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

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 _{- 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 Poisson 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 Poisson 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 Poisson error bound and χ² 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