Statistical flowgraph models have proven useful for analysis and modeling of complex systems viewed as multistate processes that lead to outcomes such as degraded operation or failure. This article provides an engineering-oriented introduction to statistical flowgraph models: system representation, setting up a flowgraph model, parameter estimation, solution of the model (using either a frequentist or Bayesian approach), and interpretation of model outputs. The method is illustrated with a model for piping reliability in a nuclear power plant, and compared with alternative solution methods.
MannNRSchaferRESingpurwallaND. Methods for statistical analysis of reliability and life data. New York: John Wiley & Sons, 1974.
2.
RausandMHøylandA. System reliability theory: models, statistical methods, and applications. 2nd ed.Hoboken, NJ: Wiley-Interscience, 2004.
3.
HamadaMSWilsonAGReeseCS. Bayesian reliability. New York: Springer, 2008.
4.
MasonSJ. Feedback theory—some properties of signal flow graphs. Proc IRE1953; 41: 1144–1156.
5.
MasonSJ. Feedback theory—further properties of signal flow graphs. Proc IRE1956; 44: 920–926.
6.
HuzurbazarAVWilliamsBJ. Flowgraph models for complex multistate system reliability. In WilsonALimniosNKeller-McNultyS. (eds) Modern statistical and mathematical methods in reliability. Singapore: World Scientific, 2005.
FlemingKN. Markov models for evaluating risk-informed in-service inspection strategies for nuclear power plant piping systems. Reliab Engng Syst Saf2004; 83: 27–45.
9.
HuzurbazarAV. Modeling and analysis of engineering systems data using flowgraph models. Technometrics2000; 42: 300–306.
10.
CollinsDHHuzurbazarAV. System reliability and safety assessment using nonparametric flowgraph models. Proc IMechE, Part O: J Risk and Reliability2008; 222: 667–674.
11.
HuzurbazarAVPetersV. Flowgraph models for prognostics and health monitoring (PHM) in complex systems. Technical Report LA-UR-07-7353, Los Alamos National Laboratory, Los Alamos, NM, 2007.
12.
CollinsDHHuzurbazarAV. Prognostic models based on statistical flowgraphs. Applied Stoch Models Bus Ind2012; 28: 141–151.
13.
ButlerRW. Reliabilities for feedback systems and their saddlepoint approximation. Stat Sci2000; 15(3): 279–298.
14.
YauCLHuzurbazarAV. Analysis of censored and incomplete survival data using flowgraph models. Stat Med2002; 21: 3727–3743.
HougaardP. Multi-state models: a review. Lifetime Data Anal1999; 5: 239–264.
17.
RossSM. Applied probability models with optimization applications. New York: Dover Publications, 1992.
18.
MontgomeryDCRungerGC. Applied statistics and probability for engineers. 3rd ed.New York: John Wiley & Sons, 2003.
19.
HuzurbazarAV. A censored data histogram. Comm Stat Simul Comp2005; 34(1): 113–120.
20.
CasellaGBergerRL. Statistical inference. 2nd ed.Pacific Grove, CA: Duxbury, 2002.
21.
AndrewsLCShivamoggiBK. Integral transforms for engineers and applied mathematicians. New York: Macmillan, 1988.
22.
WarrRLCollinsDH. A comprehensive method for solving general finite-state semi-Markov processes. Technical Report LA-UR-11-01596, Los Alamos National Laboratory, Los Alamos, NM, 2011.
23.
WarrRLHuzurbazarAV. Expanding the statistical flowgraph model framework to use any transition distribution. J Stat Theory Pract2011; 4(4): 529–539.
24.
WarrRL. Numerical approximation of probability mass functions via the inverse discrete Fourier transform. arXiv preprint2012; DOI: arXiv:1212.6546.
25.
HughettP. Error bounds for numerical inversion of a probability characteristic function. SIAM J Numer Anal1998; 35: 1368–1392.
26.
DaviesBMartinB. Numerical inversion of the Laplace transform: a survey and comparison of methods. J Comput Phys1979; 33: 1–32.
27.
DuffyDG. On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans Math Software1993; 19(3): 333–359.
28.
AbateJWhittW. The Fourier-series method for inverting transforms of probability distributions. Queueing Syst1992; 10(1): 5–88.
29.
DoetschG. Introduction to the theory and application of the Laplace transform. Berlin: Springer-Verlag, 1974.
30.
AbateJWhittW. Numerical inversion of Laplace transforms of probability distributions. ORSA J Comput1995; 1: 36–43.
31.
CohenAM. Numerical methods for Laplace transform inversion. New York: Springer, 2010.
32.
KleinJPMoeschbergerML. Survival analysis: techniques for censored and truncated data. 2nd ed.New York: Springer-Verlag, 2003.
33.
WilliamsBJHuzurbazarAV. Posterior sampling with constructed likelihood functions: an application to flowgraph models. Appl Stoch Models Bus Ind2006; 22(2): 127–137.
34.
BillingsleyP. Statistical inference for Markov processes. Chicago, IL: University of Chicago Press, 1961.
35.
WolframS. The Mathematica book. 5th ed.Champaign, IL: Wolfram Media/Cambridge University Press, 2003.
36.
MonaganMBGeddesKOHealKM. Maple 10 programming guide. Waterloo ON, Canada: Maplesoft, 2005.
37.
AbramowitzMStegunIA (eds). Handbook of mathematical functions. Washington, DC: National Bureau of Standards, 1972.
38.
R Development Core Team. R: A Language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing, 2008.
39.
ASTM. A guide for fatigue testing and the statistical analysis of fatigue data. West Conshohocken, PA: American Society for Testing and Materials, 1963.
40.
VirklerDAHillberryBMGoelPK. The statistical nature of fatigue crack propagation. Technical Report TR-78-43, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, OH, 1978.
41.
SobczykKSpencerBF. Random fatigue: from data to theory. San Diego, CA: Academic Press, 1992.
42.
EmotoSESchaferRE. On the specification of repair time requirements. IEEE Trans Reliab1980; R-29(1): 13–16.
43.
KlineMB. Suitability of the lognormal distribution for corrective maintenance repair times. Reliab Engng1984; 9: 65–80.
44.
HuzurbazarAVWilliamsBJ. Incorporating covariates in flowgraph models: applications to recurrent event data. Technometrics2010; 52(2): 198–208.
45.
SakuraiT. Numerical inversion for Laplace transforms of functions with discontinuities. Adv Appl Prob2004; 36(2): 616–642.
46.
CollinsDH2009. Nonparametric estimation of first passage time distributions in flowgraph models. PhD dissertation, University of New Mexico, Albuquerque, NM.
