The Rayleigh Model
The Rayleigh model is the special case of the Weibull model with its shape parameter fixed at 2, producing the classic rise-then-fall defect curve long used to describe how defect discovery rises through early testing, peaks, and tapers off toward release.
The defect curve behind decades of lifecycle planning
Long before it was formalized as a software reliability growth model, the Rayleigh-shaped curve was already familiar from Peter Norden's work on project staffing and defect discovery profiles: activity ramps up, peaks, and tapers off in a predictable rise-then-fall pattern. As an SRGM, it captures the same behavior in cumulative failure data.
Weibull, Shape Parameter = 2
Rayleigh is not a separate model family so much as a fixed, well-known special case of the more general Weibull curve.
Norden-Rayleigh Lineage
Traces back to Peter Norden's staffing and defect-profile research, later adapted directly into SRGM form.
Covered — Rayleigh
This model is one of the five named model families in Requs Trend's ensemble averaging.
How the Rayleigh Model works
Because Rayleigh is a fixed-shape Weibull curve, its formulas are the general Weibull equations with the shape parameter locked at 2, leaving two free parameters to fit.
Mean Value Function
The squared exponent produces the characteristic rise-then-fall shape in the failure intensity, unlike the strictly concave exponential model.
- μ(t) — expected cumulative number of failures by time t
- a — total expected number of faults (the asymptote)
- b — a scale parameter controlling where the peak of the curve falls
Failure Intensity Function
λ(t) rises from zero, peaks at t = 1/√(2b), then falls back toward zero — the signature rise-then-fall Rayleigh shape.
- λ(t) — instantaneous failure intensity at time t
- a, b — same parameters as the mean value function
- The peak location and height are both determined by b alone
Key Assumptions
Because the shape is fixed, Rayleigh assumes the data actually follows a symmetric rise-then-fall pattern rather than fitting whatever shape the data shows.
- A finite total fault count a exists in the software
- Failure intensity rises, peaks, and then falls in the specific rise-then-fall shape fixed by c = 2
- Fault repair is instantaneous and perfect
- Testing effort follows a roughly consistent ramp-up-then-taper profile, mirroring the classic Norden staffing curve
Parameter Estimation
With the shape parameter fixed, only a and b need to be estimated, typically via nonlinear least-squares regression or maximum likelihood estimation — making Rayleigh easier to fit reliably than the general Weibull model on smaller data sets.
- a — the eventual total fault count, from where the curve flattens
- b — the location of the peak in the failure intensity curve
Strengths, limitations, and best fit for the Rayleigh Model
Fewer parameters than the general Weibull
With the shape fixed, Rayleigh needs less data to produce a stable fit than the full three-parameter Weibull model.
Locked into one specific shape
If the real defect curve doesn't actually follow the classic rise-then-fall pattern, the fixed shape parameter can't adapt the way the general Weibull model can.
Lifecycle-phase defect planning
A longstanding standard for projecting defect density across development phases, from early testing through release.
Included in the ensemble
Listed directly among Requs Trend's five named model families, alongside Exponential, NHPP, Weibull, and Logarithmic.
See this model fit against real project data.
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