Reliability governed by the relative locations of random variables following a homogeneous Poisson Process in a finite domain

A powerful equation is derived for determining the probability of safe/failure states dependent on random variables, following a homogeneous Poisson process in a finite domain. The equation is generic & gives the probability of all types of relative configurations of the random variables governing reliability. The significance of the derived equation stems from the fact that the reliability problems solved are not restricted to one-dimensional problems only, or to a simple function of the relative distances between the locations of the random variables. Many intractable reliability problems can be solved easily using the derived equation which reduces complex reliability problems to problems with trivial solutions. The power of the equation is illustrated by the derivation of a number of important special cases, and numerous applications: i) determining the probability of existence of a set of minimum intervals between the locations of random variables in a finite interval, ii) determining the number density envelope of the random variables which prevents clustering within a critical distance, and iii) a closed-form relationship for determining reliability during shock loading. The new equation is at the basis of a new reliability measure which consists of a combination of a set of specified minimum free gaps before/between random variables in a finite interval, and a minimum specified probability with which they must exist. The new reliability measure is at the heart of a technology for setting quantitative reliability requirements based on minimum failure-free operating periods (MFFOP). The equation is applied to cases where failure is triggered because of clustering of two ore more demands, forces, or manufacturing/material flaws following a homogeneous Poisson process in a specified time interval/length. A number of important applications have also been considered related to the conditional case (a homogeneous Poisson process conditioned on the number of random variables in a finite interval). Solutions have been provided for common problems related to: i) collisions of demands from a given number of customers using a particular equipment for a specified time; ii) overloading of supply systems from a given number of consumers connecting independently & randomly, and iii) analysis of random failures following a homogeneous Poisson process where only the number of failures is known, but not the actual failure times. In the paper, it is demonstrated that even for a small number of random variables in a finite interval, the probability of clustering of two or more random variables is surprisingly high, and should always be accounted for in risk assessments.