138
D. Krpelík et al.
• Systems with latent failures—here we can again distinguish based on the
negligibility of maintenance length, but the main aspect is that the maintenance
does not commence directly at the time of a device failure but only at predefined
inspection times. Hence the inspection policy also becomes an aspect of the
system design.
Results of the renewal theory [12], [18, Ch. 1] directly lead to methods for
optimising the overall performance of the service by the means of influencing the
maintenance and inspection policies. The competing objectives are the availability
of the system (preferred maximal), the overall costs of the maintenance (preferred
minimal) and others relevant for the system operation, like the variability of its
performance (preferred minimal).
Approximative results can be derived from the limiting properties of N(t) and
A(t), namely: lim t→∞ A(t) and lim t→∞
N(t)
t . These describe the behaviour of
maintained devices at times after the initial fluctuations and have enough descriptive
power to reason about long-lasting or permanent systems. For the scenario with
significant repair time, but no latent failures, the asymptotic availability is given by
lim
t→∞
A(t) =
MT T F
MT T F + MT T R
,
(4.6)
where MT T F and MMT R represent the mean TTF and the mean time to
renewal, respectively.
But Eq. (4.6) only describes the asymptotic properties, and for bounded system
life duration, these might not be reached and therefore be misleading. A careful,
usually simulation-based, analysis is necessary in such cases.
4.6 Concluding Remarks
In this chapter, we have attempted to give an overview of the basic notions and
problems reliability theory deals with. For the sake of conciseness, we needed to
omit many interesting topics and sometimes also proper mathematical rigour. Nevertheless, for those interested, several pointers to further literature were provided in
relevant sections.
Regarding the optimal design problem, some aspects of dependency of reliability
on design variables has been emphasised:
• Reliability of a system depends on its structure, so the structure itself becomes a
design parameter (Sect. 4.3).
• The design may influence the working conditions of the system, the stresses on
components and the models, for this may be inferred via accelerated life tests
(Sect. 4.4.2).
• “Linear” dependencies on additional variables may be captured by the proportional hazard model (Sect. 4.4.3).
D. Krpelík et al.
• Systems with latent failures—here we can again distinguish based on the
negligibility of maintenance length, but the main aspect is that the maintenance
does not commence directly at the time of a device failure but only at predefined
inspection times. Hence the inspection policy also becomes an aspect of the
system design.
Results of the renewal theory [12], [18, Ch. 1] directly lead to methods for
optimising the overall performance of the service by the means of influencing the
maintenance and inspection policies. The competing objectives are the availability
of the system (preferred maximal), the overall costs of the maintenance (preferred
minimal) and others relevant for the system operation, like the variability of its
performance (preferred minimal).
Approximative results can be derived from the limiting properties of N(t) and
A(t), namely: lim t→∞ A(t) and lim t→∞
N(t)
t . These describe the behaviour of
maintained devices at times after the initial fluctuations and have enough descriptive
power to reason about long-lasting or permanent systems. For the scenario with
significant repair time, but no latent failures, the asymptotic availability is given by
lim
t→∞
A(t) =
MT T F
MT T F + MT T R
,
(4.6)
where MT T F and MMT R represent the mean TTF and the mean time to
renewal, respectively.
But Eq. (4.6) only describes the asymptotic properties, and for bounded system
life duration, these might not be reached and therefore be misleading. A careful,
usually simulation-based, analysis is necessary in such cases.
4.6 Concluding Remarks
In this chapter, we have attempted to give an overview of the basic notions and
problems reliability theory deals with. For the sake of conciseness, we needed to
omit many interesting topics and sometimes also proper mathematical rigour. Nevertheless, for those interested, several pointers to further literature were provided in
relevant sections.
Regarding the optimal design problem, some aspects of dependency of reliability
on design variables has been emphasised:
• Reliability of a system depends on its structure, so the structure itself becomes a
design parameter (Sect. 4.3).
• The design may influence the working conditions of the system, the stresses on
components and the models, for this may be inferred via accelerated life tests
(Sect. 4.4.2).
• “Linear” dependencies on additional variables may be captured by the proportional hazard model (Sect. 4.4.3).