Reliability Estimation
Charles Peters, Ph.D., Associate Professor, UH
Raj S. Chhikara, Ph.D., Professor, UHCL
Richard P. Heydorn, Ph.D., JSC
Huann-Sheng Chen, Ph.D., Post-Doctoral Fellow, UH and UHCL
THE STATISTICAL ANALYSIS of failure data in the reaction control system
(RCS) of the space shuttle is an important step in improving the reliability of the
system. Past studies have suggested that the two variables "soak time" and
"cycle time" are major explanatory variables for predicting reliability.
"Soak time" refers to the amount of time the oxidizer valve in the RCS is under
pressure in a N2O2 environment, while "cycle
time" is the number of cycles of operation of the valve. The reliability of the
system is expressed by the survival function R(s,t), the probability that
the system is operating when the soak time is s and the cycle time is t. In
the method currently used by JSC in estimating reliability, the two variables are treated
separately (that is, R is considered to be a function of one variable at a time)
and a Weibull model is assumed. A paradoxical result of this model is that valves appear
to wear out with increasing soak time but improve with increasing cycle time. Figure 1 shows quantile-quantile plots of uncensored failure
times with fitted Weibull distributions. The estimated parameter values of 2.9 (>1!)
and 0.737 (<1!) bear out the observations mentioned above about the wear of oxidizer
valves. The shapes of the plots indicate that the Weibull model is reasonable but capable
of improvement. The majority of the observations in the RCS data are censored; that is,
units are still operating or were routinely replaced before failure, and we can say only
that their soak and cycle times to failure are greater than their current values. A
difficulty with the Weibull model is that the large fraction of censored observations
causes a serious discrepancy between the fitted survival function and the empirical
survival function (Fig. 1). Possible avenues of improvement are modeling the interactions
of soak and cycle times with a multivariate survival function and considering such models
as frailty models and the Cox regression model.
Above. Reliable thrusters are critical to the orientation of the Space Shuttle and
the International Space Station.
Let p(s,t) denote the bivariate failure rate. We consider models in which
this function is of logistic form:
Logistic models have been widely employed in the analysis of categorical data. We have
concentrated on the logit transformation above, which has numerous properties that make it
appropriate for binary data (e.g., "failure" and "survival"),
but other possible choices have been considered, among them the probit and complementary
log log transformations. All of these transformations seem to leave open the possibility
of nonlinear relationships between the transformed failure rate and the covariates.
Therefore, we also consider polynomials with higher order and interaction terms and other
nonlinear functions. We use maximum likelihood to estimate parameters of the various
models. Diagnostics on fitted models utilize nonparametric estimates and sensitivity
studies to determine robustness. It appears that the main effect of cycle time is less
important than that of soak time, but the logistic model clearly provides more accurate
predictions of reliability than the currently employed method does.
Various other extensions are possible. Since the oxidizer valve may experience repeated
failures and repairs, Poisson processes and renewal processes for recurrent events can be
applied. Suppose that the ith valve is observed over time periods [0,di]x[0,li]. Let Ni(s,t)
denote the number of failures with repair over 
. The cumulative mean function (CMF) for the process Ni(s,t)
is
Mi(s,t) = ENi(s,t).
If it is assumed that the valves all have the same CMF, the goal is then to estimate
the trend of this counting process. Other approaches, such as Poisson or renewal processes
with random effects can also be taken into consideration. We intend to continue the
investigation of the effectiveness of these approaches and the statistical properties of
the fitted models developed from them.
Empirical and Hypothesized Weibull CDFs
References
D. Clayton and J. Cuzick. "Multivariate Generalizations of the Proportional Hazards
Model," J. Royal Statistical Society A 148 (1985): 82-117.
D. R. Cox. Renewal Theory. John Wiley (1962).
H. S. Chen, R. Chhikara, R. Heydorn, and C. Peters. "Modeling the Reliability of the
Shuttle Reaction Control System," (1997). (In preparation.)
D. Oaks. "Bivariate Survival Models Induced by Frailties," J. American
Statistical Association 84 (1989): 487-93.
R. D. Siddhartha, B. F. Edward, and H. Bruce. "Risk Analysis of the Space Shuttle:
Pre-Challenger Prediction of Failure," J. American Statistical Association 84
(1989): 945-57.
Contents
ISSO -- Institute for Space Systems Operations
1996-1997 Annual Report
