Computer‐aided selection of prior distributions for generating monte carlo confidence bounds on system reliability

A description is given of results of preliminary investigations (by a group at North American Rockwell Corporation) related to the Monte Carlo generation of lower Confidence bounds on the reliability of a logically complex system. In calculating system confidence bounds by use of a Monte Carlo procedure, one must generate the distribution of each independent subsystem reliability, given the life‐test failure data for that subsystem. Therefore, an assumption of a specified a priori distribution for each subsystem reliability is implicit in the procedure. In order that clues may be obtained as to optimum prior assumptions to be used in calculating Monte Carlo bounds for a complex system, the model has been restricted to a series system wherein each independent subsystem has exponentially distributed failure time and prototypes of each subsystem are tested until a fixed (but not necessarily the same for each subsystem) number of failures occurs. For this model, optimum (uniformly most accurate unbiased) exact classical confidence bounds on the reliability R (tm) at a specified mission time, tm, are available, although not easily calculated (El Mawaziny [12] and Lenter and Buehler [27]). Computer programs for calculating the optimum classical bounds and the Bayesian Monte Carlo bounds were written, and a means of numerically comparing various forms of prior distributions against an optimum standard was thus provided. One prior distribution widely used in obtaining Monte Carlo and general Bayesian exact lower confidence bounds on system reliability is thereby shown numerically to yield bounds which are conservative in the classical sense for this series‐system model. Another suggested prior distribution is shown to give bounds which are usually conservative but under certain conditions are liberal, and hence not truly confidence bounds. Moreover, it is demonstrated by a combination of numerical and analytical results, that for a series system containing more than one independent subsystem there exists no prior distribution for subsystem reliability which is independent of the data and which yields the optimum lower bounds. Other numerical results related to the selection of optimum methods for generating the bounds and evaluation of certain approximate methods are described.