1. Foreword
Experience proves the interdependence of systems’ reliability and repairability. They are steadily coupled during the whole system life. In details, a young system, free from errors, is reliable and repairable, thereafter, the more a system becomes old and unreliable and the more repairs are difficult.
These parallel phenomena, well known in practice [
1], are not theoretically justified. Current literature does not calculate how they are joined, how they evolve together by time passing.
Reliability is the probability of a system performing a defined job and a fault closes its performance. Authors ground the Reliability Theory upon the stochastic model, which neglects the inner structure of a general system.
Chains figure a physical system that works steadily and stops, they do not detail how in general its internal parts evolve and break down.
In substance, the stochastic model does not cast light into the internal causes that generate a failure and enable a system to restart. The Markov chains prevent us from tackling the aforementioned problems and consequently
Failure analysis, repairability and maintenance researches give sharp and practical results that however have a specific meaning. They are not included within a general framework.
Today theories scarcely formalize the relations between repairability and maintenance tenets.
Theoretical investigations upon reliability and maintenance are separated, thus specialists double their efforts.
Some years ago we aimed at solving these problems, even if the scientific community devotes few resources to wide ranging questions today [
2] and in the future [
3]. We did not find a significant help in bibliography, instead we have taken an inspiration from thermodynamics. The Boltzmann entropy calculates the evolution of a thermodynamical system [
4] and we supposed that this kind of function could symmetrically detail the internal of a system and complement the stochastic model. In particular we guessed that the entropy could explicate the inner evolution of a general system when it deteriorates and/or when it improves after a repair.
In this article, we introduce the Boltzmann-like entropy and we discuss it with respect to problems 1 and 2.
2. Entropy Function
Let the system
S assumes
n states
The generic state
Ai is complex and consists of
r cooperating substates
By definition each stochastic state/substate depends on the probability, notably from (2) we have
A system, that assumes the state
Ai, stays steady in it or moves easily from
Ai. In the former situation
S rarely evolves from
Ai, this state is rather stable and we say that the system
S assumes an
irreversible state. When
S easily abandons
Ai, this state is somewhat unstable and we shall say the state
Ai is reversible. E.g. A man/woman goes into an irreversible coma, then he/she does not leave this state and no longer
recovers. E.g. The machine
S is immediately repaired, the failure was light and the failure state was very reversible.
We introduce the entropy function
That calculates the attitude of evolving using this criterion
2.1) – The more Ai is irreversible and the higher is H(Ai). Vice versa the more the state reversible, the lower is the entropy.
The ensuing conditions specify point 2.1).
2.1A) – Each state in a stochastic system depends on the probability by definition thus the entropy depends on the probability
2.1B) - Once
S assumes an irreversible state, this state is rather stable; conversely a reversible state appears unstable and somewhat infrequent. We reasonably conclude
2.1) – The reversibility/irreversibility of each substate influences the aptitude of
Ai. E.g. If the part
q of the equipment
Q breaks down and assumes an irreversible state, then
q affects the whole machine. In practice,
q must be substituted since its entropy spoils the overall operational state of
Q. As a further example, let all the
r parts of (3) work steadily. The system fairly runs since its reliability depends on each part, namely the entropy of the operational substate influence the global operational state. We conclude that the global entropy is the summation of the entropy of each substate
This is the most meaningful assumption as it relates the reversibility/irreversibility with the system complexity and disorder. Summation (7) calculate the internal structure, which we make explicit in (1) and (2).
Theorem 2.1: If (5), (6) and (7) are true, we get
where
a, b, c are nonnegative constants.
Proof : In order to simplify the inferences from the formal point of view, let
A is equipped with two substates. From (7) we get
From (3) we obtain
Now we write (9) in function of the probability
Differentiating with respect to
Pi2
and differentiating (12) with respect to
Pi1 we obtain
(10) leads to
and so
This expression yields
which we write as
Where
a* is some nonnegative constant as yet undetermined. We calculate (17) and get
Since
we can write (18) such as
The entropy varies in the open range (- ∞, 0) because S is strictly stochastic and the extreme values of probability are excluded.
