Rare Failure-State in a Markov Chain Model for Software Reliability

• Authors: Michael G. Thomason;James A. Whittaker
• Venue: ISSRE '99 Proceedings of the 10th International Symposium on Software Reliability Engineering
• Year: 1999

Software systems composed of highly reliable components may experience few if any failures while undergoing heavy testing or field-usage. Kaufman et al [14, 15] have applied statistics of the extremes [13] 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 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 [4], yield: an explicit error-bound on a Poisson Approximation for counts of failures as rare events in long realizations of the chain, and an approximate exponential distribution for the inter-occurrence time of failure as a rare event.We compute both the Poisson error-bound and 驴 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.