# Decomposition of the Failure Histogram

## Abstract

Peterson and Arellano (2002) have shown in a seminal paper published in *Reliability Review* that, in general, the normalized defect rate (normalized failure rate) of software modules usually varies with time according to a Rayleigh distribution (Peterson and Arellano in Reliability Review, 22, 2002). However, the normalized failure rates *observed* in the “tails” (large values of time) are higher than those predicted by the Rayleigh model. And, the normalized failure rate histograms (normalized failure rate vs. time) of some data sets do not seem to fit the Rayleigh model very well at all. Furthermore, normalized software failure rates observed in the field after development follow an exponential decay law. How are all these difficulties to be resolved? The observations by Peterson and Arellano have profound implications, not just for complex software, but for complex systems in general. This chapter will take their analysis one step further. A unified theory of (normalized) failure rates will be presented that removes the paradoxes of the simple Rayleigh model. It will be shown that the behavior of real-world complex systems can be understood in terms of multiple Rayleigh failure rate laws, one for each subsystem, such that the modes of these Rayleigh distributions have values that are themselves distributed according to a probability density function law. Furthermore, subsequent chapters will demonstrate that this theorem is a fundamental building block of system safety science because it will allow a priori computation of the “Black Swan” probability of unlikely events represented by the area under the tail of real-world (normalized) failure rate laws.

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