Ohba Inflection S-Shaped Model
Intended to capture the case in which some faults mask others. An overly complex model is not recommended by IEEE 1633.
An S-curve caused by faults hiding behind other faults
It assumes some faults are effectively invisible to testing until other faults are found and fixed first — a mutual dependency, or masking, effect that's common in complex systems with layered logic.
This model is overkill because you can simply use different models when inflections occur. The academic community drives for one model that fits all assumptions, when in fact you can simply use a selection of models.
This model is not one of the software reliability growth models recommended in the IEEE 1633. Requs Trend employs models that are recommended.
S-Shaped, Inflection NHPP
Adds an inflection parameter to the Goel-Okumoto model to represent mutual dependency between faults.
1984 — Mitsuru Ohba
Developed to address cases where purely exponential SRGMs proved insufficient for data showing fault masking effects.
Standard S-Shaped Option
A recognized, standards-consistent alternative when the data shows a fault-masking-driven S-curve rather than a testing ramp-up.
How the Inflection S-Shaped Model works
The model introduces an inflection parameter c that represents the degree of mutual dependency among faults — how much some faults are hidden behind others until those others are fixed.
Mean Value Function
As c approaches zero, this reduces to the standard Goel-Okumoto exponential curve; larger values of c produce a more pronounced S-shape.
- m(t) — expected cumulative number of failures by time t
- a — total expected number of faults (the asymptote)
- b — the fault detection rate
- c — the inflection (mutual dependency) parameter — higher values mean more fault masking
Failure Intensity Function
Like the Delayed S-Shaped model, intensity rises before it falls — but here the shape is driven by the inflection parameter c rather than a separate ramp-up term.
- λ(t) — instantaneous failure intensity at time t
- a, b, c — same parameters as the mean value function
- Larger c values push the peak of λ(t) later in the testing period
Key Assumptions
The core NHPP assumptions still apply, with an added assumption specifically about fault interdependency.
- A finite total fault count a exists in the software
- Some faults are mutually dependent — undetectable until other faults are found and removed first
- The degree of this masking effect is constant and captured by a single inflection parameter c
- Fault repair is instantaneous and perfect — no new faults introduced
Parameter Estimation
a, b, and c are estimated using maximum likelihood estimation or nonlinear least-squares regression. With three parameters instead of two, the model needs a data set large enough to distinguish the inflection effect from simple exponential decay.
- a — the eventual total fault count
- b — the underlying fault detection rate
- c — the strength of the fault-masking/inflection effect
Strengths, limitations, and best fit for the Inflection S-Shaped Model
Models a real, distinct failure pattern
Captures fault masking directly, a mechanism the Delayed S-Shaped model doesn't represent, since its S-curve comes from a different cause (testing ramp-up, not fault dependency).
Overly complex
Multiple models that handle different inflections is a better idea.
Not recommended by IEEE 1633
Faults hide other faults when there is an increasing fault rate. Using different models is a better idea.
Needs enough data for three parameters
As a three-parameter model, it requires a larger, cleaner data set than the two-parameter exponential models to fit reliably.
See this model fit against real project data.
Start with the online demo of Requs Trend or a discussion of your current reliability growth modeling approach.