The Weibull Model
Borrowed from classical reliability engineering, the Weibull model adds a shape parameter to the basic exponential growth curve, letting failure intensity increase, stay flat, or decrease depending on what the data actually shows, instead of forcing a single fixed shape.
A flexible curve for failure rates that don't behave simply
Most classical SRGMs assume failure intensity decreases monotonically throughout testing. In practice, failure intensity often rises briefly at the start of testing (as testers ramp up) before falling. The Weibull model's shape parameter directly accommodates that behavior, making it one of the more flexible members of the NHPP family.
Shape-Flexible NHPP
Reduces to other well-known shapes at specific parameter values — including the Rayleigh model when the shape parameter equals 2.
Adapted from Weibull (1951)
Originally a general-purpose statistical distribution for time-to-failure data, later adapted into an NHPP mean value function for software.
Covered — Weibull
This model is one of the five named model families in Requs Trend's ensemble averaging.
How the Weibull Model works
The Weibull mean value function adds a shape parameter c to the basic exponential form, letting the curve bend differently depending on what the data requires.
Mean Value Function
When c = 1, this reduces to the same exponential shape as the Goel-Okumoto model. When c = 2, it becomes the Rayleigh model.
- μ(t) — expected cumulative number of failures by time t
- a — total expected number of faults (the asymptote)
- b — a scale parameter governing how quickly failures accumulate
- c — the shape parameter; c < 1 concave, c = 1 exponential, c > 1 S-shaped/increasing-then-decreasing
Failure Intensity Function
The extra t^(c−1) term is what allows intensity to rise before it falls, unlike the strictly decreasing exponential model.
- λ(t) — instantaneous failure intensity at time t
- a, b, c — same parameters as the mean value function
- The peak of λ(t), if one exists, occurs earlier or later in testing depending on c
Key Assumptions
The added flexibility comes with an added parameter to estimate, which requires enough data to constrain reliably.
- A finite total fault count a exists in the software
- Failure intensity may rise, stay flat, or fall depending on the fitted shape parameter c
- Fault repair is instantaneous and perfect
- The data set is large enough to estimate three parameters (a, b, c) reliably rather than just two
Parameter Estimation
a, b, and c are typically estimated with nonlinear least-squares regression or maximum likelihood estimation. Because there are three parameters instead of two, the Weibull model requires more failure data points than simpler exponential models to produce a stable fit.
- a — the eventual total fault count, from where the curve flattens
- b — the overall time scale of the growth curve
- c — the shape of the curve, from whether intensity rises before it falls
Strengths, limitations, and best fit for the Weibull Model
Adapts to non-monotonic failure data
Captures the common real-world pattern of failure intensity rising briefly before falling, which purely concave models can't represent.
Needs more data to fit reliably
The extra shape parameter means small or noisy data sets can produce unstable or overfit parameter estimates.
Data with an early ramp-up in defects
Well suited to projects where testers take time to ramp up, causing failure intensity to increase before it decreases.
Included in the ensemble
Listed directly among Requs Trend's five named model families, alongside Exponential, NHPP, Rayleigh, and Logarithmic.
See this model fit against real project data.
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