3. Reliability and Repairability Are Coupled
We suppose that
S either runs or is repaired after the failure. We reject that a system is capable of running and is idle. We exclude also that
S is broken-down and is not maintained, hence
S assumes either the
operational state Ah or the
failure state Aj during which the fault of
S is remedied
The states are mutually exclusive and their probabilities verify
This means the states
Ah and
Aj are joined
Let us use (20) and we obtain that the entropies of the operational state and the failure state are coupled
In particular we derive two pairs of connected results
The former result holds that if the operational state is highly irreversible, than the failure state is very reversible (and vice versa). In other words S is stable in Ah ,namely it is reliable and contemporary is repairable.
Expression (26) affirms that the operational state is unstable while Aj is irreversible. The system S is unreliable and irreparable at the same time.
Results (23) and (23bis) explicate how the probability of working and the probability of failing are joined; (24) and (24bis) detail how the entropies are coupled, lastly (25) and (26) calculate the limit values. In such a way we give an answer to question 1 in Foreword.
4. Reliability function
Both (24) and (24bis) are valid on the theoretical plane. We wonder which of them is usually verified in practice, to wit we search for the direction of the reversibility/irreversibility.
We assume the system S executes the plainest job, it works regularly and continuously during the time. We exclude any acceleration, any stop and restart that can stress the system. We prevent S also from external attacks or disturbances. In these conditions, a system free from errors reaches the high value of Ph at its birth by definition. The probability cannot do but decrease by time passing. As a consequence the reliability entropy H(Ph) diminishes. The trend (24bis) is true in the physical world whereas (24) is false. However (24bis) does not give any explicit relationship between the entropy H(Ph) and the time and we look for this relation.
If H(Ph) is “high”, S goes on working. Conversely if H(Ph) is “low”, the system abandons Ah and switches to Aj since it fails. The reliability entropy quantifies the system attitude to running. We reasonably assume that H(Ph) decreases linearly when S works regularly. In particular we define the simplest evolution in these terms.
4.1A) The reliability entropy decreases constantly with respect to time
where
d is a positive constant. Intuitively
d expresses the speed of the entropy descent during the time. The turbine in the power plant that night and day runs, provides a perfect example. The machine wears continuously hour by hour and (27) is true. On the contrary, the turbine destroyed by a single psychopathic being does not follow (27).
Biological systems do not follow (27) in the first period of their life and offer a second counter-example. A cell, a body, a tree etc. grow from their birth to the youth and the number
s of the operational substates
scales up
During system’s childhood the reliability entropy raises due to the gain of
s
Trend (24bis) is false and (24) is true. Experience confirms how a body becomes more robust from the birth to the youngness. The growth of a biological system comes to the end in the maturity when
From this period onward the biological system entropy follows (27).
What does the formula (27) mean in substance?
Regular running produces several degenerative dynamics such as attrition, oxidation, grinding etc. All of them are summarized by the constant reduction of entropy H(Ph) that explains the gradual decaying of systems working continuously. Expression (27) claims that the longer a system works, the more S degenerates and the lower is H(Ph). The system degrades due to its mere job and we highlight this explanatory quality of (27). Regular running is the first and most general reason for reliability shortage. Long investigations on incidents, accurate and complicate accounts of events make clear outcomes due to simple working. This is the origin of an unlimited list of faults.
Now we calculate
Ph in the most simple and common conditions held by (27). We choose these values for (20)
Thus (27) is
We assume
S starts at time
then
and
From this we obtain the
reliability function
where ν and λ are positive constants depending on the specific system. This results confirms the correctness and consistency of the Boltzmann-like entropy introduced in Reliability Theory.
Today authors calculate the failure rate
as an empirical function and from the facts they derive the meaning of its constant trend
On the contrary we derive the result (37) from (27). This explicit hypothesis clarifies the reliability function theoretically and details the internal reasons for the most common failures as point 2 demands.