Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T10:52:55.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Sensitivity Analysis for the System Reliability Function

Published online by Cambridge University Press:  27 July 2009

George S. Fishman
George S. Fishman
Affiliation:
Department of Operations ResearchUniversity of North Carolina, Chapel Hill, North Carolina 27599
Get access Rights & Permissions [Opens in a new window]

Abstract

Sensitivity analysis is an integral part of virtually every study of system reliability. This paper describes a Monte Carlo sampling plan for estimating this sensitivity in system reliability to changes in component reliabilities. The unique feature of the approach is that sample data collected on K independent replications using a specified component reliability vector p are transformed by an importance function into unbiased estimates of system reliability for each component reliability vector q in a set of vectors Q. Moreover, this importance function together with available prior information about the given system enables one to produce estimates that require considerably less computing time to achieve a specified accuracy for all |Q| reliability estimates than a set of |Q| crude Monte Carlo sampling experiments would require to estimate each of the |Q| system reliabilities separately. As the number of components in the system grows, the relative efficiency continues to favor the proposed method.

The paper shows the intimate relationship between the proposal and the method of control variates. Next, it relates the proposal to the estimation of coefficients in a reliability polynomial and indicates how this concept can be used to improve computing efficiency in certain cases. It also describes a procedure that determines the p vector, to be used in the sampling experiment, that minimizes a bound on the worst-case variance. The paper also derives individual and simultaneous confidence intervals that hold for every fixed sample size K. An example illustrates how the proposal works in an s-t connectedness problem.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 5 , Issue 2 , April 1991 , pp. 185 - 213
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrawal, A. & Satyanarayana, A. (1984). An 0|E| time algorithm for computing the reliability of a class of directed networks. Operations Research, 32: 493–515.CrossRefGoogle Scholar
Aho, A.V., Hopcroft, J. E., & Ullman, J. D. (1974). The design and analysis of computer algorithms. Reading, MA: Addison-Wesley.Google Scholar
Ball, M.O. & Provan, J.S. (1983). Calculating bounds on reachability and connectedness in stochastic networks. Networks 13: 253–278.CrossRefGoogle Scholar
Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing probability models. Silver Spring, MD: To Begin With.Google Scholar
Dantzig, G.B. (1963). Linear programming and extensions. Princeton, NJ: Princeton University Press.Google Scholar
Easton, M.C. & Wong, C.K. (1980). Sequential destruction method for Monte Carlo evaluation of system reliability. IEEE Transactions on Reliability 29: 27–32.CrossRefGoogle Scholar
Esary, J.D. & Proschan, F. (1966). Coherent structures of non-identical components. Technometrics 5: 191–209.CrossRefGoogle Scholar
Fishman, G.S. (1986). A Monte Carlo sampling plan for estimating network reliability. Operations Research 34: 581–594.CrossRefGoogle Scholar
Fishman, G.S. (1987a). A Monte Carlo sampling plan for estimating reliability parameters and related functions. Networks 17: 169–186.CrossRefGoogle Scholar
Fishman, G.S. (1987b). Sensitivity analysis for the system reliability function, UNCøORøTR-87ø6, Department of Operations Research, University of North Carolina at Chapel Hill.Google Scholar
Fishman, G. S. (1988). Confidence intervals for means and proportions in the bounded case. Technical Report No. UNCøORSAøTR-86ø19, Department of Operations Research, University of North Carolina at Chapel Hill, revised 01 1988.Google Scholar
Kahn, H. (1950). Random sampling (Monte Carlo) techniques in neutron attenuation problems-I. Nucleonics 6: 27–37.Google ScholarPubMed
Kahn, H. & Harris, T.E. (1951). Estimation of particle transmission by random sampling. Monte Carlo Methods. National Bureau of Standards, Applied Mathematics Series, 12.Google Scholar
Karp, R. & Luby, M.G. (1983). A new Monte Carlo method for estimating the failure probability of an n-component system. Computer Science Division, University of California, Berkeley.Google Scholar
Kumamoto, H., Tanaka, K. & Inoue, K. (1977). Efficient evaluation of system reliability by Monte Carlo method. IEEE Transactions on Reliability 26: 311–315.CrossRefGoogle Scholar
Kumamoto, H., Tanaka, K., Inoue, K. & Henley, E.J. (1980). Dagger-sampling Monte Carlo for system unavailability evaluation. IEEE Transactions on Reliability 29: 122–125.CrossRefGoogle Scholar
Miller, R. (1981). Simultaneous statistical inference, 2nd ed. Springer Verlag.CrossRefGoogle Scholar
Page, L. B. & Perry, J. E. (1988). A practical implementation of the factoring theorem for network reliability. IEEE Transactions on Reliability, 37: 259–267.CrossRefGoogle Scholar
Shier, D. (1988). Algebraic aspects of computing network reliability. In Ringeisen, R. & Roberts, F. (Eds.) Proceedings of the Third Conference on Discrete Mathematics, published by SIAM in Applications of Discrete Mathematics, Philadelphia, pp. 135–147.Google Scholar
Valiant, L.G. (1979). The complexity of enumeration and reliability problems. SIAM Journal of Computing, 8: 410–421.CrossRefGoogle Scholar
Van, Slyke R. M. & Frank, H. (1972). Network reliability analysis: Part I. Networks 1: 279–290.Google Scholar
Wagner, H. M. (1975). Principles of operations research, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar