Yamada Delayed S-Shaped Model
Where the exponential models assume testers are fully effective from day one, the Yamada Delayed S-Shaped model builds in a learning curve: testers need time to understand the software before fault detection accelerates, producing the characteristic S-shaped rise in cumulative failures.
This is an example of an overly complex model produced by academia. In the real world, the effect of a learning curve is minor in comparison to other events and factors.
A model that accounts for the testing learning curve
Yamada and colleagues observed that testers typically aren't at full effectiveness from the start of a test campaign — they need time to build familiarity with the software before fault detection accelerates. The Delayed S-Shaped model captures that ramp-up directly, producing a cumulative failure curve that starts slow, accelerates, and then tapers off like the concave models.
In reality, such a ramp-up curve is typically in the first program increment and is temporary. This is an example of an overly complex model developed in the isolated environment of an academic setting. Requs Trend doesn't support this model, and neither does the IEEE 1633.
S-Shaped, Finite-Failure NHPP
Derived from the Goel-Okumoto model by adding a factor that delays the effective start of fault detection.
1983-1984 — Yamada, Ohba, Osaki
Part of a body of work extending exponential NHPP models to account for testing effectiveness ramping up over time.
Standard S-Shaped Option
A recognized, standards-consistent alternative when the data shows a clear early ramp-up before the concave decline.
How the Delayed S-Shaped Model works
The model modifies the Goel-Okumoto mean value function with an additional (1 + bt) term that produces the initial S-shaped ramp-up before the curve behaves like a standard exponential model.
Mean Value Function
The (1 + bt) term is what produces the initial slow start before the curve accelerates and eventually flattens like a standard exponential model.
- m(t) — expected cumulative number of failures by time t
- a — total expected number of faults (the asymptote)
- b — the fault detection rate, controlling both the ramp-up and the eventual decay
Failure Intensity Function
Failure intensity starts at zero, rises to a peak at t = 1/b, then decreases — the signature S-shaped behavior in the underlying cumulative curve.
- λ(t) — instantaneous failure intensity at time t
- a, b — same parameters as the mean value function
- The peak of λ(t) occurs at t = 1/b, marking the point of maximum testing effectiveness
Key Assumptions
The core NHPP assumptions still apply, with an added assumption about how testing effectiveness changes over time.
- A finite total fault count a exists in the software
- Assumes a waterfall environment as opposed to an Agile environment in which the learning curve affects only the first program increment.
- Testing effectiveness (the rate of fault detection) is low initially and increases as testers gain familiarity
- Fault repair is instantaneous and perfect — no new faults introduced
- Time to failure for a given fault effectively follows a gamma distribution rather than a simple exponential
Parameter Estimation
a and b are estimated using maximum likelihood estimation or nonlinear least-squares regression, the same general approach as the Goel-Okumoto model. The S-shape means enough early-stage data is needed to distinguish the ramp-up phase from simple exponential decay.
- a — the eventual total fault count, from where the curve flattens
- b — both the steepness of the initial ramp-up and the rate of the later decline
Strengths, limitations, and best fit for the Delayed S-Shaped Model
Can misfit data without a genuine ramp-up
If the software actually shows simple exponential decay from the start, forcing an S-shaped fit can distort the estimated total fault count.
Needs enough early-stage data points
Requires more work and data for a situation that is largely temporary.
See this model fit against real project data.
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