The Negative Binomial Process with Applications to Reliability

Abstract

The negative binomial distribution is frequently used as an alternative to the Poisson distribution for count data. One interpretation of the negative binomial distribution is that it is a compound Poisson distribution with means chosen at random from a gamma distribution. This appears to be a useful model for failure data, particularly for data from a number of repairable systems all of which follow a Poisson process but with different intensities. In this framework the times to failure are of importance and the distributions of these are derived for both a single system and a collection of systems. The distributions are shown to be related to Snedecor's F distribution. The usefulness of these concepts is exemplified through evaluations of reliability and spare parts requirements based on some airplane component data. Method of moments estimation of the parameters and confidence interval estimation of the average failure intensity are also mentioned.