4 Reliability Theory
139
• Reliability of a system may be further improved by adding redundant components
and planning for maintenance, which introduces additional design variables
(Sect. 4.5).
Reliability theory also suffers from the same drawback like the general uncertainty quantification, as introduced in Chap. 2. Namely, the available information
is often too scarce to construct precise stochastic models. In reliability engineering,
knowledge often comes in the form of distribution summaries (mean TTF and other
moments supplied by the manufacturer) or only via limited amount of samples (as
with testing highly reliable components). It has been argued [23, 24] that imprecise
probability models are necessary in order to obtain reliable predictions.
The problem of inference with limited assumptions was already tackled also in
the treatment of Barlow and Proschan [4], who had derived several inequalities for
bounding the survival functions based on combination of quantitative (moments
of the lifetime distributions) and qualitative judgements (whether the failure time
distribution has increasing or decreasing failure rate). Further extensions to imprecise probability framework have been achieved in the field of robust Bayesian
inference [26], in analysing censored datasets via NPI [8], in ALT through imprecise
transformation of observations to the base level [27] and more in the field of system
reliability where imprecise failure distributions of component lifetimes may be
extended to imprecise reliability of some basic systems [19, 25].
References
1. J.D. Andrews, D.R. Prescott, R. Remenyte-Prescott, A systems reliability approach to decision
making in autonomous multi-platform systems operating a phased mission, in 2008 Annual
Reliability and Maintainability Symposium (2008), pp. 8–14
2. J.D. Andrews, J. Poole, W.-H. Chen, Fast mission reliability prediction for Unmanned Aerial
Vehicles. Reliab. Eng. Syst. Saf. 120, 3–9 (2013)
3. T. Augustin et al. (eds.), Introduction to Imprecise Probabilities (Wiley, New York, 2014)
4. R.E. Barlow, F. Proschan, Mathematical Theory of Reliability/Richard E. Barlow, Frank
Proschan, with contributions by Larry C. Hunter [English] (Wiley, New York, 1967)
5. P. Bessière et al., Bayesian Programming (CRC Press, Boca Raton, 2013)
6. G. Casella, R.L. Berger, Statistical Inference (Thomson Learning, Pacific Grove, 2002)
7. F.P.A. Coolen, T. Coolen-Maturi, Generalizing the signature to systems with multiple types
of components, in Complex Systems and Dependability, ed. by W. Zamojski et al. (Springer,
Berlin, 2012), pp. 115–130
8. F.P.A. Coolen, K.-J. Yan, Nonparametric predictive inference with right-censored data. J. Stat.
Plan. Inference 126, 25–54 (2004)
9. D.R. Cox, Regression models and life-tables. J. R. Stat. Soc. Series B Methodol. 34, 187–220
(1972)
10. J.D. Esary, H. Ziehms, Reliability analysis of phased missions. Tech. rep., Naval postgraduate
school, Monterey, California, 1975
11. D.F. Haasl et al., Fault Tree Handbook (US Nuclear Regulatory Commission, Washington,
1981)
12. A.K.S. Jardine, Maintenance, Replacement and Reliability (Halsted Press, Wiley, New York,
1973)
139
• Reliability of a system may be further improved by adding redundant components
and planning for maintenance, which introduces additional design variables
(Sect. 4.5).
Reliability theory also suffers from the same drawback like the general uncertainty quantification, as introduced in Chap. 2. Namely, the available information
is often too scarce to construct precise stochastic models. In reliability engineering,
knowledge often comes in the form of distribution summaries (mean TTF and other
moments supplied by the manufacturer) or only via limited amount of samples (as
with testing highly reliable components). It has been argued [23, 24] that imprecise
probability models are necessary in order to obtain reliable predictions.
The problem of inference with limited assumptions was already tackled also in
the treatment of Barlow and Proschan [4], who had derived several inequalities for
bounding the survival functions based on combination of quantitative (moments
of the lifetime distributions) and qualitative judgements (whether the failure time
distribution has increasing or decreasing failure rate). Further extensions to imprecise probability framework have been achieved in the field of robust Bayesian
inference [26], in analysing censored datasets via NPI [8], in ALT through imprecise
transformation of observations to the base level [27] and more in the field of system
reliability where imprecise failure distributions of component lifetimes may be
extended to imprecise reliability of some basic systems [19, 25].
References
1. J.D. Andrews, D.R. Prescott, R. Remenyte-Prescott, A systems reliability approach to decision
making in autonomous multi-platform systems operating a phased mission, in 2008 Annual
Reliability and Maintainability Symposium (2008), pp. 8–14
2. J.D. Andrews, J. Poole, W.-H. Chen, Fast mission reliability prediction for Unmanned Aerial
Vehicles. Reliab. Eng. Syst. Saf. 120, 3–9 (2013)
3. T. Augustin et al. (eds.), Introduction to Imprecise Probabilities (Wiley, New York, 2014)
4. R.E. Barlow, F. Proschan, Mathematical Theory of Reliability/Richard E. Barlow, Frank
Proschan, with contributions by Larry C. Hunter [English] (Wiley, New York, 1967)
5. P. Bessière et al., Bayesian Programming (CRC Press, Boca Raton, 2013)
6. G. Casella, R.L. Berger, Statistical Inference (Thomson Learning, Pacific Grove, 2002)
7. F.P.A. Coolen, T. Coolen-Maturi, Generalizing the signature to systems with multiple types
of components, in Complex Systems and Dependability, ed. by W. Zamojski et al. (Springer,
Berlin, 2012), pp. 115–130
8. F.P.A. Coolen, K.-J. Yan, Nonparametric predictive inference with right-censored data. J. Stat.
Plan. Inference 126, 25–54 (2004)
9. D.R. Cox, Regression models and life-tables. J. R. Stat. Soc. Series B Methodol. 34, 187–220
(1972)
10. J.D. Esary, H. Ziehms, Reliability analysis of phased missions. Tech. rep., Naval postgraduate
school, Monterey, California, 1975
11. D.F. Haasl et al., Fault Tree Handbook (US Nuclear Regulatory Commission, Washington,
1981)
12. A.K.S. Jardine, Maintenance, Replacement and Reliability (Halsted Press, Wiley, New York,
1973)