The Maximum Error in System Reliability Calculations by Using a Subset of the Minimal States

The reliability of a system is evaluated from information about the minimal states, using Poincare's method (inclusion-exclusion). An equation is derived using the minimal paths, which gives the reliability (the probability of system success) as a function of the reliabilities of the components; the unreliability or probability of system failure is obtained by subtracting this from one. Dually, with the minimal cuts, an equation is derived which gives the system unreliability as a function of the unreliabilities of the components; the reliability is obtained by subtracting from one. If some of the minimal states are missing, unknown, or unused, an error is made in the calculation. The probability which is calculated from the minimal states is underestimated and its complement overestimated. In this paper a method is described for determining the maximum error, both absolute and relative, when: 1) the minimal states are all statistically independent (e.g., they have no states in common); and 2) every minimal state has the same probability p, where p is selected so as to maximize the error. It is shown that the error and its associated system probability depend only upon the ratio of the number of minimal states used to the total number of minimal states. A table of errors, system probabilities, and relative errors is given for values of this ratio 0.01 (0.01) 0.99.