Rare failure-state in a Markov chain model for software reliability

Abstract

Software systems composed of highly reliable components may experience few, if any, failures while undergoing heavy testing or field usage. L.M. Kaufman et al. (1997, 1998) applied statistics of the extremes to software reliability analysis for failure as an infrequent, unlikely occurrence /sub -/ /sub a/ so-called rare event. This paper combines (i) software failure as a rare event with (ii) a finite-state, discrete-parameter recurrent Markov chain that models both the failures (as transitions to a rare fail state) and the software usage probabilities (as transitions among usage states not involving the fail state). When conditions for rare events are met, reliability analysis in greater detail, with fewer assumptions, may be possible, and there may be additional justification for using the popular and exponential distributions for certain random variables. We describe how the Markov chain and the Poisson law of small numbers, which has a central role in the study of rare events and extreme values, yield (a) an explicit error bound on a approximation for counts of failures as rare events in long realizations of the chain, and (b) an approximate exponential distribution for the interoccurrence time of failures as rare events. We compute both the error bound and /spl chi//sup 2/ goodness-of-fit tests for samples and the approximate distributions for a small Markov chain. A typical application of these results would be in the analysis of software reliability for systems of high-quality COTS components.