repeated a great
number of times. Only then can the measured value,
the probability, be used to assess the occurrence of
each individual functionability event, which is one of
the possible outcomes of the test.
The unusual features of the laws of probability
have a natural explanation. In fact, most probabilistic
events are results of quite complex physical
processes, which in many cases cannot be studied or
understood in all of its intricacy. Such inability takes
its toll, as it is only possible to predict with certainty
the average result of numerous identical tests. Thus,
for each functionability event it is only possible to
indicate its likely outcome.
Probabilistic predictions of the functionability
trajectory are based on the framework of the
sequence of occurrences of Positive and Negative
Functionability Events, whose individual and
cumulative times are measured as shown in the
Figure below.
Figure 2: Individual and Cumulative Times to
Functionability Events
Based on the Figure 2, the following functions are
used:
Negative Function, Fi(t), which defines the
probability that the ith NFE will take place before or
at instant of time t is defined in the following way:
( ) ( ), 1,
ii
F t P TNE t i
. 2.
Positive Function, Oi(t), which define the
probability that the ith PFE will take place before or at
instant of time t is defined by the following
expression:
( ) ( ), 1,
ii
O t P TPE t i
. 3.
Probability distribution that defines this event is
uniquely determined by the physical properties of the
process that generate positive functionability event
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(replacement, repair, calibration, modification and
similar) [9].
Sequential Negative Function, Fi(t), which
defines the probability that the ith sequential NFE will
take place before or at instant of time t, is defined as:
i
( ) ( ), 1,
i
F t P TNE t i
. 4.
Sequential Positive Function, Oi(t), which
defines the probability that the ith sequential PFE will
take place before or at instant of time t: is presented
in the following manner:
i
( ) ( ), 1,
i
O t P TPE t i
. 5.
Equations E3 and E4 define the sequence of
functionability events for any maintainable system.
Having determined the probability distribution and its
governing parameters of the times to subsequent
functionability event, positive and negative, it is
possible to develop a mathematical scheme that will
provide opportunity to predict the future sequence of
functionability events for any given system. This is
the essence of the Mirce Mechanics, which is the
only theory available to design engineers to
quantitatively predict the consequences of all of their
decisions on in-service behaviour of their future
systems.
X. Mirce Functionability Equation
The trajectory of functionability is uniquely
defined by the sequence of functionability events,
from the birth of the system to its decommissioning.
Thus, the fundamental equation of Mirce Mechanics,
the functionability equation y(t), that defines the
probability of a system being functionable at a given
instant of time t is defined as:
0
1
0
( ) ( @ )
( @ )
i
i
ii
i
y t P PFS t
P PFS t
P TPE t P TNE t
Making use of equations 3 and 4, while bearing in
mind that
0(0) 1O
, as a system starts its life in
positive functionability state, the above expression of
functionability equation could be presented in its
final form:
( ) 1 ( ) ( )y t t t
6.
where:
1
( ) ( )
i
i
t P TNE t
is the expected number of
negative functionability events that will take place
from the birth of a system and a given instant of time
t.
1
( ) ( )
i
i
t P TPE t
is the expected number of
positive functionability events that will take place
from the birth of a system and a given instant of time
t. This expression is developed by the author and it
is named Mirce Functionability Equation. It defines
the trajectory of a functionability through the
probability of a system being in positive
functionability state at a given instant of time t.
The unit of functionability determined in
accordance to the Mirce Functionability Equation, .is
1 Senna [1S]. It is quantified by the probability of
maintainable system being in PFS at a given instant
of time.
Making use of existing observational data related
to the in-service behaviour of a sample of 497
systems, operating in similar environmental and
utilisation conditions, the probability laws that drive
shapes of positive and negative functions defined by
the equations 2-5 where determined. The obtained
functions are shown in Figure 3, where the green
lines represent positive functions and the read lines
represents negative functions.
The functionability trajectory, calculated in
accordance to the expression 6 is shown with a black
line in the Figure 3.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0100 200 300 400 500 600 700 800 900 1000
Figure 3: Functionability profile calculated by Mirce
Functionability Equation for the Example shown in
Figure 1.
Analytical solutions for the Mirce
Functionability Equation are seldom possible due to
inability of mathematics to deal with the large
number of functions and their interactions. These
types of problems are not specifically related to the
Mirce Mechanics; they are common to all scientific
disciplines of this nature, as it is a known
mathematical fact that the integral equations do not
have analytical solutions. [10]
However, it is necessary to develop
computational methods to deal with the mathematical
difficulties as it is unacceptable to simplify observed
reality of system in-service behaviour in order to
cope with mathematical limitations. [11]
For the numerical example used in this paper the
result of the application of the Monte Carlo
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simulation method performed to obtained quantitative
solution of the Mirce Functionability Equation is
shown in Figure 4 as dots.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0100 200 300 400 500 600 700 800 900 1000
Figure 4: Functionability profile calculated by Mirce
Functionability Equation
XI. The Impact of Mirce Functionability
Equation on System Engineering and
Management
Although science has to be truthful, rather then
useful, the body of knowledge of Mirce Mechanics is
essential for scientists, mathematicians, engineers,
managers, technicians and analysts in industry,
government and academia to predict functionability
trajectories of the future systems, for a given
configurations, in-service rules and conditions, in
order to manage functionability events in the way
that the functionability performance is delivered
through the life of system, at least investment in
resources and environmental impact. For that to
happened, the science proven method is needed, very
much different from the classical scientific
knowledge, described trough the type of the
equations mentioned in the introductory part of the
paper, because functionability performance are
defined in the following way:
Every scheduled flight will leave on time with a
probability of at least 0.97 or in other words, it is
acceptable to have no more than three delays, on
average, out of 100 flights;
The direct maintenance cost will not exceed 25
% of the purchase cost with a probability of 0.95;
The probability that the production line will be
fully operational during the specified in-service
time will be not less than 0.91;
In system consisting of several systems, at least
90% of them will be operational at all times with
a probability not less than 0.925;
The mission reliability will be greater than 0.98
for missions shorter than 500 hours;
There should be 5 NFEs among 1000 systems,
on average, during the first 10 years of service,
with a probability of 0.95.
Each 10 hour flight will be successfully
completed with probability of 0.995, during the
first 20 years of operation
Consequently, the only way to address
performance targets formulated in the way above is
to use concept and principles of Mirce Mechanics to
evaluate engineering and management options, at the
time when fundamental and irreversible decision are
made regarding future systems.
XII. Conclusion
This paper clearly demonstrates that
functionability performance of any maintainable
system is very much different from its functionality
performance, in physical, technical, engineering and
management sense.
This paper also demonstrates that functionability
performance is the time dependent property of the
system and its motion is manifested through the
sequence of transitions through positive and negative
functionability states.
Like in the classical mechanics, where the
continuous uniform motion is natural state of the
macro world that is fully defined and predictable by
Newton’s equations, or in quantum mechanics where
the continuous motion is also natural state of a micro
world fully described and predictable by Schrodinger
equation, in Mirce Mechanics continuous change in
the functionability states is a natural state of
maintainable systems during they in-service life,
which is fully defined and predictable by Mirce
Functionability Equation.
Finally, Mirce Functionability Equation is the
scientific foundation of the System Engineering and
Management predictions and analysis regarding the
motion of functionability through the life of
maintainable system.
References
[1] Knezevic, J., Functionability in Motion,
Proceedings 10th International Conference
on Dependability and Quality, DQM
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