A Simple Model with Applications in Structural Reliability, Extinction of Species, Inventory Depletion, and Urn Sampling. I. Cut Set Failure.

摘要

A series-parallel system consists of (k+1) subsystems C(0), C(1), ..., C(k), also called cut sets. Cut set C(i) contains n(i) components arranged in parallel, i = 0, 1, ..., k. No two cut sets have a component in common. It is assumed that after t components have failed, each of the remaining components in equally likely to fail, t = 0, 1, ... . Consider the probability that the system fails because a specified cut set C(0), say fails first. Several alternative expressions and recurrence relations are obtained for this probability. Some of these formulae are useful in computations while others permit us to derive qualificative features like monotonicity, Schur-concavity, asymptotic limits, etc. These results are extended to the situation where some cut set size is first reduced to a, where a is a specified positive integer. The above model has applications in the study of reliability, extinction of species, inventory depletion, urn sampling, among others.